| 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 |
| 4.1-a1 |
4.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$4.34877$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$349.7445014$ |
2.263635480 |
\( 1076 a^{2} - 4192 a - 2724 \) |
\( \bigl[a^{2} - a - 6\) , \( -a - 1\) , \( 0\) , \( -2 a^{2} + 3 a + 14\) , \( -a^{2} + 2 a + 9\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a^{2}+3a+14\right){x}-a^{2}+2a+9$ |
| 4.1-a2 |
4.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$4.34877$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$349.7445014$ |
2.263635480 |
\( 12632070 a^{2} - 35848202 a - 27409712 \) |
\( \bigl[a^{2} - a - 6\) , \( -a - 1\) , \( 0\) , \( -7 a^{2} + 8 a + 19\) , \( 10 a^{2} - 2 a + 2\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7a^{2}+8a+19\right){x}+10a^{2}-2a+2$ |
| 4.1-b1 |
4.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$4.34877$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$96.10892833$ |
0.622041459 |
\( 12632070 a^{2} - 35848202 a - 27409712 \) |
\( \bigl[a^{2} - a - 6\) , \( -a^{2} + 2 a + 5\) , \( a^{2} - a - 6\) , \( 10 a^{2} - 8 a - 117\) , \( 156 a^{2} - 296 a - 1090\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(-a^{2}+2a+5\right){x}^{2}+\left(10a^{2}-8a-117\right){x}+156a^{2}-296a-1090$ |
| 4.1-b2 |
4.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$4.34877$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$96.10892833$ |
0.622041459 |
\( 1076 a^{2} - 4192 a - 2724 \) |
\( \bigl[a^{2} - a - 6\) , \( -a^{2} + 2 a + 5\) , \( a^{2} - a - 6\) , \( -5 a^{2} + 7 a + 38\) , \( -4 a^{2} + 5 a + 34\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(-a^{2}+2a+5\right){x}^{2}+\left(-5a^{2}+7a+38\right){x}-4a^{2}+5a+34$ |
| 10.1-a1 |
10.1-a |
$1$ |
$1$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{3} \) |
$5.06629$ |
$(-a-1), (-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$1$ |
$40.56854489$ |
3.150839446 |
\( \frac{265668}{125} a^{2} - \frac{331}{250} a - \frac{1168752}{125} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 7\) , \( a\) , \( 2 a + 5\) , \( a^{2} + a - 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(2a+5\right){x}+a^{2}+a-5$ |
| 10.1-b1 |
10.1-b |
$1$ |
$1$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{3} \) |
$5.06629$ |
$(-a-1), (-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3^{2} \) |
$0.034466930$ |
$185.6901871$ |
4.473741775 |
\( \frac{265668}{125} a^{2} - \frac{331}{250} a - \frac{1168752}{125} \) |
\( \bigl[a^{2} - a - 5\) , \( -a^{2} + 3 a + 7\) , \( a\) , \( -23 a^{2} - 51 a + 1\) , \( 101 a^{2} + 296 a + 170\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-23a^{2}-51a+1\right){x}+101a^{2}+296a+170$ |
| 14.1-a1 |
14.1-a |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{3} \cdot 7^{6} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$8.639864589$ |
1.342065693 |
\( \frac{3997411383388001288125}{235298} a^{2} - \frac{12498115170624193076125}{235298} a - \frac{9398808715124539071625}{235298} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( -a - 1\) , \( a^{2} - a - 5\) , \( 8134 a^{2} - 13229 a - 64917\) , \( -686217 a^{2} + 1116121 a + 5476714\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(8134a^{2}-13229a-64917\right){x}-686217a^{2}+1116121a+5476714$ |
| 14.1-a2 |
14.1-a |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$34.55945835$ |
1.342065693 |
\( -\frac{181461689375}{1372} a^{2} + \frac{192740013125}{686} a + \frac{277317708375}{343} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( -a - 1\) , \( a^{2} - a - 5\) , \( 8054 a^{2} - 13099 a - 64277\) , \( -701267 a^{2} + 1140599 a + 5596830\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(8054a^{2}-13099a-64277\right){x}-701267a^{2}+1140599a+5596830$ |
| 14.1-a3 |
14.1-a |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2 \cdot 7^{2} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$25.91959376$ |
1.342065693 |
\( \frac{23669636875}{98} a^{2} + \frac{32568443125}{49} a + \frac{31529708625}{98} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( -a - 1\) , \( a^{2} - a - 5\) , \( 1214 a^{2} - 1974 a - 9687\) , \( 43178 a^{2} - 70230 a - 344608\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1214a^{2}-1974a-9687\right){x}+43178a^{2}-70230a-344608$ |
| 14.1-a4 |
14.1-a |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$103.6783750$ |
1.342065693 |
\( \frac{19500}{7} a^{2} + \frac{126625}{14} a + \frac{89875}{14} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( -a - 1\) , \( a^{2} - a - 5\) , \( 129 a^{2} - 209 a - 1027\) , \( -177 a^{2} + 286 a + 1409\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(129a^{2}-209a-1027\right){x}-177a^{2}+286a+1409$ |
| 14.1-b1 |
14.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1.050675045$ |
$61.09965081$ |
2.492952439 |
\( \frac{452889307}{686} a^{2} + \frac{1284061053}{686} a + \frac{1318968795}{1372} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{2} - a - 5\) , \( -5 a^{2} + 15 a + 13\) , \( 10 a^{2} - 32 a - 28\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a^{2}+15a+13\right){x}+10a^{2}-32a-28$ |
| 14.1-b2 |
14.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{3} \cdot 7^{6} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$2.101350090$ |
$30.54982540$ |
2.492952439 |
\( -\frac{47776174685999}{235298} a^{2} + \frac{15216030349402}{117649} a + \frac{279364928462390}{117649} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{2} - a - 5\) , \( -75 a^{2} + 235 a + 173\) , \( 716 a^{2} - 2238 a - 1692\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-75a^{2}+235a+173\right){x}+716a^{2}-2238a-1692$ |
| 14.1-c1 |
14.1-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{7} \cdot 7^{10} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 5 \) |
$1$ |
$9.317632379$ |
2.412243584 |
\( -\frac{1698047523124255}{2259801992} a^{2} + \frac{765068361399869}{564950498} a + \frac{14314410212379621}{2259801992} \) |
\( \bigl[a\) , \( a^{2} - 3 a - 6\) , \( a\) , \( 2471 a^{2} - 4021 a - 19721\) , \( 118199 a^{2} - 192248 a - 943357\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-6\right){x}^{2}+\left(2471a^{2}-4021a-19721\right){x}+118199a^{2}-192248a-943357$ |
| 14.1-c2 |
14.1-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{14} \cdot 7^{5} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$37.27052951$ |
2.412243584 |
\( -\frac{107586091}{268912} a^{2} + \frac{523687385}{537824} a + \frac{2086704197}{537824} \) |
\( \bigl[a\) , \( a^{2} - 3 a - 6\) , \( a\) , \( 171 a^{2} - 281 a - 1361\) , \( 1263 a^{2} - 2056 a - 10077\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-6\right){x}^{2}+\left(171a^{2}-281a-1361\right){x}+1263a^{2}-2056a-10077$ |
| 14.1-d1 |
14.1-d |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{7} \cdot 7^{10} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$0.093375148$ |
$49.74772395$ |
2.705843205 |
\( -\frac{1698047523124255}{2259801992} a^{2} + \frac{765068361399869}{564950498} a + \frac{14314410212379621}{2259801992} \) |
\( \bigl[a^{2} - a - 5\) , \( a - 1\) , \( a^{2} - a - 5\) , \( -20 a^{2} - 65 a - 50\) , \( 141 a^{2} + 390 a + 191\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-20a^{2}-65a-50\right){x}+141a^{2}+390a+191$ |
| 14.1-d2 |
14.1-d |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{14} \cdot 7^{5} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$0.093375148$ |
$49.74772395$ |
2.705843205 |
\( -\frac{107586091}{268912} a^{2} + \frac{523687385}{537824} a + \frac{2086704197}{537824} \) |
\( \bigl[a^{2} - a - 5\) , \( a - 1\) , \( a^{2} - a - 5\) , \( -5 a - 10\) , \( -3 a^{2} - 10 a - 9\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a-10\right){x}-3a^{2}-10a-9$ |
| 14.1-e1 |
14.1-e |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{3} \cdot 7^{6} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$0.414765604$ |
$138.7504926$ |
2.234828059 |
\( -\frac{47776174685999}{235298} a^{2} + \frac{15216030349402}{117649} a + \frac{279364928462390}{117649} \) |
\( \bigl[a^{2} - a - 5\) , \( -a^{2} + a + 6\) , \( 1\) , \( 4509 a^{2} - 7334 a - 35988\) , \( -285705 a^{2} + 464695 a + 2280220\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(4509a^{2}-7334a-35988\right){x}-285705a^{2}+464695a+2280220$ |
| 14.1-e2 |
14.1-e |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$0.207382802$ |
$277.5009852$ |
2.234828059 |
\( \frac{452889307}{686} a^{2} + \frac{1284061053}{686} a + \frac{1318968795}{1372} \) |
\( \bigl[a^{2} - a - 5\) , \( -a^{2} + a + 6\) , \( 1\) , \( 279 a^{2} - 454 a - 2228\) , \( -3783 a^{2} + 6153 a + 30192\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(279a^{2}-454a-2228\right){x}-3783a^{2}+6153a+30192$ |
| 14.1-f1 |
14.1-f |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7 \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$173.7918027$ |
0.999844271 |
\( \frac{19500}{7} a^{2} + \frac{126625}{14} a + \frac{89875}{14} \) |
\( \bigl[a^{2} - a - 5\) , \( a^{2} - a - 5\) , \( a^{2} - a - 6\) , \( 3 a^{2} + 2 a - 10\) , \( 3 a^{2} + 7 a - 3\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(3a^{2}+2a-10\right){x}+3a^{2}+7a-3$ |
| 14.1-f2 |
14.1-f |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2 \cdot 7^{2} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$173.7918027$ |
0.999844271 |
\( \frac{23669636875}{98} a^{2} + \frac{32568443125}{49} a + \frac{31529708625}{98} \) |
\( \bigl[a^{2} - a - 5\) , \( a^{2} - a - 5\) , \( a^{2} - a - 6\) , \( -2 a^{2} + 17 a\) , \( -15 a^{2} + 60 a + 38\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-2a^{2}+17a\right){x}-15a^{2}+60a+38$ |
| 14.1-f3 |
14.1-f |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$6.436733435$ |
0.999844271 |
\( -\frac{181461689375}{1372} a^{2} + \frac{192740013125}{686} a + \frac{277317708375}{343} \) |
\( \bigl[a^{2} - a - 5\) , \( a^{2} - a - 5\) , \( a^{2} - a - 6\) , \( -22 a^{2} + 82 a + 40\) , \( -173 a^{2} + 558 a + 399\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-22a^{2}+82a+40\right){x}-173a^{2}+558a+399$ |
| 14.1-f4 |
14.1-f |
$4$ |
$6$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{3} \cdot 7^{6} \) |
$5.35851$ |
$(-a-1), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$6.436733435$ |
0.999844271 |
\( \frac{3997411383388001288125}{235298} a^{2} - \frac{12498115170624193076125}{235298} a - \frac{9398808715124539071625}{235298} \) |
\( \bigl[a^{2} - a - 5\) , \( a^{2} - a - 5\) , \( a^{2} - a - 6\) , \( -422 a^{2} + 1332 a + 980\) , \( -10397 a^{2} + 32522 a + 24437\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-422a^{2}+1332a+980\right){x}-10397a^{2}+32522a+24437$ |
| 14.2-a1 |
14.2-a |
$1$ |
$1$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2 \cdot 7^{3} \) |
$5.35851$ |
$(-a-1), (-2a^2+4a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$170.7369617$ |
4.420212386 |
\( -\frac{197489}{686} a^{2} + \frac{143495}{343} a + \frac{1707049}{686} \) |
\( \bigl[a^{2} - a - 5\) , \( -a^{2} + a + 7\) , \( a^{2} - 2 a - 6\) , \( 3 a^{2} - 4 a - 24\) , \( 25 a^{2} - 40 a - 200\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(-a^{2}+a+7\right){x}^{2}+\left(3a^{2}-4a-24\right){x}+25a^{2}-40a-200$ |
| 14.2-b1 |
14.2-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{5} \cdot 7^{4} \) |
$5.35851$ |
$(-a-1), (-2a^2+4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 5 \) |
$1$ |
$13.08312409$ |
3.387092437 |
\( \frac{1757923381954}{2401} a^{2} + \frac{19356386812047}{9604} a + \frac{9368540463215}{9604} \) |
\( \bigl[a^{2} - a - 5\) , \( a^{2} - a - 7\) , \( a^{2} - a - 5\) , \( -9 a^{2} - 20 a - 17\) , \( -89 a^{2} - 233 a - 115\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-a-7\right){x}^{2}+\left(-9a^{2}-20a-17\right){x}-89a^{2}-233a-115$ |
| 14.2-b2 |
14.2-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{10} \cdot 7^{2} \) |
$5.35851$ |
$(-a-1), (-2a^2+4a+13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$26.16624818$ |
3.387092437 |
\( -\frac{3994191}{784} a^{2} - \frac{5975415}{392} a - \frac{5952575}{784} \) |
\( \bigl[a^{2} - a - 5\) , \( a^{2} - a - 7\) , \( a^{2} - a - 5\) , \( a^{2} - 7\) , \( -a^{2} - 5 a - 7\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-a-7\right){x}^{2}+\left(a^{2}-7\right){x}-a^{2}-5a-7$ |
| 14.2-c1 |
14.2-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{5} \cdot 7^{4} \) |
$5.35851$ |
$(-a-1), (-2a^2+4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.121893065$ |
$95.47576644$ |
4.519381482 |
\( \frac{1757923381954}{2401} a^{2} + \frac{19356386812047}{9604} a + \frac{9368540463215}{9604} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 320 a^{2} - 512 a - 2563\) , \( -5635 a^{2} + 9178 a + 44946\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(320a^{2}-512a-2563\right){x}-5635a^{2}+9178a+44946$ |
| 14.2-c2 |
14.2-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{10} \cdot 7^{2} \) |
$5.35851$ |
$(-a-1), (-2a^2+4a+13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.060946532$ |
$190.9515328$ |
4.519381482 |
\( -\frac{3994191}{784} a^{2} - \frac{5975415}{392} a - \frac{5952575}{784} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 10 a^{2} - 12 a - 73\) , \( -181 a^{2} + 298 a + 1452\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10a^{2}-12a-73\right){x}-181a^{2}+298a+1452$ |
| 14.2-d1 |
14.2-d |
$1$ |
$1$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2 \cdot 7^{3} \) |
$5.35851$ |
$(-a-1), (-2a^2+4a+13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.164414693$ |
$116.6903532$ |
4.470269220 |
\( -\frac{197489}{686} a^{2} + \frac{143495}{343} a + \frac{1707049}{686} \) |
\( \bigl[1\) , \( -a^{2} + a + 5\) , \( a\) , \( a^{2} - 3 a + 4\) , \( 3 a^{2} - 12 a - 6\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(a^{2}-3a+4\right){x}+3a^{2}-12a-6$ |
| 14.3-a1 |
14.3-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{5} \cdot 7^{12} \) |
$5.35851$ |
$(-a-1), (-a^2+2a+6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \cdot 5 \) |
$1$ |
$19.10542638$ |
4.946207399 |
\( -\frac{1342172484230361085}{27682574402} a^{2} + \frac{625125823824394281}{7909306972} a + \frac{21445020670433570627}{55365148804} \) |
\( \bigl[1\) , \( a^{2} - 3 a - 6\) , \( a^{2} - a - 6\) , \( 628 a^{2} - 1022 a - 5012\) , \( 14832 a^{2} - 24124 a - 118376\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(a^{2}-3a-6\right){x}^{2}+\left(628a^{2}-1022a-5012\right){x}+14832a^{2}-24124a-118376$ |
| 14.3-a2 |
14.3-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{10} \cdot 7^{6} \) |
$5.35851$ |
$(-a-1), (-a^2+2a+6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$38.21085276$ |
4.946207399 |
\( \frac{3646450602897}{1882384} a^{2} - \frac{813832378251}{134456} a - \frac{8601362456303}{1882384} \) |
\( \bigl[1\) , \( a^{2} - 3 a - 6\) , \( a^{2} - a - 6\) , \( 38 a^{2} - 62 a - 302\) , \( 206 a^{2} - 336 a - 1646\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}+\left(a^{2}-3a-6\right){x}^{2}+\left(38a^{2}-62a-302\right){x}+206a^{2}-336a-1646$ |
| 14.3-b1 |
14.3-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{5} \cdot 7^{12} \) |
$5.35851$ |
$(-a-1), (-a^2+2a+6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$0.096643802$ |
$26.98494469$ |
3.038250946 |
\( -\frac{1342172484230361085}{27682574402} a^{2} + \frac{625125823824394281}{7909306972} a + \frac{21445020670433570627}{55365148804} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( a^{2} - 3 a - 7\) , \( a\) , \( -303 a^{2} - 844 a - 414\) , \( -9980 a^{2} - 27475 a - 13294\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-7\right){x}^{2}+\left(-303a^{2}-844a-414\right){x}-9980a^{2}-27475a-13294$ |
| 14.3-b2 |
14.3-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{10} \cdot 7^{6} \) |
$5.35851$ |
$(-a-1), (-a^2+2a+6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$0.048321901$ |
$53.96988939$ |
3.038250946 |
\( \frac{3646450602897}{1882384} a^{2} - \frac{813832378251}{134456} a - \frac{8601362456303}{1882384} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( a^{2} - 3 a - 7\) , \( a\) , \( -13 a^{2} - 44 a - 24\) , \( -220 a^{2} - 607 a - 294\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-7\right){x}^{2}+\left(-13a^{2}-44a-24\right){x}-220a^{2}-607a-294$ |
| 16.1-a1 |
16.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{11} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$5.875276403$ |
1.368946263 |
\( -643928257795 a^{2} + 1049079504293 a + 5142910167338 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 66 a^{2} - 70 a - 632\) , \( 694 a^{2} - 972 a - 6090\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(66a^{2}-70a-632\right){x}+694a^{2}-972a-6090$ |
| 16.1-a2 |
16.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{10} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$5.875276403$ |
1.368946263 |
\( 70234953 a^{2} - 219146556 a - 166914917 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -14 a^{2} + 60 a + 8\) , \( -88 a^{2} + 304 a + 140\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-14a^{2}+60a+8\right){x}-88a^{2}+304a+140$ |
| 16.1-b1 |
16.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{8} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$212.8151395$ |
2.754787558 |
\( 22765256684 a^{2} + 62673649730 a + 30329193250 \) |
\( \bigl[a^{2} - a - 6\) , \( 1\) , \( 0\) , \( 53 a^{2} - 85 a - 420\) , \( -269 a^{2} + 438 a + 2148\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}={x}^{3}+{x}^{2}+\left(53a^{2}-85a-420\right){x}-269a^{2}+438a+2148$ |
| 16.1-b2 |
16.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$425.6302790$ |
2.754787558 |
\( -26300 a^{2} - 92424 a - 47580 \) |
\( \bigl[a^{2} - a - 6\) , \( 1\) , \( 0\) , \( -2 a^{2} + 5 a + 20\) , \( -19 a^{2} + 32 a + 154\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}={x}^{3}+{x}^{2}+\left(-2a^{2}+5a+20\right){x}-19a^{2}+32a+154$ |
| 16.1-c1 |
16.1-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{8} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$28.19422359$ |
0.364960390 |
\( 22765256684 a^{2} + 62673649730 a + 30329193250 \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 7\) , \( a^{2} - 2 a - 5\) , \( 5 a^{2} - 18 a - 3\) , \( 315 a^{2} - 990 a - 741\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(5a^{2}-18a-3\right){x}+315a^{2}-990a-741$ |
| 16.1-c2 |
16.1-c |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$56.38844718$ |
0.364960390 |
\( -26300 a^{2} - 92424 a - 47580 \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 7\) , \( a^{2} - 2 a - 5\) , \( -20 a^{2} + 62 a + 57\) , \( 101 a^{2} - 315 a - 234\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(-20a^{2}+62a+57\right){x}+101a^{2}-315a-234$ |
| 16.1-d1 |
16.1-d |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{11} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$99.72686981$ |
2.581830792 |
\( -643928257795 a^{2} + 1049079504293 a + 5142910167338 \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 7\) , \( a^{2} - 2 a - 5\) , \( -138 a^{2} - 316 a - 136\) , \( 2382 a^{2} + 6714 a + 3272\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-138a^{2}-316a-136\right){x}+2382a^{2}+6714a+3272$ |
| 16.1-d2 |
16.1-d |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{10} \) |
$5.47911$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$99.72686981$ |
2.581830792 |
\( 70234953 a^{2} - 219146556 a - 166914917 \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 7\) , \( a^{2} - 2 a - 5\) , \( -18 a^{2} + 14 a + 24\) , \( -8 a^{2} + 140 a + 92\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-18a^{2}+14a+24\right){x}-8a^{2}+140a+92$ |
| 19.1-a1 |
19.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{2} \) |
$5.63831$ |
$(a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.252501772$ |
$347.4500582$ |
3.406933580 |
\( \frac{7955904}{361} a^{2} + \frac{27971264}{361} a + \frac{23479088}{361} \) |
\( \bigl[a^{2} - a - 6\) , \( a^{2} - 3 a - 6\) , \( a^{2} - 2 a - 6\) , \( 2 a^{2} - 5 a - 13\) , \( a^{2} - 2 a - 6\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(a^{2}-3a-6\right){x}^{2}+\left(2a^{2}-5a-13\right){x}+a^{2}-2a-6$ |
| 19.1-a2 |
19.1-a |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$5.63831$ |
$(a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.505003544$ |
$347.4500582$ |
3.406933580 |
\( \frac{524288}{19} a^{2} - \frac{1654784}{19} a - \frac{1179648}{19} \) |
\( \bigl[0\) , \( -a^{2} + 3 a + 5\) , \( a\) , \( -2 a^{2} - 10 a - 3\) , \( 21 a^{2} + 60 a + 30\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a^{2}+3a+5\right){x}^{2}+\left(-2a^{2}-10a-3\right){x}+21a^{2}+60a+30$ |
| 19.1-b1 |
19.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{2} \) |
$5.63831$ |
$(a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$101.2663753$ |
1.310843540 |
\( \frac{7955904}{361} a^{2} + \frac{27971264}{361} a + \frac{23479088}{361} \) |
\( \bigl[a^{2} - a - 6\) , \( a^{2} - a - 5\) , \( a^{2} - 2 a - 6\) , \( 65 a^{2} - 100 a - 508\) , \( 519 a^{2} - 838 a - 4128\bigr] \) |
${y}^2+\left(a^{2}-a-6\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(65a^{2}-100a-508\right){x}+519a^{2}-838a-4128$ |
| 19.1-b2 |
19.1-b |
$2$ |
$2$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$5.63831$ |
$(a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$202.5327507$ |
1.310843540 |
\( \frac{524288}{19} a^{2} - \frac{1654784}{19} a - \frac{1179648}{19} \) |
\( \bigl[0\) , \( a + 1\) , \( a^{2} - a - 5\) , \( 2 a\) , \( a^{2} - 3 a - 6\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a+1\right){x}^{2}+2a{x}+a^{2}-3a-6$ |
| 20.1-a1 |
20.1-a |
$2$ |
$3$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{5} \) |
$5.68671$ |
$(-a-1), (-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \cdot 5 \) |
$0.083095059$ |
$42.98159526$ |
4.160886155 |
\( -\frac{13104524288}{3125} a^{2} + \frac{39822965248}{3125} a + \frac{30126674432}{3125} \) |
\( \bigl[0\) , \( a^{2} - a - 6\) , \( a^{2} - 2 a - 5\) , \( -442 a^{2} - 1176 a - 555\) , \( -18340 a^{2} - 50585 a - 24500\bigr] \) |
${y}^2+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-442a^{2}-1176a-555\right){x}-18340a^{2}-50585a-24500$ |
| 20.1-a2 |
20.1-a |
$2$ |
$3$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{15} \) |
$5.68671$ |
$(-a-1), (-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \cdot 5 \) |
$0.249285177$ |
$14.32719842$ |
4.160886155 |
\( -\frac{5206366482548384852992}{30517578125} a^{2} + \frac{8468079724997819269632}{30517578125} a + \frac{41552168966842387071488}{30517578125} \) |
\( \bigl[0\) , \( a^{2} - a - 6\) , \( a^{2} - 2 a - 5\) , \( 388 a^{2} + 1084 a + 535\) , \( -73254 a^{2} - 201907 a - 97748\bigr] \) |
${y}^2+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(388a^{2}+1084a+535\right){x}-73254a^{2}-201907a-97748$ |
| 20.1-b1 |
20.1-b |
$2$ |
$3$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{15} \) |
$5.68671$ |
$(-a-1), (-a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$4.917880956$ |
1.145872006 |
\( -\frac{5206366482548384852992}{30517578125} a^{2} + \frac{8468079724997819269632}{30517578125} a + \frac{41552168966842387071488}{30517578125} \) |
\( \bigl[0\) , \( a^{2} - a - 5\) , \( a^{2} - 2 a - 5\) , \( 18 a^{2} - 22 a - 158\) , \( 180 a^{2} - 450 a - 823\bigr] \) |
${y}^2+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(18a^{2}-22a-158\right){x}+180a^{2}-450a-823$ |
| 20.1-b2 |
20.1-b |
$2$ |
$3$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{5} \) |
$5.68671$ |
$(-a-1), (-a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$132.7827858$ |
1.145872006 |
\( -\frac{13104524288}{3125} a^{2} + \frac{39822965248}{3125} a + \frac{30126674432}{3125} \) |
\( \bigl[0\) , \( a^{2} - a - 5\) , \( a^{2} - 2 a - 5\) , \( -12 a^{2} + 38 a + 32\) , \( 40 a^{2} - 124 a - 97\bigr] \) |
${y}^2+\left(a^{2}-2a-5\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-12a^{2}+38a+32\right){x}+40a^{2}-124a-97$ |
| 22.1-a1 |
22.1-a |
$1$ |
$1$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{10} \cdot 11^{9} \) |
$5.77777$ |
$(-a-1), (a^2+3a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$19.21190540$ |
4.973773773 |
\( -\frac{1236193789551325371}{37727163056} a^{2} + \frac{392480348944629325}{4715895382} a + \frac{2745506146683394201}{37727163056} \) |
\( \bigl[a^{2} - a - 5\) , \( -a\) , \( a^{2} - a - 5\) , \( 1398 a^{2} - 2248 a - 11269\) , \( 54878 a^{2} - 89375 a - 437552\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}-a{x}^{2}+\left(1398a^{2}-2248a-11269\right){x}+54878a^{2}-89375a-437552$ |
| 22.1-b1 |
22.1-b |
$2$ |
$5$ |
3.3.1492.1 |
$3$ |
$[3, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11^{3} \) |
$5.77777$ |
$(-a-1), (a^2+3a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$1$ |
$31.10558977$ |
4.026465969 |
\( -\frac{107362298}{1331} a^{2} + \frac{694281135}{5324} a + \frac{3424025161}{5324} \) |
\( \bigl[1\) , \( 1\) , \( a^{2} - 2 a - 6\) , \( 3 a^{2} - 10 a - 5\) , \( 8 a^{2} - 24 a - 23\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+{x}^{2}+\left(3a^{2}-10a-5\right){x}+8a^{2}-24a-23$ |
*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.
