| 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 |
| 2.1-a1 |
2.1-a |
$1$ |
$1$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
2.1 |
\( 2 \) |
\( 2^{11} \) |
$11.39585$ |
$(a^3+a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$174.6561251$ |
1.493496725 |
\( \frac{1103617}{64} a^{3} - \frac{282057}{32} a^{2} - \frac{1908787}{32} a - \frac{480401}{64} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} + a - 3\) , \( a^{2} - 3\) , \( 2 a^{3} + 3 a^{2} - 6 a - 5\) , \( 2 a^{3} + 3 a^{2} - 6 a - 5\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(2a^{3}+3a^{2}-6a-5\right){x}+2a^{3}+3a^{2}-6a-5$ |
| 5.1-a1 |
5.1-a |
$2$ |
$2$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$12.77878$ |
$(a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$190.8058436$ |
0.815797048 |
\( \frac{88428292}{15625} a^{3} + \frac{280112876}{15625} a^{2} - \frac{2264399873}{15625} a + \frac{2387866552}{15625} \) |
\( \bigl[a^{3} - 4 a + 2\) , \( -a^{2} + a + 4\) , \( a^{2} + a - 2\) , \( 6 a^{3} - 33 a + 8\) , \( -11 a^{3} - 6 a^{2} + 59 a + 12\bigr] \) |
${y}^2+\left(a^{3}-4a+2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(6a^{3}-33a+8\right){x}-11a^{3}-6a^{2}+59a+12$ |
| 5.1-a2 |
5.1-a |
$2$ |
$2$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$12.77878$ |
$(a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$381.6116872$ |
0.815797048 |
\( \frac{6026746876}{125} a^{3} + \frac{7201451003}{125} a^{2} - \frac{20405861244}{125} a - \frac{2737943594}{125} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} + a^{2} + 4 a - 4\) , \( a + 1\) , \( -6 a^{3} - 4 a^{2} + 19 a - 1\) , \( -12 a^{3} - 13 a^{2} + 38 a + 6\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-4\right){x}^{2}+\left(-6a^{3}-4a^{2}+19a-1\right){x}-12a^{3}-13a^{2}+38a+6$ |
| 5.1-b1 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.282963297$ |
$1940.651542$ |
1.760878505 |
\( \frac{15108569973068}{15625} a^{3} + \frac{17881399284329}{15625} a^{2} - \frac{51603394402592}{15625} a - \frac{6913408451892}{15625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 6 a - 5\) , \( a^{2} - 2\) , \( 109 a^{3} + 45 a^{2} - 590 a - 71\) , \( -689 a^{3} - 295 a^{2} + 3711 a + 477\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-5\right){x}^{2}+\left(109a^{3}+45a^{2}-590a-71\right){x}-689a^{3}-295a^{2}+3711a+477$ |
| 5.1-b2 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$2.263706383$ |
$30.32268035$ |
1.760878505 |
\( -\frac{400746610369089}{125} a^{3} + \frac{1396299571446858}{125} a^{2} - \frac{1064271251542409}{125} a - \frac{161315221341009}{125} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{2} - 3\) , \( a^{2} + a - 2\) , \( 48 a^{3} + 34 a^{2} - 224 a - 27\) , \( 170 a^{3} + 92 a^{2} - 867 a - 115\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(48a^{3}+34a^{2}-224a-27\right){x}+170a^{3}+92a^{2}-867a-115$ |
| 5.1-b3 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{6} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$1.131853191$ |
$121.2907214$ |
1.760878505 |
\( -\frac{455848566818}{15625} a^{3} - \frac{179763284329}{15625} a^{2} + \frac{2420671996342}{15625} a + \frac{317655014392}{15625} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a\) , \( -28 a^{3} + 43 a^{2} + 178 a - 259\) , \( -202 a^{3} + 242 a^{2} + 1199 a - 1619\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-4a-1\right){x}^{2}+\left(-28a^{3}+43a^{2}+178a-259\right){x}-202a^{3}+242a^{2}+1199a-1619$ |
| 5.1-b4 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.141481648$ |
$1940.651542$ |
1.760878505 |
\( \frac{2020821614698}{125} a^{3} - \frac{2282320776656}{125} a^{2} - \frac{11831018728437}{125} a + \frac{15676279119138}{125} \) |
\( \bigl[a\) , \( a^{3} - 5 a + 2\) , \( 0\) , \( -15 a^{3} - 15 a^{2} + 54 a - 3\) , \( 47 a^{3} + 54 a^{2} - 162 a - 15\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-5a+2\right){x}^{2}+\left(-15a^{3}-15a^{2}+54a-3\right){x}+47a^{3}+54a^{2}-162a-15$ |
| 5.1-b5 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.565926595$ |
$485.1628857$ |
1.760878505 |
\( \frac{951264459084193785302}{125} a^{3} + \frac{1125943894828802059656}{125} a^{2} - \frac{3248943375132704727563}{125} a - \frac{435634715898896362138}{125} \) |
\( \bigl[a\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} - 5 a + 2\) , \( -61 a^{3} - 16 a^{2} + 330 a - 16\) , \( -517 a^{3} - 336 a^{2} + 2754 a + 989\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(-61a^{3}-16a^{2}+330a-16\right){x}-517a^{3}-336a^{2}+2754a+989$ |
| 5.1-b6 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 3 \) |
$2.263706383$ |
$7.580670089$ |
1.760878505 |
\( -\frac{481125092163573911}{125} a^{3} - \frac{206676631511074858}{125} a^{2} + \frac{2591293738450970409}{125} a + \frac{336559202058340009}{125} \) |
\( \bigl[1\) , \( -a^{3} + 5 a - 1\) , \( a^{2} - 2\) , \( -212 a^{3} - 275 a^{2} + 756 a + 102\) , \( -3442 a^{3} - 4269 a^{2} + 12008 a + 1611\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+5a-1\right){x}^{2}+\left(-212a^{3}-275a^{2}+756a+102\right){x}-3442a^{3}-4269a^{2}+12008a+1611$ |
| 5.1-b7 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{12} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.565926595$ |
$485.1628857$ |
1.760878505 |
\( \frac{2056552059833}{244140625} a^{3} + \frac{2922785273374}{244140625} a^{2} - \frac{6005493015127}{244140625} a - \frac{394249422177}{244140625} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( a\) , \( a^{3} - 5 a + 3\) , \( -a^{2} + 6 a - 7\) , \( -a^{2} + 5 a - 6\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-5a+3\right){y}={x}^{3}+a{x}^{2}+\left(-a^{2}+6a-7\right){x}-a^{2}+5a-6$ |
| 5.1-b8 |
5.1-b |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{24} \) |
$12.77878$ |
$(a^3-5a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.131853191$ |
$30.32268035$ |
1.760878505 |
\( \frac{9887791437762799088}{59604644775390625} a^{3} - \frac{11236346643322151361}{59604644775390625} a^{2} - \frac{65791710660436623972}{59604644775390625} a + \frac{94119463815817212228}{59604644775390625} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( a\) , \( a^{3} - 5 a + 3\) , \( 4 a^{2} - 14 a + 13\) , \( 11 a^{3} - 32 a^{2} + 8 a + 21\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-5a+3\right){y}={x}^{3}+a{x}^{2}+\left(4a^{2}-14a+13\right){x}+11a^{3}-32a^{2}+8a+21$ |
| 8.1-a1 |
8.1-a |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$131.2391395$ |
2.244470098 |
\( -\frac{586495}{16} a^{3} + \frac{2380535}{16} a^{2} - \frac{2698031}{16} a + \frac{80895}{2} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} - a^{2} - 5 a + 5\) , \( a^{2} - 3\) , \( a^{3} + 3 a^{2} - 3 a - 6\) , \( -4 a^{3} + 8 a^{2} + 28 a - 45\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+5\right){x}^{2}+\left(a^{3}+3a^{2}-3a-6\right){x}-4a^{3}+8a^{2}+28a-45$ |
| 8.1-a2 |
8.1-a |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{17} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$524.9565582$ |
2.244470098 |
\( -\frac{225}{32} a^{3} + \frac{63215}{256} a^{2} + \frac{6611}{16} a - \frac{154953}{256} \) |
\( \bigl[a\) , \( a^{3} - 6 a + 3\) , \( 0\) , \( -2 a^{3} - a^{2} + 10 a + 6\) , \( 7 a^{3} + 3 a^{2} - 38 a - 3\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-6a+3\right){x}^{2}+\left(-2a^{3}-a^{2}+10a+6\right){x}+7a^{3}+3a^{2}-38a-3$ |
| 8.1-a3 |
8.1-a |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{4} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$32.80978488$ |
2.244470098 |
\( -\frac{154976771899405}{2} a^{3} - \frac{66572027166713}{2} a^{2} + 417346253094964 a + \frac{108408471029123}{2} \) |
\( \bigl[1\) , \( a^{3} + a^{2} - 6 a - 2\) , \( a^{2} - 3\) , \( -157 a^{3} + 540 a^{2} - 414 a - 43\) , \( 2672 a^{3} - 9307 a^{2} + 7084 a + 1081\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}+a^{2}-6a-2\right){x}^{2}+\left(-157a^{3}+540a^{2}-414a-43\right){x}+2672a^{3}-9307a^{2}+7084a+1081$ |
| 8.1-a4 |
8.1-a |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{5} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2099.826232$ |
2.244470098 |
\( \frac{583864811501}{2} a^{3} - \frac{1317654757391}{4} a^{2} - 1709684162508 a + \frac{9057690979355}{4} \) |
\( \bigl[a^{3} - 4 a + 2\) , \( a^{3} + a^{2} - 6 a - 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( -242 a^{3} + 267 a^{2} + 1425 a - 1870\) , \( 4097 a^{3} - 4622 a^{2} - 24000 a + 31782\bigr] \) |
${y}^2+\left(a^{3}-4a+2\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-6a-1\right){x}^{2}+\left(-242a^{3}+267a^{2}+1425a-1870\right){x}+4097a^{3}-4622a^{2}-24000a+31782$ |
| 8.1-a5 |
8.1-a |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$2099.826232$ |
2.244470098 |
\( \frac{2934593}{16} a^{3} - 205173 a^{2} - \frac{17353647}{16} a + \frac{2878209}{2} \) |
\( \bigl[a^{3} - 4 a + 2\) , \( a^{3} + a^{2} - 6 a - 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( -17 a^{3} + 17 a^{2} + 95 a - 115\) , \( 59 a^{3} - 66 a^{2} - 348 a + 456\bigr] \) |
${y}^2+\left(a^{3}-4a+2\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-6a-1\right){x}^{2}+\left(-17a^{3}+17a^{2}+95a-115\right){x}+59a^{3}-66a^{2}-348a+456$ |
| 8.1-a6 |
8.1-a |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{8} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$524.9565582$ |
2.244470098 |
\( -\frac{16383457}{4} a^{3} - 1753154 a^{2} + \frac{88305335}{4} a + \frac{11477151}{4} \) |
\( \bigl[a^{3} - 4 a + 2\) , \( a^{3} - 4 a + 1\) , \( 0\) , \( 335 a^{3} + 144 a^{2} - 1803 a - 231\) , \( 4856 a^{3} + 2085 a^{2} - 26156 a - 3396\bigr] \) |
${y}^2+\left(a^{3}-4a+2\right){x}{y}={x}^{3}+\left(a^{3}-4a+1\right){x}^{2}+\left(335a^{3}+144a^{2}-1803a-231\right){x}+4856a^{3}+2085a^{2}-26156a-3396$ |
| 8.1-b1 |
8.1-b |
$2$ |
$2$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{10} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.031765078$ |
$1173.639725$ |
1.912742172 |
\( \frac{3326905}{4} a^{3} + \frac{7882387}{8} a^{2} - \frac{11355779}{4} a - \frac{3045427}{8} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 6 a - 4\) , \( a\) , \( -4 a^{3} - 4 a^{2} + 11 a + 9\) , \( a^{3} + 7 a^{2} - 7 a - 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+6a-4\right){x}^{2}+\left(-4a^{3}-4a^{2}+11a+9\right){x}+a^{3}+7a^{2}-7a-5$ |
| 8.1-b2 |
8.1-b |
$2$ |
$2$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{14} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.015882539$ |
$1173.639725$ |
1.912742172 |
\( \frac{206153}{64} a^{3} - \frac{33925}{8} a^{2} - \frac{1280511}{64} a + \frac{113759}{4} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{3} - 4 a + 3\) , \( -3 a^{3} + 13 a - 5\) , \( -a^{3} + 4 a - 2\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-4a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-4\right){x}^{2}+\left(-3a^{3}+13a-5\right){x}-a^{3}+4a-2$ |
| 8.1-c1 |
8.1-c |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{15} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$698.5747421$ |
2.986780707 |
\( \frac{1901}{8} a^{3} + \frac{63855}{64} a^{2} + \frac{3803}{4} a - \frac{2009}{64} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 5 a + 2\) , \( a^{3} + a^{2} - 5 a\) , \( -27 a^{3} + 27 a^{2} + 155 a - 192\) , \( -543 a^{3} + 613 a^{2} + 3171 a - 4198\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-5a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+2\right){x}^{2}+\left(-27a^{3}+27a^{2}+155a-192\right){x}-543a^{3}+613a^{2}+3171a-4198$ |
| 8.1-c2 |
8.1-c |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{20} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$8.624379532$ |
2.986780707 |
\( \frac{3264488656593704935}{512} a^{3} - \frac{3685445265889772599}{512} a^{2} - \frac{19111692872743975497}{512} a + \frac{25315878058321694857}{512} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( a^{3} + a^{2} - 5 a - 1\) , \( 4 a^{3} + 8 a^{2} - 20 a - 45\) , \( 25 a^{3} + 43 a^{2} - 127 a - 202\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-5a-1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(4a^{3}+8a^{2}-20a-45\right){x}+25a^{3}+43a^{2}-127a-202$ |
| 8.1-c3 |
8.1-c |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{12} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$698.5747421$ |
2.986780707 |
\( \frac{354133}{8} a^{3} - \frac{396215}{8} a^{2} - 261398 a + \frac{2769789}{8} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( a^{3} + a^{2} - 5 a - 1\) , \( -a^{3} - 2 a^{2} + 5 a + 5\) , \( -a^{3} - a^{2} + 5 a + 2\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-5a-1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{3}-2a^{2}+5a+5\right){x}-a^{3}-a^{2}+5a+2$ |
| 8.1-c4 |
8.1-c |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{37} \) |
$13.55203$ |
$(a^3+a^2-4a), (a^3+a^2-3a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$8.624379532$ |
2.986780707 |
\( -\frac{7299219398521}{262144} a^{3} + \frac{8073100974761}{262144} a^{2} + \frac{42709589308247}{262144} a - \frac{55629569388695}{262144} \) |
\( \bigl[a^{3} - 4 a + 2\) , \( 0\) , \( a^{2} + a - 2\) , \( 53 a^{3} - 204 a^{2} + 169 a + 21\) , \( 4248 a^{3} - 14852 a^{2} + 11356 a + 1707\bigr] \) |
${y}^2+\left(a^{3}-4a+2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(53a^{3}-204a^{2}+169a+21\right){x}+4248a^{3}-14852a^{2}+11356a+1707$ |
| 10.1-a1 |
10.1-a |
$2$ |
$3$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{6} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.380585844$ |
$530.7670600$ |
4.606231698 |
\( -\frac{18188281}{62500} a^{3} - \frac{48745359}{31250} a^{2} - \frac{30235093}{31250} a + \frac{27277289}{62500} \) |
\( \bigl[a^{3} + a^{2} - 5 a\) , \( -a^{3} - a^{2} + 5 a + 1\) , \( a^{3} + a^{2} - 4 a\) , \( 4 a^{3} - 25 a - 2\) , \( -9 a^{3} - 5 a^{2} + 46 a + 7\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a\right){x}{y}+\left(a^{3}+a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+1\right){x}^{2}+\left(4a^{3}-25a-2\right){x}-9a^{3}-5a^{2}+46a+7$ |
| 10.1-a2 |
10.1-a |
$2$ |
$3$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{18} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$1.141757533$ |
$6.552679754$ |
4.606231698 |
\( \frac{4815023773732035286193}{7629394531250} a^{3} - \frac{8388364426256937388148}{3814697265625} a^{2} + \frac{6393688314570263378704}{3814697265625} a + \frac{1938220698820990286233}{7629394531250} \) |
\( \bigl[a^{3} + a^{2} - 5 a\) , \( -a^{3} - a^{2} + 5 a + 1\) , \( a^{3} + a^{2} - 4 a\) , \( -51 a^{3} - 20 a^{2} + 275 a + 23\) , \( 17 a^{3} + 13 a^{2} - 91 a - 46\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a\right){x}{y}+\left(a^{3}+a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+1\right){x}^{2}+\left(-51a^{3}-20a^{2}+275a+23\right){x}+17a^{3}+13a^{2}-91a-46$ |
| 10.1-b1 |
10.1-b |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{6} \cdot 5^{6} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$10.17886170$ |
0.783361416 |
\( \frac{6120829065239101}{125000} a^{3} - \frac{6910073960898697}{125000} a^{2} - \frac{35833984066955519}{125000} a + \frac{11866642329683439}{31250} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} + a^{2} + 4 a - 6\) , \( a^{2} + a - 3\) , \( -182 a^{3} + 191 a^{2} + 1110 a - 1445\) , \( -2642 a^{3} + 2877 a^{2} + 15844 a - 20811\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-6\right){x}^{2}+\left(-182a^{3}+191a^{2}+1110a-1445\right){x}-2642a^{3}+2877a^{2}+15844a-20811$ |
| 10.1-b2 |
10.1-b |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{2} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$824.4877978$ |
0.783361416 |
\( \frac{49919}{50} a^{3} - \frac{30593}{50} a^{2} - \frac{306661}{50} a + \frac{186382}{25} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} + a^{2} + 4 a - 6\) , \( a^{2} + a - 3\) , \( -2 a^{3} + a^{2} + 10 a - 10\) , \( -2 a^{3} + a^{2} + 11 a - 12\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-6\right){x}^{2}+\left(-2a^{3}+a^{2}+10a-10\right){x}-2a^{3}+a^{2}+11a-12$ |
| 10.1-b3 |
10.1-b |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$20.35772340$ |
0.783361416 |
\( -\frac{277419899158931}{8000} a^{3} - \frac{119148593356293}{8000} a^{2} + \frac{1494216556758789}{8000} a + \frac{12129445669329}{500} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} + a^{2} + 6 a - 4\) , \( a^{2} + a - 3\) , \( 719 a^{3} + 298 a^{2} - 3899 a - 500\) , \( 17289 a^{3} + 7369 a^{2} - 93269 a - 12115\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-4\right){x}^{2}+\left(719a^{3}+298a^{2}-3899a-500\right){x}+17289a^{3}+7369a^{2}-93269a-12115$ |
| 10.1-b4 |
10.1-b |
$4$ |
$6$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1648.975595$ |
0.783361416 |
\( -\frac{4259}{20} a^{3} - \frac{25417}{20} a^{2} + \frac{20901}{20} a + \frac{39649}{5} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} + a^{2} + 6 a - 4\) , \( a^{2} + a - 3\) , \( 9 a^{3} + 3 a^{2} - 49 a\) , \( 32 a^{3} + 14 a^{2} - 173 a - 25\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-4\right){x}^{2}+\left(9a^{3}+3a^{2}-49a\right){x}+32a^{3}+14a^{2}-173a-25$ |
| 10.1-c1 |
10.1-c |
$2$ |
$2$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{32} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.049353810$ |
$227.0607591$ |
3.066431084 |
\( \frac{3384529939}{327680} a^{3} + \frac{1942801277}{327680} a^{2} - \frac{12455145341}{327680} a + \frac{581806253}{40960} \) |
\( \bigl[a\) , \( a^{3} - a^{2} - 4 a + 5\) , \( a^{2} + a - 3\) , \( -a^{3} - 8 a^{2} + 7 a + 9\) , \( -14 a^{3} + 9 a^{2} + 15 a + 1\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+5\right){x}^{2}+\left(-a^{3}-8a^{2}+7a+9\right){x}-14a^{3}+9a^{2}+15a+1$ |
| 10.1-c2 |
10.1-c |
$2$ |
$2$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5^{2} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.098707621$ |
$113.5303795$ |
3.066431084 |
\( \frac{56854439373}{1600} a^{3} - \frac{189412415049}{6400} a^{2} - \frac{1574690865423}{6400} a + \frac{1973817153827}{6400} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{3} - a^{2} + 6 a\) , \( a^{3} - 5 a + 2\) , \( -11 a^{3} + 7 a^{2} + 60 a - 52\) , \( -28 a^{3} + 26 a^{2} + 162 a - 194\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+6a\right){x}^{2}+\left(-11a^{3}+7a^{2}+60a-52\right){x}-28a^{3}+26a^{2}+162a-194$ |
| 10.1-d1 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$256$ |
\( 2^{2} \) |
$1$ |
$0.651828984$ |
1.426901804 |
\( \frac{215727208600480024273929}{20} a^{3} + \frac{255340910881155050234847}{20} a^{2} - \frac{736793515750653515750391}{20} a - \frac{24698247772403392435269}{5} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 6\) , \( a^{3} - 4 a + 3\) , \( -1058 a^{3} + 867 a^{2} + 6709 a - 8273\) , \( -36079 a^{3} + 34756 a^{2} + 227338 a - 290618\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-6\right){x}^{2}+\left(-1058a^{3}+867a^{2}+6709a-8273\right){x}-36079a^{3}+34756a^{2}+227338a-290618$ |
| 10.1-d2 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{4} \) |
$1$ |
$10.42926374$ |
1.426901804 |
\( -\frac{92268402809240943}{400} a^{3} + \frac{321553734539671071}{400} a^{2} - \frac{245127549262237983}{400} a - \frac{4644243560817351}{50} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 6\) , \( a^{3} - 4 a + 3\) , \( 7 a^{3} - 203 a^{2} + 614 a - 483\) , \( 1556 a^{3} - 6577 a^{2} + 8225 a - 2863\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-6\right){x}^{2}+\left(7a^{3}-203a^{2}+614a-483\right){x}+1556a^{3}-6577a^{2}+8225a-2863$ |
| 10.1-d3 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{16} \cdot 5^{4} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$166.8682199$ |
1.426901804 |
\( -\frac{359758454967}{160000} a^{3} + \frac{1303428694599}{160000} a^{2} - \frac{1020608793927}{160000} a - \frac{19173030669}{20000} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 6\) , \( a^{3} - 4 a + 3\) , \( -8 a^{3} - 3 a^{2} + 79 a - 83\) , \( 103 a^{3} - 220 a^{2} - 236 a + 486\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-6\right){x}^{2}+\left(-8a^{3}-3a^{2}+79a-83\right){x}+103a^{3}-220a^{2}-236a+486$ |
| 10.1-d4 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{32} \cdot 5^{2} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$41.71705498$ |
1.426901804 |
\( -\frac{135260847}{1638400} a^{3} - \frac{1659443841}{1638400} a^{2} + \frac{3418001793}{1638400} a + \frac{186442371}{204800} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 6\) , \( a^{3} - 4 a + 3\) , \( -34 a^{3} - 15 a^{2} + 172 a + 36\) , \( -40 a^{3} - 15 a^{2} + 203 a + 17\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-6\right){x}^{2}+\left(-34a^{3}-15a^{2}+172a+36\right){x}-40a^{3}-15a^{2}+203a+17$ |
| 10.1-d5 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$256$ |
\( 2^{2} \) |
$1$ |
$0.651828984$ |
1.426901804 |
\( -\frac{45805496695244070114990447682569}{20} a^{3} + \frac{159597595321072009052081650129953}{20} a^{2} - \frac{121646626371171997610631332520969}{20} a - \frac{4609598453749295997174707166891}{5} \) |
\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{3} + a^{2} - 4 a\) , \( -3362 a^{3} - 4000 a^{2} + 11495 a + 1565\) , \( -214234 a^{3} - 253576 a^{2} + 731826 a + 97924\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}+a^{2}-4a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-3362a^{3}-4000a^{2}+11495a+1565\right){x}-214234a^{3}-253576a^{2}+731826a+97924$ |
| 10.1-d6 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{4} \cdot 5^{16} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$166.8682199$ |
1.426901804 |
\( -\frac{21112645044433037942412}{152587890625} a^{3} - \frac{36277177974933978612819}{610351562500} a^{2} + \frac{454842832647868475754987}{610351562500} a + \frac{59074236159578318793537}{610351562500} \) |
\( \bigl[1\) , \( -1\) , \( a^{3} - 4 a + 2\) , \( 79 a^{3} + 26 a^{2} - 449 a - 60\) , \( -695 a^{3} - 266 a^{2} + 3832 a + 496\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a+2\right){y}={x}^{3}-{x}^{2}+\left(79a^{3}+26a^{2}-449a-60\right){x}-695a^{3}-266a^{2}+3832a+496$ |
| 10.1-d7 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{8} \cdot 5^{8} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$667.4728797$ |
1.426901804 |
\( \frac{7297054829703}{6250000} a^{3} - \frac{9922176446391}{6250000} a^{2} - \frac{43238314179657}{6250000} a + \frac{8220194049771}{781250} \) |
\( \bigl[1\) , \( -1\) , \( a^{3} - 4 a + 2\) , \( 4 a^{3} + a^{2} - 24 a - 5\) , \( -13 a^{3} - 6 a^{2} + 70 a + 8\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a+2\right){y}={x}^{3}-{x}^{2}+\left(4a^{3}+a^{2}-24a-5\right){x}-13a^{3}-6a^{2}+70a+8$ |
| 10.1-d8 |
10.1-d |
$8$ |
$16$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$667.4728797$ |
1.426901804 |
\( \frac{9541989737572932}{625} a^{3} - \frac{43089726573490941}{2500} a^{2} - \frac{223451322940932507}{2500} a + \frac{295989815298440943}{2500} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( a^{3} + a^{2} - 5 a\) , \( a^{3} - 4 a + 2\) , \( 803 a^{3} + 358 a^{2} - 4292 a - 557\) , \( -45114 a^{3} - 19366 a^{2} + 243016 a + 31563\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-4a+2\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a\right){x}^{2}+\left(803a^{3}+358a^{2}-4292a-557\right){x}-45114a^{3}-19366a^{2}+243016a+31563$ |
| 10.1-e1 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$4.836060354$ |
$5.501685242$ |
2.047624965 |
\( -\frac{383183984551933565932777}{160} a^{3} - \frac{41150804073651545221179}{40} a^{2} + \frac{515948142980802738933597}{40} a + \frac{268041980152490368175163}{160} \) |
\( \bigl[a\) , \( a^{3} - 5 a + 1\) , \( 0\) , \( 2835 a^{3} + 1215 a^{2} - 15278 a - 1988\) , \( 122390 a^{3} + 52567 a^{2} - 659208 a - 85633\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(2835a^{3}+1215a^{2}-15278a-1988\right){x}+122390a^{3}+52567a^{2}-659208a-85633$ |
| 10.1-e2 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{3} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1.612020118$ |
$445.6365046$ |
2.047624965 |
\( -\frac{2472663911}{500} a^{3} - \frac{425560679}{250} a^{2} + \frac{6442022517}{250} a + \frac{1670320759}{500} \) |
\( \bigl[a\) , \( a^{3} - 5 a + 1\) , \( 0\) , \( 35 a^{3} + 15 a^{2} - 188 a - 23\) , \( 140 a^{3} + 60 a^{2} - 754 a - 97\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(35a^{3}+15a^{2}-188a-23\right){x}+140a^{3}+60a^{2}-754a-97$ |
| 10.1-e3 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{9} \cdot 5^{4} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$4.836060354$ |
$1.375421310$ |
2.047624965 |
\( \frac{441100554178271852242181}{20000} a^{3} + \frac{130524830262924055307167}{5000} a^{2} - \frac{376633116461739503270041}{5000} a - \frac{202003472744529661642639}{20000} \) |
\( \bigl[a^{3} - 4 a + 2\) , \( a^{2} + a - 2\) , \( a^{3} - 5 a + 3\) , \( -2377 a^{3} + 2635 a^{2} + 13892 a - 18304\) , \( -12046 a^{3} + 13186 a^{2} + 70255 a - 92195\bigr] \) |
${y}^2+\left(a^{3}-4a+2\right){x}{y}+\left(a^{3}-5a+3\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(-2377a^{3}+2635a^{2}+13892a-18304\right){x}-12046a^{3}+13186a^{2}+70255a-92195$ |
| 10.1-e4 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{36} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1.209015088$ |
$11.00337048$ |
2.047624965 |
\( \frac{24190930162297099}{1310720} a^{3} - \frac{27310457457069923}{1310720} a^{2} - \frac{141623901022175581}{1310720} a + \frac{45800677733559}{320} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( a^{3} - a^{2} - 4 a + 4\) , \( a^{3} + a^{2} - 5 a\) , \( 303 a^{3} + 219 a^{2} - 1609 a - 689\) , \( 4910 a^{3} + 3127 a^{2} - 26137 a - 8964\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}+a^{2}-5a\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+4\right){x}^{2}+\left(303a^{3}+219a^{2}-1609a-689\right){x}+4910a^{3}+3127a^{2}-26137a-8964$ |
| 10.1-e5 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.403005029$ |
$891.2730092$ |
2.047624965 |
\( \frac{10724451}{8000} a^{3} + \frac{15908853}{8000} a^{2} - \frac{10487669}{8000} a + \frac{2986491}{500} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( a^{3} - a^{2} - 4 a + 4\) , \( a^{3} + a^{2} - 5 a\) , \( 23 a^{3} + 9 a^{2} - 129 a - 4\) , \( -83 a^{3} - 37 a^{2} + 450 a + 54\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}+a^{2}-5a\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+4\right){x}^{2}+\left(23a^{3}+9a^{2}-129a-4\right){x}-83a^{3}-37a^{2}+450a+54$ |
| 10.1-e6 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{6} \cdot 5^{6} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.806010059$ |
$891.2730092$ |
2.047624965 |
\( \frac{5795902}{15625} a^{3} - \frac{151119977}{125000} a^{2} - \frac{295473079}{125000} a + \frac{1283739071}{125000} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{2} + 4\) , \( a^{3} - 5 a + 2\) , \( a^{3} - 3 a^{2} - 6 a + 12\) , \( a^{3} - a^{2} - 9 a + 11\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(a^{3}-3a^{2}-6a+12\right){x}+a^{3}-a^{2}-9a+11$ |
| 10.1-e7 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{3} \cdot 5^{12} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.612020118$ |
$111.4091261$ |
2.047624965 |
\( \frac{11700883577313347}{976562500} a^{3} - \frac{5821454178079417}{488281250} a^{2} - \frac{33428417874273209}{488281250} a + \frac{85186175533820757}{976562500} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{2} + 4\) , \( a^{3} - 5 a + 2\) , \( -4 a^{3} - 13 a^{2} + 19 a + 12\) , \( -14 a^{3} - 31 a^{2} + 62 a + 15\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-4a^{3}-13a^{2}+19a+12\right){x}-14a^{3}-31a^{2}+62a+15$ |
| 10.1-e8 |
10.1-e |
$8$ |
$12$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{18} \cdot 5^{2} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$2.418030177$ |
$11.00337048$ |
2.047624965 |
\( -\frac{33875812442677}{1600} a^{3} - \frac{100522898247823}{12800} a^{2} + \frac{1501120478033479}{12800} a + \frac{194665345808529}{12800} \) |
\( \bigl[a^{3} - 5 a + 3\) , \( -a^{2} + 4\) , \( a^{3} - 5 a + 2\) , \( -19 a^{3} - 58 a^{2} + 94 a + 22\) , \( -143 a^{3} - 350 a^{2} + 695 a + 95\bigr] \) |
${y}^2+\left(a^{3}-5a+3\right){x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-19a^{3}-58a^{2}+94a+22\right){x}-143a^{3}-350a^{2}+695a+95$ |
| 10.1-f1 |
10.1-f |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{8} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1039.325564$ |
2.221836373 |
\( -\frac{187482478139}{781250} a^{3} + \frac{593100477083}{781250} a^{2} - \frac{420386066459}{781250} a - \frac{31290173467}{390625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 4 a - 2\) , \( a^{3} - 5 a + 3\) , \( -4 a^{3} + 4 a^{2} + 21 a - 26\) , \( 6 a^{3} - 7 a^{2} - 36 a + 48\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-5a+3\right){y}={x}^{3}+\left(-a^{3}+4a-2\right){x}^{2}+\left(-4a^{3}+4a^{2}+21a-26\right){x}+6a^{3}-7a^{2}-36a+48$ |
| 10.1-f2 |
10.1-f |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{2} \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$259.8313910$ |
2.221836373 |
\( \frac{2016692964562557336251}{50} a^{3} + \frac{2387015629041994958053}{50} a^{2} - \frac{6887802003231142698469}{50} a - \frac{461775617853395749497}{25} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a - 1\) , \( a^{3} - 5 a + 2\) , \( -159 a^{3} + 550 a^{2} - 419 a - 62\) , \( -6046 a^{3} + 21059 a^{2} - 16030 a - 2459\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-159a^{3}+550a^{2}-419a-62\right){x}-6046a^{3}+21059a^{2}-16030a-2459$ |
| 10.1-f3 |
10.1-f |
$6$ |
$8$ |
4.4.13676.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$13.93536$ |
$(a^3+a^2-4a), (a^3-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$16.23946193$ |
2.221836373 |
\( \frac{11415742474788537239}{20} a^{3} - \frac{12887805261917715743}{20} a^{2} - \frac{66832569216947980921}{20} a + \frac{22132068370881908391}{5} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} - 5 a + 1\) , \( a^{3} + a^{2} - 5 a\) , \( 166 a^{3} + 72 a^{2} - 900 a - 116\) , \( 1653 a^{3} + 686 a^{2} - 8872 a - 1152\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-5a\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(166a^{3}+72a^{2}-900a-116\right){x}+1653a^{3}+686a^{2}-8872a-1152$ |
*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.
