| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 1.1-a1 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( -23495540774930605015040 a^{3} - 80103860217951906201600 a^{2} + 32342190266893863157760 a + 110264935508355946905600 \) |
\( \bigl[0\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{5}{8} a - \frac{7}{2}\) , \( 1\) , \( \frac{1633}{8} a^{3} + \frac{485}{2} a^{2} - \frac{19013}{8} a - \frac{5647}{2}\) , \( -\frac{23265}{4} a^{3} - 6786 a^{2} + \frac{270437}{4} a + 78896\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{5}{8}a-\frac{7}{2}\right){x}^{2}+\left(\frac{1633}{8}a^{3}+\frac{485}{2}a^{2}-\frac{19013}{8}a-\frac{5647}{2}\right){x}-\frac{23265}{4}a^{3}-6786a^{2}+\frac{270437}{4}a+78896$ |
| 1.1-a2 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$3.594937582$ |
0.308137507 |
\( -68274959951801000509440 a^{3} + 80103860217951906201600 a^{2} + 793592316273690586562560 a - 931085247325018833715200 \) |
\( \bigl[0\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{11}{4} a - 3\) , \( 1\) , \( -8 a^{3} - \frac{99}{2} a^{2} + \frac{15}{2} a + 72\) , \( \frac{2641}{4} a^{3} + 2108 a^{2} - \frac{3629}{4} a - 2904\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{11}{4}a-3\right){x}^{2}+\left(-8a^{3}-\frac{99}{2}a^{2}+\frac{15}{2}a+72\right){x}+\frac{2641}{4}a^{3}+2108a^{2}-\frac{3629}{4}a-2904$ |
| 1.1-a3 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( 23495540774930605015040 a^{3} - 80103860217951906201600 a^{2} - 32342190266893863157760 a + 110264935508355946905600 \) |
\( \bigl[0\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{11}{4} a - 2\) , \( 1\) , \( -204 a^{3} + \frac{485}{2} a^{2} + \frac{4747}{2} a - 2823\) , \( 5946 a^{3} - 6934 a^{2} - 69114 a + 80601\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{11}{4}a-2\right){x}^{2}+\left(-204a^{3}+\frac{485}{2}a^{2}+\frac{4747}{2}a-2823\right){x}+5946a^{3}-6934a^{2}-69114a+80601$ |
| 1.1-a4 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$3.594937582$ |
0.308137507 |
\( 68274959951801000509440 a^{3} + 80103860217951906201600 a^{2} - 793592316273690586562560 a - 931085247325018833715200 \) |
\( \bigl[0\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{5}{8} a - \frac{3}{2}\) , \( 1\) , \( \frac{65}{8} a^{3} - \frac{97}{2} a^{2} - \frac{93}{8} a + \frac{133}{2}\) , \( -\frac{2561}{4} a^{3} + 2052 a^{2} + \frac{3525}{4} a - 2825\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{5}{8}a-\frac{3}{2}\right){x}^{2}+\left(\frac{65}{8}a^{3}-\frac{97}{2}a^{2}-\frac{93}{8}a+\frac{133}{2}\right){x}-\frac{2561}{4}a^{3}+2052a^{2}+\frac{3525}{4}a-2825$ |
| 1.1-a5 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( -68274959951801000509440 a^{3} + 80103860217951906201600 a^{2} + 793592316273690586562560 a - 931085247325018833715200 \) |
\( \bigl[0\) , \( \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{5}{8} a + \frac{3}{2}\) , \( 1\) , \( \frac{555}{8} a^{3} - \frac{483}{2} a^{2} - \frac{703}{8} a + \frac{647}{2}\) , \( -\frac{4231}{2} a^{3} + 7176 a^{2} + \frac{5805}{2} a - 9867\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{5}{8}a+\frac{3}{2}\right){x}^{2}+\left(\frac{555}{8}a^{3}-\frac{483}{2}a^{2}-\frac{703}{8}a+\frac{647}{2}\right){x}-\frac{4231}{2}a^{3}+7176a^{2}+\frac{5805}{2}a-9867$ |
| 1.1-a6 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$3.594937582$ |
0.308137507 |
\( -23495540774930605015040 a^{3} - 80103860217951906201600 a^{2} + 32342190266893863157760 a + 110264935508355946905600 \) |
\( \bigl[0\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{11}{4} a + 2\) , \( 1\) , \( -24 a^{3} + \frac{99}{2} a^{2} + \frac{557}{2} a - 571\) , \( \frac{7537}{4} a^{3} - 2101 a^{2} - \frac{87605}{4} a + 24419\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{11}{4}a+2\right){x}^{2}+\left(-24a^{3}+\frac{99}{2}a^{2}+\frac{557}{2}a-571\right){x}+\frac{7537}{4}a^{3}-2101a^{2}-\frac{87605}{4}a+24419$ |
| 1.1-a7 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( 68274959951801000509440 a^{3} + 80103860217951906201600 a^{2} - 793592316273690586562560 a - 931085247325018833715200 \) |
\( \bigl[0\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{11}{4} a + 3\) , \( 1\) , \( -\frac{277}{4} a^{3} - \frac{485}{2} a^{2} + \frac{335}{4} a + 329\) , \( \frac{4001}{2} a^{3} + 6786 a^{2} - \frac{5483}{2} a - 9322\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{11}{4}a+3\right){x}^{2}+\left(-\frac{277}{4}a^{3}-\frac{485}{2}a^{2}+\frac{335}{4}a+329\right){x}+\frac{4001}{2}a^{3}+6786a^{2}-\frac{5483}{2}a-9322$ |
| 1.1-a8 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-315$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$3.594937582$ |
0.308137507 |
\( 23495540774930605015040 a^{3} - 80103860217951906201600 a^{2} - 32342190266893863157760 a + 110264935508355946905600 \) |
\( \bigl[0\) , \( \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{5}{8} a + \frac{7}{2}\) , \( 1\) , \( \frac{193}{8} a^{3} + \frac{99}{2} a^{2} - \frac{2253}{8} a - \frac{1143}{2}\) , \( -1919 a^{3} - 2108 a^{2} + 22306 a + 24500\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{5}{8}a+\frac{7}{2}\right){x}^{2}+\left(\frac{193}{8}a^{3}+\frac{99}{2}a^{2}-\frac{2253}{8}a-\frac{1143}{2}\right){x}-1919a^{3}-2108a^{2}+22306a+24500$ |
| 1.1-a9 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-35$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( 6594560 a^{3} - 59351040 a - 58982400 \) |
\( \bigl[0\) , \( \frac{1}{8} a^{3} + \frac{1}{2} a^{2} - \frac{13}{8} a - \frac{5}{2}\) , \( 1\) , \( \frac{175}{8} a^{3} + \frac{149}{2} a^{2} - \frac{227}{8} a - \frac{201}{2}\) , \( \frac{517}{2} a^{3} + 879 a^{2} - \frac{713}{2} a - 1211\bigr] \) |
${y}^2+{y}={x}^{3}+\left(\frac{1}{8}a^{3}+\frac{1}{2}a^{2}-\frac{13}{8}a-\frac{5}{2}\right){x}^{2}+\left(\frac{175}{8}a^{3}+\frac{149}{2}a^{2}-\frac{227}{8}a-\frac{201}{2}\right){x}+\frac{517}{2}a^{3}+879a^{2}-\frac{713}{2}a-1211$ |
| 1.1-a10 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-35$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( -6594560 a^{3} + 59351040 a - 58982400 \) |
\( \bigl[0\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{5}{2}\) , \( 1\) , \( -\frac{175}{8} a^{3} + \frac{149}{2} a^{2} + \frac{227}{8} a - \frac{201}{2}\) , \( -\frac{517}{2} a^{3} + 879 a^{2} + \frac{713}{2} a - 1211\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{5}{2}\right){x}^{2}+\left(-\frac{175}{8}a^{3}+\frac{149}{2}a^{2}+\frac{227}{8}a-\frac{201}{2}\right){x}-\frac{517}{2}a^{3}+879a^{2}+\frac{713}{2}a-1211$ |
| 1.1-a11 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-35$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( -6594560 a^{3} + 59351040 a - 58982400 \) |
\( \bigl[0\) , \( -\frac{1}{2} a^{2} - \frac{1}{2} a + 4\) , \( 1\) , \( -64 a^{3} - \frac{149}{2} a^{2} + \frac{1489}{2} a + 868\) , \( -751 a^{3} - 879 a^{2} + 8729 a + 10216\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}-\frac{1}{2}a+4\right){x}^{2}+\left(-64a^{3}-\frac{149}{2}a^{2}+\frac{1489}{2}a+868\right){x}-751a^{3}-879a^{2}+8729a+10216$ |
| 1.1-a12 |
1.1-a |
$12$ |
$315$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-35$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$291.1899442$ |
0.308137507 |
\( 6594560 a^{3} - 59351040 a - 58982400 \) |
\( \bigl[0\) , \( -\frac{1}{2} a^{2} + \frac{1}{2} a + 4\) , \( 1\) , \( 64 a^{3} - \frac{149}{2} a^{2} - \frac{1489}{2} a + 868\) , \( 751 a^{3} - 879 a^{2} - 8729 a + 10216\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{1}{2}a+4\right){x}^{2}+\left(64a^{3}-\frac{149}{2}a^{2}-\frac{1489}{2}a+868\right){x}+751a^{3}-879a^{2}-8729a+10216$ |
| 1.1-b1 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( \frac{16554983445}{8} a^{3} - \frac{148994851005}{8} a + \frac{37018076625}{2} \) |
\( \bigl[\frac{1}{8} a^{3} + \frac{1}{2} a^{2} - \frac{13}{8} a - \frac{7}{2}\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( \frac{17}{8} a^{3} + \frac{5}{2} a^{2} - \frac{221}{8} a - \frac{65}{2}\) , \( -\frac{35}{8} a^{3} - \frac{11}{2} a^{2} + \frac{383}{8} a + \frac{109}{2}\bigr] \) |
${y}^2+\left(\frac{1}{8}a^{3}+\frac{1}{2}a^{2}-\frac{13}{8}a-\frac{7}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-3\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){x}^{2}+\left(\frac{17}{8}a^{3}+\frac{5}{2}a^{2}-\frac{221}{8}a-\frac{65}{2}\right){x}-\frac{35}{8}a^{3}-\frac{11}{2}a^{2}+\frac{383}{8}a+\frac{109}{2}$ |
| 1.1-b2 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( \frac{16554983445}{8} a^{3} - \frac{148994851005}{8} a + \frac{37018076625}{2} \) |
\( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{13}{8} a + \frac{5}{2}\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{3}{2}\) , \( -3 a^{2} + 9 a + 3\) , \( -\frac{17}{8} a^{3} + 5 a^{2} + \frac{81}{8} a - \frac{29}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-3\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{3}{2}\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{13}{8}a+\frac{5}{2}\right){x}^{2}+\left(-3a^{2}+9a+3\right){x}-\frac{17}{8}a^{3}+5a^{2}+\frac{81}{8}a-\frac{29}{2}$ |
| 1.1-b3 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( -\frac{16554983445}{8} a^{3} + \frac{148994851005}{8} a + \frac{37018076625}{2} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{13}{8} a + \frac{5}{2}\) , \( \frac{1}{8} a^{3} + \frac{1}{2} a^{2} - \frac{13}{8} a - \frac{7}{2}\) , \( -\frac{5}{2} a^{2} - \frac{17}{2} a\) , \( \frac{9}{4} a^{3} + \frac{11}{2} a^{2} - \frac{47}{4} a - 17\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-3\right){x}{y}+\left(\frac{1}{8}a^{3}+\frac{1}{2}a^{2}-\frac{13}{8}a-\frac{7}{2}\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{13}{8}a+\frac{5}{2}\right){x}^{2}+\left(-\frac{5}{2}a^{2}-\frac{17}{2}a\right){x}+\frac{9}{4}a^{3}+\frac{11}{2}a^{2}-\frac{47}{4}a-17$ |
| 1.1-b4 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( -\frac{16554983445}{8} a^{3} + \frac{148994851005}{8} a + \frac{37018076625}{2} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{3}{2}\) , \( \frac{1}{8} a^{3} - \frac{9}{8} a - \frac{3}{2}\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 3\) , \( -\frac{17}{8} a^{3} + 2 a^{2} + \frac{225}{8} a - \frac{59}{2}\) , \( \frac{35}{8} a^{3} - \frac{11}{2} a^{2} - \frac{383}{8} a + \frac{109}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-3\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{9}{8}a-\frac{3}{2}\right){x}^{2}+\left(-\frac{17}{8}a^{3}+2a^{2}+\frac{225}{8}a-\frac{59}{2}\right){x}+\frac{35}{8}a^{3}-\frac{11}{2}a^{2}-\frac{383}{8}a+\frac{109}{2}$ |
| 1.1-b5 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( -\frac{85995}{8} a^{3} + \frac{773955}{8} a - \frac{191025}{2} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{11}{4} a - 2\) , \( \frac{1}{8} a^{3} + \frac{1}{2} a^{2} - \frac{13}{8} a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a + 7\) , \( -\frac{1}{8} a^{3} + \frac{1}{8} a - \frac{3}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(\frac{1}{8}a^{3}+\frac{1}{2}a^{2}-\frac{13}{8}a-\frac{7}{2}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{11}{4}a-2\right){x}^{2}+\left(-\frac{1}{2}a^{3}+\frac{9}{2}a+7\right){x}-\frac{1}{8}a^{3}+\frac{1}{8}a-\frac{3}{2}$ |
| 1.1-b6 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( \frac{85995}{8} a^{3} - \frac{773955}{8} a - \frac{191025}{2} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{5}{8} a + \frac{3}{2}\) , \( \frac{1}{8} a^{3} + \frac{1}{2} a^{2} - \frac{13}{8} a - \frac{7}{2}\) , \( -\frac{3}{8} a^{3} + \frac{1}{2} a^{2} + \frac{15}{8} a + \frac{3}{2}\) , \( \frac{5}{8} a^{3} - \frac{45}{8} a - \frac{11}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(\frac{1}{8}a^{3}+\frac{1}{2}a^{2}-\frac{13}{8}a-\frac{7}{2}\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{5}{8}a+\frac{3}{2}\right){x}^{2}+\left(-\frac{3}{8}a^{3}+\frac{1}{2}a^{2}+\frac{15}{8}a+\frac{3}{2}\right){x}+\frac{5}{8}a^{3}-\frac{45}{8}a-\frac{11}{2}$ |
| 1.1-b7 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( \frac{85995}{8} a^{3} - \frac{773955}{8} a - \frac{191025}{2} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( \frac{1}{2} a^{2} - \frac{3}{2} a - 4\) , \( 0\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{1}{2}\) , \( 0\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{3}{2}a-4\right){x}^{2}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}-\frac{3}{8}a+\frac{1}{2}\right){x}$ |
| 1.1-b8 |
1.1-b |
$8$ |
$30$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$558.1508429$ |
1.328930578 |
\( -\frac{85995}{8} a^{3} + \frac{773955}{8} a - \frac{191025}{2} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 4\) , \( 1\) , \( -\frac{9}{8} a^{3} + \frac{1}{2} a^{2} + \frac{101}{8} a + \frac{19}{2}\) , \( -\frac{7}{8} a^{3} - a^{2} + \frac{111}{8} a + \frac{21}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+4\right){x}^{2}+\left(-\frac{9}{8}a^{3}+\frac{1}{2}a^{2}+\frac{101}{8}a+\frac{19}{2}\right){x}-\frac{7}{8}a^{3}-a^{2}+\frac{111}{8}a+\frac{21}{2}$ |
| 1.1-c1 |
1.1-c |
$4$ |
$14$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$684.5475207$ |
1.629875049 |
\( 16581375 \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{8} a^{3} - \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{8} a^{3} - \frac{9}{8} a - \frac{1}{2}\) , \( -\frac{57}{2} a^{3} - \frac{213}{2} a^{2} + 38 a + 150\) , \( 530 a^{3} + \frac{3527}{2} a^{2} - \frac{1461}{2} a - 2427\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(\frac{1}{8}a^{3}-\frac{9}{8}a-\frac{1}{2}\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-\frac{57}{2}a^{3}-\frac{213}{2}a^{2}+38a+150\right){x}+530a^{3}+\frac{3527}{2}a^{2}-\frac{1461}{2}a-2427$ |
| 1.1-c2 |
1.1-c |
$4$ |
$14$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$684.5475207$ |
1.629875049 |
\( -3375 \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{8} a^{3} - \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{8} a^{3} - \frac{9}{8} a - \frac{1}{2}\) , \( -\frac{13}{8} a^{3} - \frac{13}{2} a^{2} + \frac{9}{8} a + \frac{25}{2}\) , \( \frac{75}{8} a^{3} + \frac{61}{2} a^{2} - \frac{111}{8} a - \frac{83}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(\frac{1}{8}a^{3}-\frac{9}{8}a-\frac{1}{2}\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-\frac{13}{8}a^{3}-\frac{13}{2}a^{2}+\frac{9}{8}a+\frac{25}{2}\right){x}+\frac{75}{8}a^{3}+\frac{61}{2}a^{2}-\frac{111}{8}a-\frac{83}{2}$ |
| 1.1-c3 |
1.1-c |
$4$ |
$14$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$684.5475207$ |
1.629875049 |
\( -3375 \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( 0\) , \( \frac{1}{8} a^{3} - \frac{9}{8} a - \frac{1}{2}\) , \( -\frac{41}{8} a^{3} + 6 a^{2} + \frac{481}{8} a - \frac{137}{2}\) , \( 27 a^{3} - \frac{61}{2} a^{2} - \frac{627}{2} a + 355\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+\left(\frac{1}{8}a^{3}-\frac{9}{8}a-\frac{1}{2}\right){y}={x}^{3}+\left(-\frac{41}{8}a^{3}+6a^{2}+\frac{481}{8}a-\frac{137}{2}\right){x}+27a^{3}-\frac{61}{2}a^{2}-\frac{627}{2}a+355$ |
| 1.1-c4 |
1.1-c |
$4$ |
$14$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$9.38270$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
✓ |
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$684.5475207$ |
1.629875049 |
\( 16581375 \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( 0\) , \( \frac{1}{8} a^{3} - \frac{9}{8} a + \frac{1}{2}\) , \( \frac{227}{8} a^{3} - \frac{213}{2} a^{2} - \frac{295}{8} a + \frac{299}{2}\) , \( -530 a^{3} + \frac{3527}{2} a^{2} + \frac{1461}{2} a - 2427\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+\left(\frac{1}{8}a^{3}-\frac{9}{8}a+\frac{1}{2}\right){y}={x}^{3}+\left(\frac{227}{8}a^{3}-\frac{213}{2}a^{2}-\frac{295}{8}a+\frac{299}{2}\right){x}-530a^{3}+\frac{3527}{2}a^{2}+\frac{1461}{2}a-2427$ |
| 16.1-a1 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( -\frac{1522629945065565}{64} a^{2} + 276535182623490 \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( a\) , \( \frac{181}{4} a^{3} + \frac{107}{2} a^{2} - \frac{2175}{4} a - 691\) , \( \frac{1491}{2} a^{3} + \frac{1741}{2} a^{2} - 8610 a - 9936\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(\frac{181}{4}a^{3}+\frac{107}{2}a^{2}-\frac{2175}{4}a-691\right){x}+\frac{1491}{2}a^{3}+\frac{1741}{2}a^{2}-8610a-9936$ |
| 16.1-a2 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{32} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( \frac{6240927285}{4096} a^{2} - \frac{4533829065}{256} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( a\) , \( \frac{11}{4} a^{3} + \frac{7}{2} a^{2} - \frac{125}{4} a - 41\) , \( \frac{31}{2} a^{3} + \frac{37}{2} a^{2} - 176 a - 204\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(\frac{11}{4}a^{3}+\frac{7}{2}a^{2}-\frac{125}{4}a-41\right){x}+\frac{31}{2}a^{3}+\frac{37}{2}a^{2}-176a-204$ |
| 16.1-a3 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{32} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( -\frac{6240927285}{4096} a^{2} + \frac{8590789665}{4096} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( a\) , \( \frac{1}{2} a^{3} - 4 a^{2} + \frac{13}{2} a + 13\) , \( \frac{31}{8} a^{3} - 13 a^{2} - \frac{79}{8} a + \frac{75}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}^{2}+\left(\frac{1}{2}a^{3}-4a^{2}+\frac{13}{2}a+13\right){x}+\frac{31}{8}a^{3}-13a^{2}-\frac{79}{8}a+\frac{75}{2}$ |
| 16.1-a4 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( \frac{1522629945065565}{64} a^{2} - \frac{2095937597948985}{64} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( a\) , \( \frac{21}{2} a^{3} - 54 a^{2} + \frac{93}{2} a + 13\) , \( \frac{1983}{8} a^{3} - 785 a^{2} - \frac{4815}{8} a + \frac{2763}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}^{2}+\left(\frac{21}{2}a^{3}-54a^{2}+\frac{93}{2}a+13\right){x}+\frac{1983}{8}a^{3}-785a^{2}-\frac{4815}{8}a+\frac{2763}{2}$ |
| 16.1-b1 |
16.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( \frac{1522629945065565}{64} a^{2} - \frac{2095937597948985}{64} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( a + 1\) , \( -\frac{89}{8} a^{3} - 54 a^{2} - \frac{295}{8} a + \frac{15}{2}\) , \( -\frac{2163}{8} a^{3} - 871 a^{2} + \frac{4259}{8} a + \frac{2767}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-\frac{89}{8}a^{3}-54a^{2}-\frac{295}{8}a+\frac{15}{2}\right){x}-\frac{2163}{8}a^{3}-871a^{2}+\frac{4259}{8}a+\frac{2767}{2}$ |
| 16.1-b2 |
16.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{32} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( -\frac{6240927285}{4096} a^{2} + \frac{8590789665}{4096} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( a + 1\) , \( -\frac{9}{8} a^{3} - 4 a^{2} + \frac{25}{8} a + \frac{15}{2}\) , \( -\frac{51}{8} a^{3} - 19 a^{2} + \frac{163}{8} a + \frac{79}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-\frac{9}{8}a^{3}-4a^{2}+\frac{25}{8}a+\frac{15}{2}\right){x}-\frac{51}{8}a^{3}-19a^{2}+\frac{163}{8}a+\frac{79}{2}$ |
| 16.1-b3 |
16.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( -\frac{1522629945065565}{64} a^{2} + 276535182623490 \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( a + 1\) , \( -\frac{367}{8} a^{3} + \frac{107}{2} a^{2} + \frac{4427}{8} a - \frac{1373}{2}\) , \( -\frac{5241}{8} a^{3} + \frac{1569}{2} a^{2} + \frac{60197}{8} a - \frac{17641}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}^{2}+\left(-\frac{367}{8}a^{3}+\frac{107}{2}a^{2}+\frac{4427}{8}a-\frac{1373}{2}\right){x}-\frac{5241}{8}a^{3}+\frac{1569}{2}a^{2}+\frac{60197}{8}a-\frac{17641}{2}$ |
| 16.1-b4 |
16.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{32} \) |
$13.26914$ |
$(1/4a^3-13/4a+1), (1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.23936326$ |
0.554660602 |
\( \frac{6240927285}{4096} a^{2} - \frac{4533829065}{256} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( a + 1\) , \( -\frac{27}{8} a^{3} + \frac{7}{2} a^{2} + \frac{327}{8} a - \frac{73}{2}\) , \( -\frac{81}{8} a^{3} + \frac{25}{2} a^{2} + \frac{925}{8} a - \frac{257}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}^{2}+\left(-\frac{27}{8}a^{3}+\frac{7}{2}a^{2}+\frac{327}{8}a-\frac{73}{2}\right){x}-\frac{81}{8}a^{3}+\frac{25}{2}a^{2}+\frac{925}{8}a-\frac{257}{2}$ |
| 16.2-a1 |
16.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/4a^3-13/4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$313.1141542$ |
2.982039564 |
\( 0 \) |
\( \bigl[0\) , \( \frac{1}{8} a^{3} - \frac{17}{8} a - \frac{3}{2}\) , \( a\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{5}{2}\) , \( 7 a^{3} - \frac{47}{2} a^{2} - \frac{21}{2} a + 31\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{17}{8}a-\frac{3}{2}\right){x}^{2}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{5}{2}\right){x}+7a^{3}-\frac{47}{2}a^{2}-\frac{21}{2}a+31$ |
| 16.2-a2 |
16.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/4a^3-13/4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$313.1141542$ |
2.982039564 |
\( 0 \) |
\( \bigl[0\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( a\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{5}{2}\) , \( -7 a^{3} - \frac{47}{2} a^{2} + \frac{21}{2} a + 33\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}^{2}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{5}{2}\right){x}-7a^{3}-\frac{47}{2}a^{2}+\frac{21}{2}a+33$ |
| 16.3-a1 |
16.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$313.1141542$ |
2.982039564 |
\( 0 \) |
\( \bigl[0\) , \( \frac{1}{8} a^{3} - \frac{17}{8} a - \frac{3}{2}\) , \( a + 1\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{5}{2}\) , \( -\frac{161}{8} a^{3} + 23 a^{2} + \frac{1865}{8} a - \frac{543}{2}\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{17}{8}a-\frac{3}{2}\right){x}^{2}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{5}{2}\right){x}-\frac{161}{8}a^{3}+23a^{2}+\frac{1865}{8}a-\frac{543}{2}$ |
| 16.3-a2 |
16.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
16.3 |
\( 2^{4} \) |
\( 2^{16} \) |
$13.26914$ |
$(1/8a^3+1/2a^2+3/8a-1/2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$313.1141542$ |
2.982039564 |
\( 0 \) |
\( \bigl[0\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( a + 1\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{5}{2}\) , \( \frac{161}{8} a^{3} + 23 a^{2} - \frac{1873}{8} a - \frac{539}{2}\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}^{2}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{5}{2}\right){x}+\frac{161}{8}a^{3}+23a^{2}-\frac{1873}{8}a-\frac{539}{2}$ |
| 20.1-a1 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{6} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$258.3134093$ |
1.230063853 |
\( \frac{29938059}{400} a^{3} + \frac{128057841}{500} a^{2} - \frac{200696211}{2000} a - \frac{176759631}{500} \) |
\( \bigl[1\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} - \frac{5}{2} a^{2} - 2 a + 2\) , \( \frac{7}{4} a^{3} + 5 a^{2} - \frac{27}{4} a - 13\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-\frac{1}{2}a^{3}-\frac{5}{2}a^{2}-2a+2\right){x}+\frac{7}{4}a^{3}+5a^{2}-\frac{27}{4}a-13$ |
| 20.1-a2 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{12} \cdot 5 \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$258.3134093$ |
1.230063853 |
\( \frac{2621187}{40} a^{3} + \frac{70254783}{320} a^{2} - \frac{4390443}{40} a - \frac{103402467}{320} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{3}{2} a - 2\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{173}{8} a^{3} - 72 a^{2} + \frac{237}{8} a + \frac{199}{2}\) , \( -\frac{2479}{8} a^{3} - 1056 a^{2} + \frac{3399}{8} a + \frac{2903}{2}\bigr] \) |
${y}^2+{x}{y}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{3}{2}a-2\right){x}^{2}+\left(-\frac{173}{8}a^{3}-72a^{2}+\frac{237}{8}a+\frac{199}{2}\right){x}-\frac{2479}{8}a^{3}-1056a^{2}+\frac{3399}{8}a+\frac{2903}{2}$ |
| 20.1-a3 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{4} \cdot 5^{3} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$258.3134093$ |
1.230063853 |
\( \frac{760472847}{800} a^{3} - \frac{216790911}{200} a^{2} - \frac{8953565283}{800} a + \frac{2617848639}{200} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -1\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( \frac{15}{2} a^{3} - \frac{57}{2} a^{2} - 9 a + 39\) , \( \frac{581}{8} a^{3} - 247 a^{2} - \frac{797}{8} a + \frac{679}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){y}={x}^{3}-{x}^{2}+\left(\frac{15}{2}a^{3}-\frac{57}{2}a^{2}-9a+39\right){x}+\frac{581}{8}a^{3}-247a^{2}-\frac{797}{8}a+\frac{679}{2}$ |
| 20.1-a4 |
20.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{2} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$258.3134093$ |
1.230063853 |
\( -\frac{413786772}{5} a^{3} - \frac{3883816431}{40} a^{2} + \frac{9619284519}{10} a + \frac{9028702179}{8} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{2} a^{2} + \frac{1}{2} a + 4\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} - a^{2} + \frac{1}{8} a + \frac{23}{2}\) , \( \frac{3}{8} a^{3} - a^{2} - \frac{43}{8} a + \frac{13}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-3\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{1}{2}a+4\right){x}^{2}+\left(-\frac{1}{8}a^{3}-a^{2}+\frac{1}{8}a+\frac{23}{2}\right){x}+\frac{3}{8}a^{3}-a^{2}-\frac{43}{8}a+\frac{13}{2}$ |
| 20.1-b1 |
20.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{16} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.011290155$ |
$169.3737702$ |
3.262590980 |
\( -\frac{14853918942721443}{6250000} a^{3} + \frac{12689133491253093}{1562500} a^{2} + \frac{20444491705193023}{6250000} a - \frac{3493240620731731}{312500} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{3}{2}\) , \( \frac{1}{8} a^{3} - \frac{17}{8} a - \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{195}{4} a^{3} - 290 a^{2} - \frac{255}{4} a + 396\) , \( -\frac{11461}{8} a^{3} + \frac{12851}{2} a^{2} + \frac{15833}{8} a - \frac{17711}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{3}{2}\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{17}{8}a-\frac{1}{2}\right){x}^{2}+\left(\frac{195}{4}a^{3}-290a^{2}-\frac{255}{4}a+396\right){x}-\frac{11461}{8}a^{3}+\frac{12851}{2}a^{2}+\frac{15833}{8}a-\frac{17711}{2}$ |
| 20.1-b2 |
20.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{4} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.011290155$ |
$169.3737702$ |
3.262590980 |
\( -\frac{116926849}{3200} a^{3} + \frac{34244453}{800} a^{2} + \frac{271840153}{640} a - \frac{398073249}{800} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{3}{2}\) , \( \frac{1}{8} a^{3} - \frac{17}{8} a - \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( 1\) , \( -\frac{1}{8} a^{3} + 3 a^{2} - \frac{7}{8} a - \frac{11}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{3}{2}\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{17}{8}a-\frac{1}{2}\right){x}^{2}+{x}-\frac{1}{8}a^{3}+3a^{2}-\frac{7}{8}a-\frac{11}{2}$ |
| 20.1-b3 |
20.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{8} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.022580310$ |
$169.3737702$ |
3.262590980 |
\( \frac{128458946517473}{4000} a^{3} - \frac{188393625220043}{5000} a^{2} - \frac{7465698168594497}{20000} a + \frac{2189789722359563}{5000} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{21}{8} a + \frac{9}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( 27 a^{3} - 100 a^{2} - 21 a + 155\) , \( -\frac{4389}{8} a^{3} + 1862 a^{2} + \frac{6181}{8} a - \frac{5089}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{21}{8}a+\frac{9}{2}\right){x}^{2}+\left(27a^{3}-100a^{2}-21a+155\right){x}-\frac{4389}{8}a^{3}+1862a^{2}+\frac{6181}{8}a-\frac{5089}{2}$ |
| 20.1-b4 |
20.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{4} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$4.045160621$ |
$10.58586064$ |
3.262590980 |
\( \frac{9919402510576784814896203}{400} a^{3} - \frac{2909494317952668385276541}{100} a^{2} - \frac{23059586179111455095520411}{80} a + \frac{33818435581642441798523303}{100} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( -\frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{21}{8} a + \frac{9}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( 37 a^{3} - \frac{275}{2} a^{2} - \frac{57}{2} a + 195\) , \( -\frac{1259}{8} a^{3} + \frac{1029}{2} a^{2} + \frac{1951}{8} a - \frac{1439}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{21}{8}a+\frac{9}{2}\right){x}^{2}+\left(37a^{3}-\frac{275}{2}a^{2}-\frac{57}{2}a+195\right){x}-\frac{1259}{8}a^{3}+\frac{1029}{2}a^{2}+\frac{1951}{8}a-\frac{1439}{2}$ |
| 20.1-c1 |
20.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{16} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$19.01065840$ |
2.896862233 |
\( -\frac{14853918942721443}{6250000} a^{3} + \frac{12689133491253093}{1562500} a^{2} + \frac{20444491705193023}{6250000} a - \frac{3493240620731731}{312500} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{5}{2}\) , \( -410 a^{3} - 1417 a^{2} + 566 a + 1947\) , \( -\frac{224559}{8} a^{3} - 95800 a^{2} + \frac{309087}{8} a + \frac{263729}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-3\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-410a^{3}-1417a^{2}+566a+1947\right){x}-\frac{224559}{8}a^{3}-95800a^{2}+\frac{309087}{8}a+\frac{263729}{2}$ |
| 20.1-c2 |
20.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{4} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$304.1705345$ |
2.896862233 |
\( -\frac{116926849}{3200} a^{3} + \frac{34244453}{800} a^{2} + \frac{271840153}{640} a - \frac{398073249}{800} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{5}{2}\) , \( \frac{5}{4} a^{3} + \frac{11}{2} a^{2} + \frac{9}{4} a - 8\) , \( -\frac{189}{8} a^{3} - 80 a^{2} + \frac{277}{8} a + \frac{219}{2}\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-3\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(\frac{5}{4}a^{3}+\frac{11}{2}a^{2}+\frac{9}{4}a-8\right){x}-\frac{189}{8}a^{3}-80a^{2}+\frac{277}{8}a+\frac{219}{2}$ |
| 20.1-c3 |
20.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{8} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$304.1705345$ |
2.896862233 |
\( \frac{128458946517473}{4000} a^{3} - \frac{188393625220043}{5000} a^{2} - \frac{7465698168594497}{20000} a + \frac{2189789722359563}{5000} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{15}{4} a + 4\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{5}{2}\) , \( -\frac{15}{4} a^{3} - 18 a^{2} + \frac{63}{4} a + 34\) , \( -\frac{97}{4} a^{3} - 101 a^{2} + \frac{161}{4} a + 144\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{15}{4}a+4\right){x}^{2}+\left(-\frac{15}{4}a^{3}-18a^{2}+\frac{63}{4}a+34\right){x}-\frac{97}{4}a^{3}-101a^{2}+\frac{161}{4}a+144$ |
| 20.1-c4 |
20.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{4} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$304.1705345$ |
2.896862233 |
\( \frac{9919402510576784814896203}{400} a^{3} - \frac{2909494317952668385276541}{100} a^{2} - \frac{23059586179111455095520411}{80} a + \frac{33818435581642441798523303}{100} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{15}{4} a + 4\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{5}{2}\) , \( -\frac{15}{2} a^{3} - 43 a^{2} - \frac{41}{2} a + 19\) , \( \frac{179}{2} a^{3} + \frac{653}{2} a^{2} - 71 a - 391\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(-\frac{1}{8}a^{3}+\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{5}{2}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{15}{4}a+4\right){x}^{2}+\left(-\frac{15}{2}a^{3}-43a^{2}-\frac{41}{2}a+19\right){x}+\frac{179}{2}a^{3}+\frac{653}{2}a^{2}-71a-391$ |
| 20.1-d1 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{6} \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.082272997$ |
$705.9003792$ |
3.318659435 |
\( \frac{29938059}{400} a^{3} + \frac{128057841}{500} a^{2} - \frac{200696211}{2000} a - \frac{176759631}{500} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 6 a + 4\) , \( -\frac{3}{8} a^{3} + \frac{19}{8} a + \frac{1}{2}\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-3\right){y}={x}^{3}+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}^{2}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+6a+4\right){x}-\frac{3}{8}a^{3}+\frac{19}{8}a+\frac{1}{2}$ |
| 20.1-d2 |
20.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{5}, \sqrt{21})\) |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{12} \cdot 5 \) |
$13.64447$ |
$(-1/8a^3-1/2a^2+13/8a+11/2), (1/4a^3-13/4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.123409496$ |
$1411.800758$ |
3.318659435 |
\( \frac{2621187}{40} a^{3} + \frac{70254783}{320} a^{2} - \frac{4390443}{40} a - \frac{103402467}{320} \) |
\( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{5}{8} a + \frac{3}{2}\) , \( a\) , \( \frac{17}{8} a^{3} - 14 a^{2} - \frac{25}{8} a + \frac{39}{2}\) , \( 12 a^{3} - \frac{45}{2} a^{2} - \frac{33}{2} a + 31\bigr] \) |
${y}^2+\left(-\frac{1}{8}a^{3}+\frac{17}{8}a+\frac{1}{2}\right){x}{y}+a{y}={x}^{3}+\left(\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{5}{8}a+\frac{3}{2}\right){x}^{2}+\left(\frac{17}{8}a^{3}-14a^{2}-\frac{25}{8}a+\frac{39}{2}\right){x}+12a^{3}-\frac{45}{2}a^{2}-\frac{33}{2}a+31$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
