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 |
5324.6-a1 |
5324.6-a |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{13} \cdot 11^{11} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.804718677$ |
$0.722336016$ |
1.757617297 |
\( -\frac{531165637}{1362944} a + \frac{64260511}{681472} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -45 a + 7\) , \( -280 a + 120\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-45a+7\right){x}-280a+120$ |
5324.6-a2 |
5324.6-a |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{39} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$2.414156031$ |
$0.240778672$ |
1.757617297 |
\( \frac{4070378798731}{11811160064} a + \frac{106097102447}{5905580032} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 395 a - 48\) , \( 6353 a - 210\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(395a-48\right){x}+6353a-210$ |
5324.6-b1 |
5324.6-b |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{4} \cdot 11^{10} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.482136211$ |
2.240779327 |
\( \frac{704969}{484} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 16 a - 2\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(16a-2\right){x}$ |
5324.6-b2 |
5324.6-b |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{2} \cdot 11^{14} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.741068105$ |
2.240779327 |
\( \frac{59776471}{29282} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -64 a + 8\) , \( -136 a + 138\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-64a+8\right){x}-136a+138$ |
5324.6-b3 |
5324.6-b |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{5} \cdot 11^{8} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.482136211$ |
2.240779327 |
\( -\frac{34643161}{176} a + \frac{13683239}{88} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -16 a + 62\) , \( 130 a + 30\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-16a+62\right){x}+130a+30$ |
5324.6-b4 |
5324.6-b |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{5} \cdot 11^{8} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.482136211$ |
2.240779327 |
\( \frac{34643161}{176} a - \frac{7276683}{176} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 9 a - 62\) , \( 51 a - 162\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(9a-62\right){x}+51a-162$ |
5324.6-c1 |
5324.6-c |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{3} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.036077686$ |
$4.305972120$ |
3.288129422 |
\( \frac{399175}{1408} a + \frac{5454499}{1408} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( a + 1\) , \( -a + 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a+1\right){x}-a+3$ |
5324.6-d1 |
5324.6-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{36} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{9} \) |
$0.245619348$ |
$0.288454494$ |
3.427684460 |
\( -\frac{31504873455125}{519691042816} a + \frac{132554493440375}{519691042816} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -48 a - 272\) , \( -3251 a + 4009\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-48a-272\right){x}-3251a+4009$ |
5324.6-d2 |
5324.6-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{36} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.964954788$ |
$0.288454494$ |
3.427684460 |
\( \frac{31504873455125}{519691042816} a + \frac{50524809992625}{259845521408} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -158 a + 297\) , \( -3796 a + 3141\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-158a+297\right){x}-3796a+3141$ |
5324.6-d3 |
5324.6-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{24} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$0.982477394$ |
$0.288454494$ |
3.427684460 |
\( -\frac{1133731711875}{959512576} a + \frac{4028870229625}{479756288} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -307 a - 542\) , \( -4407 a - 3414\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-307a-542\right){x}-4407a-3414$ |
5324.6-d4 |
5324.6-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{24} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.491238697$ |
$0.288454494$ |
3.427684460 |
\( \frac{1133731711875}{959512576} a + \frac{6924008747375}{959512576} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -538 a + 647\) , \( -193 a + 7937\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-538a+647\right){x}-193a+7937$ |
5324.6-d5 |
5324.6-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{12} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1.964954788$ |
$0.144227247$ |
3.427684460 |
\( -\frac{915988506230265125}{54875873536} a + \frac{1519749586080622375}{27437936768} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -4707 a - 8462\) , \( -291111 a - 234502\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-4707a-8462\right){x}-291111a-234502$ |
5324.6-d6 |
5324.6-d |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{12} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.982477394$ |
$0.144227247$ |
3.427684460 |
\( \frac{915988506230265125}{54875873536} a + \frac{2123510665930979625}{54875873536} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -8298 a + 10087\) , \( -17537 a + 545409\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-8298a+10087\right){x}-17537a+545409$ |
5324.6-e1 |
5324.6-e |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{6} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( -\frac{46830231}{234256} a + \frac{212077575}{117128} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -40 a + 50\) , \( 45 a - 83\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-40a+50\right){x}+45a-83$ |
5324.6-e2 |
5324.6-e |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{6} \cdot 11^{12} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( \frac{46830231}{234256} a + \frac{377324919}{234256} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -22 a - 43\) , \( 58 a - 9\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-22a-43\right){x}+58a-9$ |
5324.6-e3 |
5324.6-e |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{3} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.440785834$ |
2.665622171 |
\( -\frac{56046918875913}{857435524} a + \frac{93327675428637}{857435524} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -132 a - 483\) , \( -1724 a - 3749\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-132a-483\right){x}-1724a-3749$ |
5324.6-e4 |
5324.6-e |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{3} \cdot 11^{15} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.440785834$ |
2.665622171 |
\( \frac{56046918875913}{857435524} a + \frac{9320189138181}{214358881} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -330 a + 540\) , \( 1213 a + 4369\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-330a+540\right){x}+1213a+4369$ |
5324.6-e5 |
5324.6-e |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( -\frac{14003310699}{30976} a + \frac{9620888049}{15488} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -76 a - 127\) , \( -548 a - 387\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-76a-127\right){x}-548a-387$ |
5324.6-e6 |
5324.6-e |
$6$ |
$8$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
5324.6 |
\( 2^{2} \cdot 11^{3} \) |
\( 2^{9} \cdot 11^{9} \) |
$2.01952$ |
$(a), (-a+1), (-2a+3), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.881571669$ |
2.665622171 |
\( \frac{14003310699}{30976} a + \frac{5238465399}{30976} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -130 a + 152\) , \( -27 a + 1015\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-130a+152\right){x}-27a+1015$ |
*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.