| 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 |
| 13.1-a1 |
13.1-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( -13 \) |
$5.02524$ |
$(a^2-a-4)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$0.044283483$ |
$262.4039588$ |
2.851632302 |
\( \frac{190257}{13} a^{2} - \frac{35617}{13} a - \frac{1346465}{13} \) |
\( \bigl[a^{2} + a - 4\) , \( a^{2} + a - 4\) , \( a + 1\) , \( 5 a^{2} + 10 a + 3\) , \( 6 a^{2} + 30 a + 4\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(5a^{2}+10a+3\right){x}+6a^{2}+30a+4$ |
| 13.1-a2 |
13.1-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( - 13^{5} \) |
$5.02524$ |
$(a^2-a-4)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$0.044283483$ |
$52.48079176$ |
2.851632302 |
\( -\frac{1872208853}{371293} a^{2} + \frac{4761390908}{371293} a + \frac{861473885}{371293} \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( 0\) , \( -13 a^{2} + 32 a + 11\) , \( 40 a^{2} - 101 a - 20\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-13a^{2}+32a+11\right){x}+40a^{2}-101a-20$ |
| 35.2-a1 |
35.2-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( - 5^{4} \cdot 7^{2} \) |
$5.92713$ |
$(a+2), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$130.9330960$ |
3.570165058 |
\( -\frac{45033465416}{1225} a^{2} + \frac{6523344418}{1225} a + \frac{314493069659}{1225} \) |
\( \bigl[a^{2} + a - 4\) , \( -a - 1\) , \( 0\) , \( 4 a^{2} - 5 a - 6\) , \( -8 a^{2} + 22 a + 5\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a^{2}-5a-6\right){x}-8a^{2}+22a+5$ |
| 35.2-a2 |
35.2-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7 \) |
$5.92713$ |
$(a+2), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$261.8661920$ |
3.570165058 |
\( -\frac{14004}{7} a^{2} - \frac{26108}{35} a + \frac{583307}{35} \) |
\( \bigl[a^{2} + a - 4\) , \( -a - 1\) , \( 0\) , \( -a^{2} + 5 a + 4\) , \( -2 a^{2} + 7 a + 2\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a^{2}+5a+4\right){x}-2a^{2}+7a+2$ |
| 35.2-b1 |
35.2-b |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7 \) |
$5.92713$ |
$(a+2), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$297.5330937$ |
4.056431440 |
\( -\frac{1910684}{35} a^{2} - \frac{672715}{7} a + \frac{4315434}{35} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} + a + 6\) , \( a^{2} + a - 5\) , \( -8 a^{2} + 9 a + 31\) , \( -12 a^{2} + 15 a + 42\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-8a^{2}+9a+31\right){x}-12a^{2}+15a+42$ |
| 35.2-b2 |
35.2-b |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.2 |
\( 5 \cdot 7 \) |
\( - 5 \cdot 7^{2} \) |
$5.92713$ |
$(a+2), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$74.38327343$ |
4.056431440 |
\( \frac{4740912264747}{245} a^{2} + \frac{12861425252391}{245} a + \frac{1745427260714}{245} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} + a + 6\) , \( a^{2} + a - 5\) , \( -63 a^{2} + 149 a + 56\) , \( -572 a^{2} + 1447 a + 281\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-63a^{2}+149a+56\right){x}-572a^{2}+1447a+281$ |
| 35.4-a1 |
35.4-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.4 |
\( 5 \cdot 7 \) |
\( - 5 \cdot 7 \) |
$5.92713$ |
$(-a-1), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$134.1206649$ |
3.657080799 |
\( -\frac{378107689}{35} a^{2} - \frac{1026359441}{35} a - \frac{139286851}{35} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} - a + 4\) , \( a^{2} + a - 4\) , \( -16 a^{2} - 33 a + 21\) , \( 78 a^{2} + 227 a + 68\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-16a^{2}-33a+21\right){x}+78a^{2}+227a+68$ |
| 35.4-a2 |
35.4-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.4 |
\( 5 \cdot 7 \) |
\( - 5^{5} \cdot 7^{5} \) |
$5.92713$ |
$(-a-1), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$1$ |
$26.82413298$ |
3.657080799 |
\( \frac{361008131951}{52521875} a^{2} - \frac{31314057156}{52521875} a - \frac{2520635298021}{52521875} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a^{2} - 5\) , \( 5 a^{2} - 29\) , \( 10 a^{2} + a - 65\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a^{2}-29\right){x}+10a^{2}+a-65$ |
| 35.4-b1 |
35.4-b |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.4 |
\( 5 \cdot 7 \) |
\( - 5^{4} \cdot 7^{6} \) |
$5.92713$ |
$(-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$40.86888009$ |
3.343126805 |
\( \frac{874763517697611}{73530625} a^{2} + \frac{2378481300642509}{73530625} a + \frac{332307049286994}{73530625} \) |
\( \bigl[1\) , \( -a^{2} + 6\) , \( a^{2} - 4\) , \( 20 a^{2} - 5 a - 147\) , \( -66 a^{2} + 7 a + 453\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+6\right){x}^{2}+\left(20a^{2}-5a-147\right){x}-66a^{2}+7a+453$ |
| 35.4-b2 |
35.4-b |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.4 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{3} \) |
$5.92713$ |
$(-a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$81.73776019$ |
3.343126805 |
\( \frac{3451762156}{8575} a^{2} - \frac{8918762511}{8575} a - \frac{1332940226}{8575} \) |
\( \bigl[1\) , \( -a^{2} + 6\) , \( a^{2} - 4\) , \( -2\) , \( 5 a^{2} - a - 37\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+6\right){x}^{2}-2{x}+5a^{2}-a-37$ |
| 35.5-a1 |
35.5-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.5 |
\( 5 \cdot 7 \) |
\( - 5^{5} \cdot 7^{2} \) |
$5.92713$ |
$(a+2), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$50.75878616$ |
2.768089203 |
\( \frac{670902461}{6125} a^{2} - \frac{85734637}{6125} a - \frac{4700179843}{6125} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} - a - 6\) , \( a^{2} + a - 4\) , \( -93 a^{2} + 239 a + 27\) , \( -1559 a^{2} + 4006 a + 604\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-93a^{2}+239a+27\right){x}-1559a^{2}+4006a+604$ |
| 35.5-a2 |
35.5-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.5 |
\( 5 \cdot 7 \) |
\( - 5 \cdot 7^{10} \) |
$5.92713$ |
$(a+2), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$10.15175723$ |
2.768089203 |
\( -\frac{532913992124329}{1412376245} a^{2} - \frac{1445854923635462}{1412376245} a - \frac{194448967778923}{1412376245} \) |
\( \bigl[a^{2} + a - 4\) , \( a^{2} + a - 6\) , \( a^{2} + a - 5\) , \( -7 a^{2} - 22 a - 9\) , \( -95 a^{2} - 262 a - 46\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(-7a^{2}-22a-9\right){x}-95a^{2}-262a-46$ |
| 35.5-b1 |
35.5-b |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.5 |
\( 5 \cdot 7 \) |
\( - 5^{13} \cdot 7^{2} \) |
$5.92713$ |
$(a+2), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.642138520$ |
$28.83380483$ |
3.029150043 |
\( -\frac{18879838632694}{3828125} a^{2} + \frac{48543845049773}{3828125} a + \frac{7342751287322}{3828125} \) |
\( \bigl[a\) , \( -a^{2} + a + 6\) , \( 1\) , \( -10 a^{2} - 18 a + 9\) , \( 231 a^{2} + 620 a + 91\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-10a^{2}-18a+9\right){x}+231a^{2}+620a+91$ |
| 35.6-a1 |
35.6-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.6 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7 \) |
$5.92713$ |
$(-a-1), (a^2-a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.756791880$ |
$135.6042596$ |
4.197395694 |
\( -\frac{25159829}{175} a^{2} - \frac{54416601}{175} a + \frac{31667359}{175} \) |
\( \bigl[1\) , \( -a^{2} + a + 6\) , \( a^{2} - 4\) , \( 14 a^{2} - 104\) , \( 104 a^{2} - 14 a - 731\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(14a^{2}-104\right){x}+104a^{2}-14a-731$ |
| 35.6-a2 |
35.6-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
35.6 |
\( 5 \cdot 7 \) |
\( - 5^{4} \cdot 7^{2} \) |
$5.92713$ |
$(-a-1), (a^2-a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.378395940$ |
$67.80212981$ |
4.197395694 |
\( \frac{6020468824131891}{30625} a^{2} + \frac{16342434615807704}{30625} a + \frac{2217909724632489}{30625} \) |
\( \bigl[1\) , \( -a^{2} + a + 6\) , \( a^{2} - 4\) , \( 4 a^{2} + 5 a - 44\) , \( 164 a^{2} - 38 a - 1108\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(4a^{2}+5a-44\right){x}+164a^{2}-38a-1108$ |
| 37.1-a1 |
37.1-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$5.98228$ |
$(a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$115.2545158$ |
3.142655735 |
\( \frac{998668}{37} a^{2} + \frac{2737071}{37} a + \frac{435314}{37} \) |
\( \bigl[a^{2} + a - 5\) , \( 1\) , \( a^{2} - 5\) , \( 2 a + 7\) , \( a + 11\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+{x}^{2}+\left(2a+7\right){x}+a+11$ |
| 40.1-a1 |
40.1-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{3} \cdot 5^{4} \) |
$6.06052$ |
$(-a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.103092842$ |
$89.42718673$ |
3.016603195 |
\( -\frac{413381}{1250} a^{2} - \frac{333889}{1250} a + \frac{2015088}{625} \) |
\( \bigl[a^{2} - 4\) , \( -a^{2} - a + 4\) , \( a^{2} - 4\) , \( -2 a^{2} + a + 14\) , \( -4 a^{2} - 4 a + 15\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-2a^{2}+a+14\right){x}-4a^{2}-4a+15$ |
| 40.1-b1 |
40.1-b |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$6.06052$ |
$(-a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.608317174$ |
$82.16939439$ |
4.088841510 |
\( -\frac{836708429}{40} a^{2} + \frac{119783009}{40} a + \frac{5839529889}{40} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 4\) , \( a^{2} - 5\) , \( -2 a + 3\) , \( 2 a^{2} - a - 13\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-2a+3\right){x}+2a^{2}-a-13$ |
| 40.1-c1 |
40.1-c |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{15} \cdot 5^{5} \) |
$6.06052$ |
$(-a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$1$ |
$11.26223759$ |
1.535442464 |
\( -\frac{11848113401}{100000} a^{2} + \frac{1146001481}{100000} a + \frac{81393512221}{100000} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} - a + 6\) , \( a^{2} - 5\) , \( -5 a^{2} - 3 a + 37\) , \( -8 a^{2} + 52\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-a^{2}-a+6\right){x}^{2}+\left(-5a^{2}-3a+37\right){x}-8a^{2}+52$ |
| 40.1-c2 |
40.1-c |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( 2^{3} \cdot 5 \) |
$6.06052$ |
$(-a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$56.31118797$ |
1.535442464 |
\( -\frac{41429383}{5} a^{2} + \frac{206035021}{10} a + \frac{25629488}{5} \) |
\( \bigl[a\) , \( a^{2} - a - 5\) , \( 0\) , \( -a^{2} + 2 a + 8\) , \( -a^{2} + 14 a - 2\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-a^{2}+2a+8\right){x}-a^{2}+14a-2$ |
| 40.1-d1 |
40.1-d |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
40.1 |
\( 2^{3} \cdot 5 \) |
\( - 2^{27} \cdot 5^{4} \) |
$6.06052$ |
$(-a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$5.428864458$ |
2.664528450 |
\( -\frac{10286139}{20000} a^{2} + \frac{59409393}{40000} a - \frac{70548721}{320000} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} + 5\) , \( a^{2} + a - 4\) , \( 2 a^{2} + 19 a + 36\) , \( -267 a^{2} - 704 a - 45\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(2a^{2}+19a+36\right){x}-267a^{2}-704a-45$ |
| 43.1-a1 |
43.1-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
43.1 |
\( 43 \) |
\( -43 \) |
$6.13401$ |
$(-a^2-3a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$183.3290259$ |
4.998849812 |
\( \frac{336015233087}{43} a^{2} + \frac{912106156667}{43} a + \frac{123786286374}{43} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} - a - 6\) , \( a^{2} + a - 4\) , \( -4 a^{2} - 4 a + 24\) , \( -20 a^{2} + a + 139\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-4a^{2}-4a+24\right){x}-20a^{2}+a+139$ |
| 43.1-b1 |
43.1-b |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
43.1 |
\( 43 \) |
\( -43 \) |
$6.13401$ |
$(-a^2-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$2.035055050$ |
$26.76178727$ |
4.455037751 |
\( \frac{812947539197441}{43} a^{2} - \frac{2090251868923810}{43} a - \frac{316174194780136}{43} \) |
\( \bigl[a + 1\) , \( -a^{2} + 6\) , \( a^{2} - 5\) , \( -10 a^{2} - 23 a + 7\) , \( -40 a^{2} - 109 a - 17\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-a^{2}+6\right){x}^{2}+\left(-10a^{2}-23a+7\right){x}-40a^{2}-109a-17$ |
| 49.1-a1 |
49.1-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{7} \) |
$6.26901$ |
$(a^2-a-5), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$54.06700858$ |
2.211375320 |
\( -\frac{3987072}{2401} a^{2} - \frac{3762793}{2401} a + \frac{43729410}{2401} \) |
\( \bigl[1\) , \( -a^{2} + 5\) , \( a\) , \( -5 a^{2} + 10 a + 10\) , \( -9 a^{2} + 21 a + 7\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-5a^{2}+10a+10\right){x}-9a^{2}+21a+7$ |
| 49.1-a2 |
49.1-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{14} \) |
$6.26901$ |
$(a^2-a-5), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$27.03350429$ |
2.211375320 |
\( \frac{3252492505}{5764801} a^{2} - \frac{4939702623}{5764801} a - \frac{566808740}{5764801} \) |
\( \bigl[1\) , \( -a^{2} + 5\) , \( a\) , \( -15 a^{2} + 35 a + 15\) , \( 64 a^{2} - 167 a - 20\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-15a^{2}+35a+15\right){x}+64a^{2}-167a-20$ |
| 49.1-b1 |
49.1-b |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{8} \) |
$6.26901$ |
$(a^2-a-5), (a-2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.083405119$ |
$71.82734083$ |
4.410472524 |
\( \frac{1110896383775}{117649} a^{2} + \frac{3034427608750}{117649} a + \frac{457852185222}{117649} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 4\) , \( a^{2} - 4\) , \( 22 a^{2} - 8 a - 153\) , \( 83 a^{2} - 13 a - 577\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(22a^{2}-8a-153\right){x}+83a^{2}-13a-577$ |
| 49.1-b2 |
49.1-b |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$6.26901$ |
$(a^2-a-5), (a-2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.083405119$ |
$287.3093633$ |
4.410472524 |
\( \frac{27105081}{343} a^{2} - \frac{68372781}{343} a - \frac{9755048}{343} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 4\) , \( a^{2} - 4\) , \( 2 a^{2} - 3 a - 8\) , \( -a^{2} - a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(2a^{2}-3a-8\right){x}-a^{2}-a+9$ |
| 53.1-a1 |
53.1-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
53.1 |
\( 53 \) |
\( -53 \) |
$6.35154$ |
$(2a^2-a-11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.362147169$ |
$28.48361978$ |
0.843801684 |
\( \frac{56664300}{53} a^{2} + \frac{149077808}{53} a + \frac{19217207}{53} \) |
\( \bigl[a^{2} - 4\) , \( -a^{2} + a + 6\) , \( 1\) , \( -5 a^{2} + a + 33\) , \( -5 a^{2} - a + 30\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-5a^{2}+a+33\right){x}-5a^{2}-a+30$ |
| 56.1-a1 |
56.1-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{3} \cdot 7^{2} \) |
$6.41009$ |
$(a^2-a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.699308248$ |
$34.14855876$ |
3.906889589 |
\( -\frac{101275755985949}{98} a^{2} - \frac{273232294700669}{98} a - \frac{16496734272528}{49} \) |
\( \bigl[1\) , \( a^{2} - 5\) , \( a\) , \( -1089 a^{2} - 2954 a - 394\) , \( 52752 a^{2} + 143191 a + 19430\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(-1089a^{2}-2954a-394\right){x}+52752a^{2}+143191a+19430$ |
| 56.1-b1 |
56.1-b |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{21} \cdot 7 \) |
$6.41009$ |
$(a^2-a-5), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.025356032$ |
$119.8794119$ |
5.221620508 |
\( \frac{1496391429}{896} a^{2} + \frac{253863661}{56} a + \frac{137732501}{224} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 6\) , \( 1\) , \( -11 a^{2} - 26 a + 8\) , \( 30 a^{2} + 84 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-11a^{2}-26a+8\right){x}+30a^{2}+84a+19$ |
| 56.1-c1 |
56.1-c |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{15} \cdot 7^{10} \) |
$6.41009$ |
$(a^2-a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.716689741$ |
1.286104232 |
\( -\frac{1294442215613329}{4519603984} a^{2} - \frac{7021784221915147}{9039207968} a - \frac{29080501888920}{282475249} \) |
\( \bigl[a^{2} + a - 4\) , \( -a + 1\) , \( a\) , \( -128 a^{2} - 353 a - 57\) , \( -2547 a^{2} - 6932 a - 985\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-128a^{2}-353a-57\right){x}-2547a^{2}-6932a-985$ |
| 56.1-c2 |
56.1-c |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{3} \cdot 7^{2} \) |
$6.41009$ |
$(a^2-a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$23.58344870$ |
1.286104232 |
\( -\frac{2972186405389}{49} a^{2} + \frac{855761659293}{98} a + \frac{41477539360685}{98} \) |
\( \bigl[1\) , \( a^{2} - a - 6\) , \( a + 1\) , \( 243 a^{2} + 660 a + 95\) , \( 508 a^{2} + 1381 a + 193\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(243a^{2}+660a+95\right){x}+508a^{2}+1381a+193$ |
| 56.1-d1 |
56.1-d |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{21} \cdot 7 \) |
$6.41009$ |
$(a^2-a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.340699657$ |
$36.57529299$ |
4.011247181 |
\( \frac{1440450120964717}{112} a^{2} - \frac{1651070684311621}{896} a - \frac{40214322893674249}{448} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 6 a^{2} - 24 a - 5\) , \( -112 a^{2} + 296 a + 45\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(6a^{2}-24a-5\right){x}-112a^{2}+296a+45$ |
| 65.2-a1 |
65.2-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
65.2 |
\( 5 \cdot 13 \) |
\( - 5^{5} \cdot 13^{2} \) |
$6.57131$ |
$(a+2), (a^2-a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$11.06430014$ |
3.016913137 |
\( -\frac{7396029785004}{21125} a^{2} - \frac{28564158674062}{21125} a - \frac{24548464851653}{21125} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 7 a^{2} - 10 a - 67\) , \( -39 a^{2} - 29 a + 178\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a^{2}-10a-67\right){x}-39a^{2}-29a+178$ |
| 89.1-a1 |
89.1-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
89.1 |
\( 89 \) |
\( 89 \) |
$6.92465$ |
$(a^2+2a+2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$5.883625779$ |
$221.3341317$ |
4.261019653 |
\( \frac{799290}{89} a^{2} + \frac{2076145}{89} a + \frac{73601}{89} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} + a - 6\) , \( 0\) , \( 2 a^{2} - a - 2\) , \( -4 a^{2} + 12 a + 3\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(2a^{2}-a-2\right){x}-4a^{2}+12a+3$ |
| 89.1-a2 |
89.1-a |
$2$ |
$5$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
89.1 |
\( 89 \) |
\( 89^{5} \) |
$6.92465$ |
$(a^2+2a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$29.41812889$ |
$1.770673054$ |
4.261019653 |
\( -\frac{2106814494166409227}{5584059449} a^{2} + \frac{5454467756788131324}{5584059449} a + \frac{824750724006451487}{5584059449} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} + a - 6\) , \( 0\) , \( -428 a^{2} + 1109 a + 153\) , \( -12123 a^{2} + 31215 a + 4600\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(-428a^{2}+1109a+153\right){x}-12123a^{2}+31215a+4600$ |
| 89.3-a1 |
89.3-a |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
89.3 |
\( 89 \) |
\( -89 \) |
$6.92465$ |
$(a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$157.4417098$ |
4.292977927 |
\( -\frac{2358112}{89} a^{2} - \frac{1848548}{89} a + \frac{22527455}{89} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 4 a^{2} + 2 a - 20\) , \( -3 a^{2} + 3 a + 27\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(4a^{2}+2a-20\right){x}-3a^{2}+3a+27$ |
| 89.3-b1 |
89.3-b |
$1$ |
$1$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
89.3 |
\( 89 \) |
\( -89 \) |
$6.92465$ |
$(a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$35.13486027$ |
0.958025543 |
\( -\frac{8148673}{89} a^{2} - \frac{26891494}{89} a - \frac{2442625}{89} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} - 5\) , \( 0\) , \( -40 a^{2} + 106 a + 16\) , \( -403 a^{2} + 1038 a + 157\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(-40a^{2}+106a+16\right){x}-403a^{2}+1038a+157$ |
| 91.2-a1 |
91.2-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
91.2 |
\( 7 \cdot 13 \) |
\( 7^{10} \cdot 13 \) |
$6.95035$ |
$(a^2-a-5), (a^2-a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.075146759$ |
$34.94894666$ |
4.395720344 |
\( -\frac{350983454860981}{3672178237} a^{2} + \frac{43864297557921}{3672178237} a + \frac{2687363605785317}{3672178237} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} - a + 6\) , \( a^{2} + a - 4\) , \( 4 a^{2} - 7 a - 44\) , \( -6 a^{2} - 10 a + 13\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(-a^{2}-a+6\right){x}^{2}+\left(4a^{2}-7a-44\right){x}-6a^{2}-10a+13$ |
| 91.2-a2 |
91.2-a |
$2$ |
$2$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
91.2 |
\( 7 \cdot 13 \) |
\( - 7^{5} \cdot 13^{2} \) |
$6.95035$ |
$(a^2-a-5), (a^2-a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.537573379$ |
$69.89789333$ |
4.395720344 |
\( -\frac{290624926}{2840383} a^{2} - \frac{1100177381}{2840383} a + \frac{5127769625}{2840383} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} - a + 6\) , \( a^{2} + a - 4\) , \( -6 a^{2} - 2 a + 36\) , \( -7 a^{2} - a + 44\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a^{2}+a-4\right){y}={x}^{3}+\left(-a^{2}-a+6\right){x}^{2}+\left(-6a^{2}-2a+36\right){x}-7a^{2}-a+44$ |
| 91.3-a1 |
91.3-a |
$2$ |
$3$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
91.3 |
\( 7 \cdot 13 \) |
\( - 7^{4} \cdot 13 \) |
$6.95035$ |
$(a-1), (a^2-a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.439963692$ |
$218.3345286$ |
3.492342711 |
\( -\frac{27349015}{31213} a^{2} - \frac{1842296}{31213} a - \frac{31186}{31213} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -a^{2}\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a^{2}{x}$ |
| 91.3-a2 |
91.3-a |
$2$ |
$3$ |
3.3.1345.1 |
$3$ |
$[3, 0]$ |
91.3 |
\( 7 \cdot 13 \) |
\( - 7^{12} \cdot 13^{3} \) |
$6.95035$ |
$(a-1), (a^2-a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.146654564$ |
$8.086464024$ |
3.492342711 |
\( \frac{7059889639483973}{30409307980597} a^{2} - \frac{724469684900355}{30409307980597} a + \frac{70071048175392}{30409307980597} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 4 a^{2}\) , \( a\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+4a^{2}{x}+a$ |
*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.
