| 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 |
| 5324.7-a1 |
5324.7-a |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{13} \cdot 11^{11} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.804718677$ |
$0.722336016$ |
1.757617297 |
\( \frac{531165637}{1362944} a - \frac{402644615}{1362944} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 44 a - 37\) , \( 279 a - 159\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(44a-37\right){x}+279a-159$ |
| 5324.7-a2 |
5324.7-a |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{39} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$2.414156031$ |
$0.240778672$ |
1.757617297 |
\( -\frac{4070378798731}{11811160064} a + \frac{4282573003625}{11811160064} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -396 a + 348\) , \( -6354 a + 6144\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-396a+348\right){x}-6354a+6144$ |
| 5324.7-b1 |
5324.7-b |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{3} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.036077686$ |
$4.305972120$ |
3.288129422 |
\( -\frac{399175}{1408} a + \frac{2926837}{704} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -a + 2\) , \( a + 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-a+2\right){x}+a+2$ |
| 5324.7-c1 |
5324.7-c |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{36} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.964954788$ |
$0.288454494$ |
3.427684460 |
\( -\frac{31504873455125}{519691042816} a + \frac{132554493440375}{519691042816} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 157 a + 140\) , \( 3795 a - 654\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(157a+140\right){x}+3795a-654$ |
| 5324.7-c2 |
5324.7-c |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{36} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{9} \) |
$0.245619348$ |
$0.288454494$ |
3.427684460 |
\( \frac{31504873455125}{519691042816} a + \frac{50524809992625}{259845521408} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 48 a - 320\) , \( 3251 a + 758\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(48a-320\right){x}+3251a+758$ |
| 5324.7-c3 |
5324.7-c |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{24} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.491238697$ |
$0.288454494$ |
3.427684460 |
\( -\frac{1133731711875}{959512576} a + \frac{4028870229625}{479756288} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 537 a + 109\) , \( 192 a + 7744\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(537a+109\right){x}+192a+7744$ |
| 5324.7-c4 |
5324.7-c |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{24} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$0.982477394$ |
$0.288454494$ |
3.427684460 |
\( \frac{1133731711875}{959512576} a + \frac{6924008747375}{959512576} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 307 a - 849\) , \( 4407 a - 7821\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(307a-849\right){x}+4407a-7821$ |
| 5324.7-c5 |
5324.7-c |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{12} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.982477394$ |
$0.144227247$ |
3.427684460 |
\( -\frac{915988506230265125}{54875873536} a + \frac{1519749586080622375}{27437936768} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 8297 a + 1789\) , \( 17536 a + 527872\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(8297a+1789\right){x}+17536a+527872$ |
| 5324.7-c6 |
5324.7-c |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{12} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1.964954788$ |
$0.144227247$ |
3.427684460 |
\( \frac{915988506230265125}{54875873536} a + \frac{2123510665930979625}{54875873536} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 4707 a - 13169\) , \( 291111 a - 525613\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4707a-13169\right){x}+291111a-525613$ |
| 5324.7-d1 |
5324.7-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{6} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( -\frac{46830231}{234256} a + \frac{212077575}{117128} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 21 a - 64\) , \( -59 a + 50\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(21a-64\right){x}-59a+50$ |
| 5324.7-d2 |
5324.7-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{6} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( \frac{46830231}{234256} a + \frac{377324919}{234256} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 39 a + 11\) , \( -46 a - 37\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(39a+11\right){x}-46a-37$ |
| 5324.7-d3 |
5324.7-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{3} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.440785834$ |
2.665622171 |
\( -\frac{56046918875913}{857435524} a + \frac{93327675428637}{857435524} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 329 a + 211\) , \( -1214 a + 5583\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(329a+211\right){x}-1214a+5583$ |
| 5324.7-d4 |
5324.7-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{3} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.440785834$ |
2.665622171 |
\( \frac{56046918875913}{857435524} a + \frac{9320189138181}{214358881} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 131 a - 614\) , \( 1723 a - 5472\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(131a-614\right){x}+1723a-5472$ |
| 5324.7-d5 |
5324.7-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( -\frac{14003310699}{30976} a + \frac{9620888049}{15488} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 129 a + 23\) , \( 26 a + 989\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(129a+23\right){x}+26a+989$ |
| 5324.7-d6 |
5324.7-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( \frac{14003310699}{30976} a + \frac{5238465399}{30976} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 75 a - 202\) , \( 547 a - 934\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(75a-202\right){x}+547a-934$ |
| 5324.7-e1 |
5324.7-e |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{4} \cdot 11^{10} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.482136211$ |
2.240779327 |
\( \frac{704969}{484} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -14 a + 12\) , \( 15 a - 13\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-14a+12\right){x}+15a-13$ |
| 5324.7-e2 |
5324.7-e |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{2} \cdot 11^{14} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.741068105$ |
2.240779327 |
\( \frac{59776471}{29282} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 66 a - 58\) , \( 71 a + 59\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(66a-58\right){x}+71a+59$ |
| 5324.7-e3 |
5324.7-e |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{5} \cdot 11^{8} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.482136211$ |
2.240779327 |
\( -\frac{34643161}{176} a + \frac{13683239}{88} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -7 a - 54\) , \( -59 a - 165\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-7a-54\right){x}-59a-165$ |
| 5324.7-e4 |
5324.7-e |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.7 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{5} \cdot 11^{8} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.482136211$ |
2.240779327 |
\( \frac{34643161}{176} a - \frac{7276683}{176} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 16 a + 46\) , \( -130 a + 160\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(16a+46\right){x}-130a+160$ |
*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.
