| 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 |
| 27104.13-a1 |
27104.13-a |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{12} \cdot 7 \cdot 11^{2} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.136882134$ |
$3.595509661$ |
2.976310195 |
\( \frac{151031}{112} a + \frac{895283}{112} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -3 a + 4\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-3a+4\right){x}-a$ |
| 27104.13-a2 |
27104.13-a |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{20} \cdot 7^{3} \cdot 11^{2} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.410646403$ |
$1.198503220$ |
2.976310195 |
\( -\frac{24989591975}{200704} a + \frac{110583852637}{200704} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -48 a + 99\) , \( 183 a + 296\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-48a+99\right){x}+183a+296$ |
| 27104.13-b1 |
27104.13-b |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{10} \cdot 7^{5} \cdot 11^{8} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$0.693813856$ |
1.048947953 |
\( \frac{680543}{1372} a - \frac{362437}{1372} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 53 a - 15\) , \( -257 a - 25\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(53a-15\right){x}-257a-25$ |
| 27104.13-c1 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{48} \cdot 7^{2} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{6} \) |
$4.166277711$ |
$0.131974098$ |
3.325124285 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 2217 a - 8014\) , \( 107002 a - 256732\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2217a-8014\right){x}+107002a-256732$ |
| 27104.13-c2 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{27} \cdot 7^{3} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$2.777518474$ |
$0.395922296$ |
3.325124285 |
\( \frac{10538337875}{200704} a - \frac{36575498625}{200704} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 262 a + 547\) , \( 5209 a - 7769\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(262a+547\right){x}+5209a-7769$ |
| 27104.13-c3 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{27} \cdot 7^{3} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$2.777518474$ |
$0.395922296$ |
3.325124285 |
\( -\frac{10538337875}{200704} a - \frac{13018580375}{100352} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -550 a + 499\) , \( 1911 a - 8645\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-550a+499\right){x}+1911a-8645$ |
| 27104.13-c4 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{17} \cdot 7 \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.925839491$ |
$1.187766888$ |
3.325124285 |
\( \frac{831875}{112} a - \frac{499125}{56} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -15 a + 54\) , \( 86 a + 38\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a+54\right){x}+86a+38$ |
| 27104.13-c5 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{17} \cdot 7 \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.925839491$ |
$1.187766888$ |
3.325124285 |
\( -\frac{831875}{112} a - \frac{166375}{112} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -13 a + 52\) , \( -66 a - 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-13a+52\right){x}-66a-16$ |
| 27104.13-c6 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{16} \cdot 7^{2} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{6} \) |
$0.462919745$ |
$1.187766888$ |
3.325124285 |
\( -\frac{15625}{28} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 7 a - 24\) , \( -38 a + 100\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(7a-24\right){x}-38a+100$ |
| 27104.13-c7 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{24} \cdot 7^{6} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Cs |
$1$ |
\( 2^{6} \) |
$1.388759237$ |
$0.395922296$ |
3.325124285 |
\( \frac{9938375}{21952} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -58 a + 211\) , \( 777 a - 1953\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-58a+211\right){x}+777a-1953$ |
| 27104.13-c8 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{57} \cdot 7 \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$8.332555422$ |
$0.131974098$ |
3.325124285 |
\( \frac{70135314719125}{481036337152} a + \frac{288417990127625}{481036337152} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -903 a - 1198\) , \( 23634 a - 40428\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-903a-1198\right){x}+23634a-40428$ |
| 27104.13-c9 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{57} \cdot 7 \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$8.332555422$ |
$0.131974098$ |
3.325124285 |
\( -\frac{70135314719125}{481036337152} a + \frac{179276652423375}{240518168576} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 1525 a - 1046\) , \( 11576 a - 46864\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1525a-1046\right){x}+11576a-46864$ |
| 27104.13-c10 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{18} \cdot 7^{12} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \) |
$2.777518474$ |
$0.197961148$ |
3.325124285 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 462 a - 1669\) , \( 8257 a - 19465\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(462a-1669\right){x}+8257a-19465$ |
| 27104.13-c11 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{14} \cdot 7^{4} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.925839491$ |
$0.593883444$ |
3.325124285 |
\( \frac{128787625}{98} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 137 a - 494\) , \( -1668 a + 4206\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(137a-494\right){x}-1668a+4206$ |
| 27104.13-c12 |
27104.13-c |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{30} \cdot 7^{4} \cdot 11^{6} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$8.332555422$ |
$0.065987049$ |
3.325124285 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 35497 a - 128334\) , \( 6857722 a - 16580828\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(35497a-128334\right){x}+6857722a-16580828$ |
| 27104.13-d1 |
27104.13-d |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{28} \cdot 7^{3} \cdot 11^{8} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$0.227832765$ |
$0.326594068$ |
6.749734578 |
\( \frac{58211797249}{51380224} a + \frac{140999388325}{51380224} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -363 a + 397\) , \( -a - 3433\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-363a+397\right){x}-a-3433$ |
| 27104.13-e1 |
27104.13-e |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{14} \cdot 7^{3} \cdot 11^{10} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.504539270$ |
4.576750071 |
\( -\frac{1896471}{3136} a - \frac{1181651}{3136} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 95 a - 133\) , \( 811 a - 161\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(95a-133\right){x}+811a-161$ |
| 27104.13-e2 |
27104.13-e |
$2$ |
$3$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{10} \cdot 7^{9} \cdot 11^{10} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$9$ |
\( 2^{2} \) |
$1$ |
$0.168179756$ |
4.576750071 |
\( -\frac{3507393849}{67228} a + \frac{26103442243}{67228} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 715 a + 4327\) , \( 90931 a - 84873\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(715a+4327\right){x}+90931a-84873$ |
| 27104.13-f1 |
27104.13-f |
$1$ |
$1$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{22} \cdot 7 \cdot 11^{8} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$0.236904872$ |
$0.670889151$ |
7.208700609 |
\( \frac{386167}{7168} a + \frac{2566579}{7168} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -51 a + 21\) , \( 143 a + 239\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-51a+21\right){x}+143a+239$ |
| 27104.13-g1 |
27104.13-g |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{8} \cdot 7^{4} \cdot 11^{9} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.651481602$ |
3.939790407 |
\( -\frac{4807755}{784} a - \frac{11276631}{784} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -71 a - 128\) , \( 467 a + 485\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-71a-128\right){x}+467a+485$ |
| 27104.13-g2 |
27104.13-g |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
27104.13 |
\( 2^{5} \cdot 7 \cdot 11^{2} \) |
\( 2^{16} \cdot 7^{2} \cdot 11^{9} \) |
$3.03351$ |
$(a), (-a+1), (-2a+1), (-2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.651481602$ |
3.939790407 |
\( -\frac{783675}{1792} a - \frac{238599}{1792} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -41 a - 41\) , \( 211 a + 275\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-41a-41\right){x}+211a+275$ |
*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.
