| 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 |
| 3.1-a1 |
3.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{8} \) |
$0.97688$ |
$(a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.01298155$ |
0.662903589 |
\( -\frac{2924207}{81} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -8 a - 34\) , \( -50 a - 186\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-8a-34\right){x}-50a-186$ |
| 3.1-a2 |
3.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{4} \) |
$0.97688$ |
$(a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.01298155$ |
0.662903589 |
\( \frac{12214672127}{9} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -143 a - 529\) , \( -2255 a - 8241\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-143a-529\right){x}-2255a-8241$ |
| 3.1-b1 |
3.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{8} \) |
$0.97688$ |
$(a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.01298155$ |
0.662903589 |
\( -\frac{2924207}{81} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 6 a - 41\) , \( 49 a - 236\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(6a-41\right){x}+49a-236$ |
| 3.1-b2 |
3.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{4} \) |
$0.97688$ |
$(a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.01298155$ |
0.662903589 |
\( \frac{12214672127}{9} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 141 a - 671\) , \( 2254 a - 10496\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(141a-671\right){x}+2254a-10496$ |
| 15.1-a1 |
15.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.46078$ |
$(a+4), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.157212265$ |
$57.85829530$ |
2.190067553 |
\( -\frac{32768}{15} a + \frac{65536}{15} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( 2 a - 9\) , \( -3 a + 6\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2a-9\right){x}-3a+6$ |
| 15.1-b1 |
15.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.46078$ |
$(a+4), (-a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$13.89855772$ |
1.673189728 |
\( -\frac{32768}{15} a + \frac{65536}{15} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -2 a - 7\) , \( -5 a - 20\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-2a-7\right){x}-5a-20$ |
| 15.1-c1 |
15.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$20.52581431$ |
2.471017666 |
\( \frac{1234571}{5625} a - \frac{1399274}{1875} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 3 a - 7\) , \( -6 a + 14\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3a-7\right){x}-6a+14$ |
| 15.1-c2 |
15.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$20.52581431$ |
2.471017666 |
\( -\frac{686395653229}{1171875} a + \frac{3202515927403}{1171875} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 48 a - 217\) , \( -321 a + 1481\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(48a-217\right){x}-321a+1481$ |
| 15.1-c3 |
15.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{16} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.131453578$ |
2.471017666 |
\( \frac{319450720644289}{457763671875} a + \frac{2560911940710052}{457763671875} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 53 a - 242\) , \( -231 a + 1064\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(53a-242\right){x}-231a+1064$ |
| 15.1-c4 |
15.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$20.52581431$ |
2.471017666 |
\( -\frac{5357636444159857}{1875} a + \frac{24930832344480524}{1875} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 763 a - 3552\) , \( -22671 a + 105506\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(763a-3552\right){x}-22671a+105506$ |
| 15.1-d1 |
15.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$14.26530951$ |
1.717341456 |
\( -\frac{2920448}{9375} a + \frac{20340736}{9375} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 5\) , \( -a - 6\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+5{x}-a-6$ |
| 15.1-e1 |
15.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.354224316$ |
$5.995714898$ |
2.421076605 |
\( \frac{1234571}{5625} a - \frac{1399274}{1875} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 9 a + 33\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a+33\right){x}$ |
| 15.1-e2 |
15.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$6.708448632$ |
$5.995714898$ |
2.421076605 |
\( -\frac{686395653229}{1171875} a + \frac{3202515927403}{1171875} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -36 a - 132\) , \( -399 a - 1458\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-36a-132\right){x}-399a-1458$ |
| 15.1-e3 |
15.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{16} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.354224316$ |
$5.995714898$ |
2.421076605 |
\( \frac{319450720644289}{457763671875} a + \frac{2560911940710052}{457763671875} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -166 a - 607\) , \( 1600 a + 5845\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-166a-607\right){x}+1600a+5845$ |
| 15.1-e4 |
15.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$13.41689726$ |
$1.498928724$ |
2.421076605 |
\( -\frac{5357636444159857}{1875} a + \frac{24930832344480524}{1875} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -626 a - 2297\) , \( -19954 a - 72913\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-626a-2297\right){x}-19954a-72913$ |
| 15.1-f1 |
15.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$1.46078$ |
$(a+4), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.081964889$ |
$22.55016089$ |
2.225117533 |
\( -\frac{2920448}{9375} a + \frac{20340736}{9375} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 41 a + 157\) , \( 297 a + 1087\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(41a+157\right){x}+297a+1087$ |
| 15.2-a1 |
15.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.46078$ |
$(a+4), (-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.157212265$ |
$57.85829530$ |
2.190067553 |
\( \frac{32768}{15} a + \frac{32768}{15} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -2 a - 7\) , \( 2 a + 4\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-2a-7\right){x}+2a+4$ |
| 15.2-b1 |
15.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$1.46078$ |
$(a+4), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$13.89855772$ |
1.673189728 |
\( \frac{32768}{15} a + \frac{32768}{15} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( 2 a - 9\) , \( 4 a - 24\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(2a-9\right){x}+4a-24$ |
| 15.2-c1 |
15.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$20.52581431$ |
2.471017666 |
\( -\frac{1234571}{5625} a - \frac{2963251}{5625} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -2 a - 4\) , \( 2 a + 5\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a-4\right){x}+2a+5$ |
| 15.2-c2 |
15.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{16} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.131453578$ |
2.471017666 |
\( -\frac{319450720644289}{457763671875} a + \frac{2880362661354341}{457763671875} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -52 a - 189\) , \( 177 a + 645\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-52a-189\right){x}+177a+645$ |
| 15.2-c3 |
15.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$20.52581431$ |
2.471017666 |
\( \frac{686395653229}{1171875} a + \frac{838706758058}{390625} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -47 a - 169\) , \( 272 a + 992\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-47a-169\right){x}+272a+992$ |
| 15.2-c4 |
15.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$20.52581431$ |
2.471017666 |
\( \frac{5357636444159857}{1875} a + \frac{19573195900320667}{1875} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -762 a - 2789\) , \( 21907 a + 80047\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-762a-2789\right){x}+21907a+80047$ |
| 15.2-d1 |
15.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$14.26530951$ |
1.717341456 |
\( \frac{2920448}{9375} a + \frac{17420288}{9375} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( 5\) , \( -6\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+5{x}-6$ |
| 15.2-e1 |
15.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.354224316$ |
$5.995714898$ |
2.421076605 |
\( -\frac{1234571}{5625} a - \frac{2963251}{5625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 32\) , \( 32 a - 96\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+32{x}+32a-96$ |
| 15.2-e2 |
15.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{16} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.354224316$ |
$5.995714898$ |
2.421076605 |
\( -\frac{319450720644289}{457763671875} a + \frac{2880362661354341}{457763671875} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 175 a - 783\) , \( -2383 a + 11139\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(175a-783\right){x}-2383a+11139$ |
| 15.2-e3 |
15.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$6.708448632$ |
$5.995714898$ |
2.421076605 |
\( \frac{686395653229}{1171875} a + \frac{838706758058}{390625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 45 a - 178\) , \( 221 a - 978\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(45a-178\right){x}+221a-978$ |
| 15.2-e4 |
15.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5^{4} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$13.41689726$ |
$1.498928724$ |
2.421076605 |
\( \frac{5357636444159857}{1875} a + \frac{19573195900320667}{1875} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 635 a - 2933\) , \( 17021 a - 79203\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(635a-2933\right){x}+17021a-79203$ |
| 15.2-f1 |
15.2-f |
$1$ |
$1$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{5} \) |
$1.46078$ |
$(a+4), (-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.081964889$ |
$22.55016089$ |
2.225117533 |
\( \frac{2920448}{9375} a + \frac{17420288}{9375} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -41 a + 198\) , \( -298 a + 1385\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-41a+198\right){x}-298a+1385$ |
| 20.1-a1 |
20.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{6} \cdot 5^{3} \) |
$1.56971$ |
$(-a+4), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$19.72995721$ |
2.375207731 |
\( \frac{117621}{500} a + \frac{1288531}{1000} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 3 a + 9\) , \( 2 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a+9\right){x}+2a+3$ |
| 20.1-a2 |
20.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{2} \cdot 5^{9} \) |
$1.56971$ |
$(-a+4), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$19.72995721$ |
2.375207731 |
\( \frac{5242924526487}{3906250} a + \frac{17908232933941}{3906250} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -37 a - 141\) , \( 142 a + 521\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-37a-141\right){x}+142a+521$ |
| 20.1-a3 |
20.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{18} \cdot 5 \) |
$1.56971$ |
$(-a+4), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.192217468$ |
2.375207731 |
\( \frac{517891128481}{1280} a + \frac{3784034339991}{2560} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -72 a - 266\) , \( -1027 a - 3756\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-72a-266\right){x}-1027a-3756$ |
| 20.1-b1 |
20.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{6} \cdot 5^{3} \) |
$1.56971$ |
$(-a+4), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$2.046777100$ |
$17.07236171$ |
2.804454178 |
\( \frac{117621}{500} a + \frac{1288531}{1000} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -7 a + 40\) , \( 30 a - 134\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-7a+40\right){x}+30a-134$ |
| 20.1-b2 |
20.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{2} \cdot 5^{9} \) |
$1.56971$ |
$(-a+4), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.140331300$ |
$1.896929079$ |
2.804454178 |
\( \frac{5242924526487}{3906250} a + \frac{17908232933941}{3906250} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 283 a - 1310\) , \( 5606 a - 26082\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(283a-1310\right){x}+5606a-26082$ |
| 20.1-b3 |
20.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{18} \cdot 5 \) |
$1.56971$ |
$(-a+4), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.682259033$ |
$17.07236171$ |
2.804454178 |
\( \frac{517891128481}{1280} a + \frac{3784034339991}{2560} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 68 a - 310\) , \( -1212 a + 5644\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(68a-310\right){x}-1212a+5644$ |
| 20.2-a1 |
20.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( - 2^{2} \cdot 5^{9} \) |
$1.56971$ |
$(-a-3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$19.72995721$ |
2.375207731 |
\( -\frac{5242924526487}{3906250} a + \frac{11575578730214}{1953125} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 46 a - 177\) , \( -283 a + 1356\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(46a-177\right){x}-283a+1356$ |
| 20.2-a2 |
20.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( - 2^{18} \cdot 5 \) |
$1.56971$ |
$(-a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.192217468$ |
2.375207731 |
\( -\frac{517891128481}{1280} a + \frac{4819816596953}{2560} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 81 a - 337\) , \( 761 a - 3495\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(81a-337\right){x}+761a-3495$ |
| 20.2-a3 |
20.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( - 2^{6} \cdot 5^{3} \) |
$1.56971$ |
$(-a-3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$19.72995721$ |
2.375207731 |
\( -\frac{117621}{500} a + \frac{1523773}{1000} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 6 a + 13\) , \( 7 a + 18\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a+13\right){x}+7a+18$ |
| 20.2-b1 |
20.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( - 2^{2} \cdot 5^{9} \) |
$1.56971$ |
$(-a-3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.140331300$ |
$1.896929079$ |
2.804454178 |
\( -\frac{5242924526487}{3906250} a + \frac{11575578730214}{1953125} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -284 a - 1027\) , \( -5606 a - 20476\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-284a-1027\right){x}-5606a-20476$ |
| 20.2-b2 |
20.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( - 2^{18} \cdot 5 \) |
$1.56971$ |
$(-a-3), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.682259033$ |
$17.07236171$ |
2.804454178 |
\( -\frac{517891128481}{1280} a + \frac{4819816596953}{2560} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -69 a - 242\) , \( 1212 a + 4432\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-69a-242\right){x}+1212a+4432$ |
| 20.2-b3 |
20.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( - 2^{6} \cdot 5^{3} \) |
$1.56971$ |
$(-a-3), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$2.046777100$ |
$17.07236171$ |
2.804454178 |
\( -\frac{117621}{500} a + \frac{1523773}{1000} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 6 a + 33\) , \( -30 a - 104\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(6a+33\right){x}-30a-104$ |
| 23.1-a1 |
23.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$1.62553$ |
$(-3a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$32.10629687$ |
0.966285984 |
\( -\frac{14013}{23} a + \frac{65691}{23} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 8 a + 22\) , \( 12 a + 38\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a+22\right){x}+12a+38$ |
| 23.1-a2 |
23.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( 23^{2} \) |
$1.62553$ |
$(-3a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.05314843$ |
0.966285984 |
\( -\frac{50921541}{23} a + \frac{237454362}{23} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 23 a - 48\) , \( 78 a - 270\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(23a-48\right){x}+78a-270$ |
| 23.1-b1 |
23.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$1.62553$ |
$(-3a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.557942273$ |
$26.98204712$ |
2.530298262 |
\( \frac{14013}{23} a + \frac{51678}{23} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -a + 8\) , \( 2 a - 11\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+8\right){x}+2a-11$ |
| 23.1-b2 |
23.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( 23^{2} \) |
$1.62553$ |
$(-3a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.778971136$ |
$26.98204712$ |
2.530298262 |
\( \frac{50921541}{23} a + \frac{186532821}{23} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 14 a - 62\) , \( 82 a - 383\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(14a-62\right){x}+82a-383$ |
| 23.1-c1 |
23.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$1.62553$ |
$(-3a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$32.10629687$ |
0.966285984 |
\( \frac{14013}{23} a + \frac{51678}{23} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 3 a + 11\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(3a+11\right){x}$ |
| 23.1-c2 |
23.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( 23^{2} \) |
$1.62553$ |
$(-3a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.05314843$ |
0.966285984 |
\( \frac{50921541}{23} a + \frac{186532821}{23} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -12 a - 44\) , \( -121 a - 442\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-12a-44\right){x}-121a-442$ |
| 23.1-d1 |
23.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$1.62553$ |
$(-3a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.557942273$ |
$26.98204712$ |
2.530298262 |
\( -\frac{14013}{23} a + \frac{65691}{23} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( a + 7\) , \( -2 a - 9\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(a+7\right){x}-2a-9$ |
| 23.1-d2 |
23.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
23.1 |
\( 23 \) |
\( 23^{2} \) |
$1.62553$ |
$(-3a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.778971136$ |
$26.98204712$ |
2.530298262 |
\( -\frac{50921541}{23} a + \frac{237454362}{23} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -14 a - 48\) , \( -82 a - 301\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-14a-48\right){x}-82a-301$ |
| 25.1-a1 |
25.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{10} \) |
$1.65977$ |
$(-a+4), (-a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.366207254$ |
$12.39496478$ |
1.092893117 |
\( -\frac{115732691419136}{1953125} a - \frac{422807160782848}{1953125} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -232 a - 847\) , \( 3842 a + 14036\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-232a-847\right){x}+3842a+14036$ |
| 25.1-a2 |
25.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{69}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{6} \) |
$1.65977$ |
$(-a+4), (-a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1.098621763$ |
$12.39496478$ |
1.092893117 |
\( -\frac{32768}{125} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -2 a - 7\) , \( 8 a + 29\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-2a-7\right){x}+8a+29$ |
*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.
