| 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 |
| 81225.1-a1 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{38} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$8.055414916$ |
$0.074031481$ |
2.754442525 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -1651 a - 5281\) , \( -142860 a - 266580\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-1651a-5281\right){x}-142860a-266580$ |
| 81225.1-a2 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.503463432$ |
$1.184503703$ |
2.754442525 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -a - 1\) , \( 30 a + 60\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-a-1\right){x}+30a+60$ |
| 81225.1-a3 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{10} \cdot 5^{16} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.027707458$ |
$0.148062962$ |
2.754442525 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 524 a + 1679\) , \( -6249 a - 13968\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(524a+1679\right){x}-6249a-13968$ |
| 81225.1-a4 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{14} \cdot 5^{8} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.013853729$ |
$0.296125925$ |
2.754442525 |
\( \frac{111284641}{50625} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -151 a - 481\) , \( -1200 a - 1710\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-151a-481\right){x}-1200a-1710$ |
| 81225.1-a5 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{10} \cdot 5^{4} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.006926864$ |
$0.592251851$ |
2.754442525 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -76 a - 241\) , \( 591 a + 1422\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-76a-241\right){x}+591a+1422$ |
| 81225.1-a6 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{22} \cdot 5^{4} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.027707458$ |
$0.148062962$ |
2.754442525 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -2026 a - 6481\) , \( -104775 a - 192360\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-2026a-6481\right){x}-104775a-192360$ |
| 81225.1-a7 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.013853729$ |
$0.296125925$ |
2.754442525 |
\( \frac{56667352321}{15} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -1201 a - 3841\) , \( 44286 a + 89262\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-1201a-3841\right){x}+44286a+89262$ |
| 81225.1-a8 |
81225.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{14} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$8.055414916$ |
$0.074031481$ |
2.754442525 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -32401 a - 103681\) , \( -6545490 a - 12381240\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-32401a-103681\right){x}-6545490a-12381240$ |
| 81225.1-b1 |
81225.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 19^{2} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \) |
$0.161093105$ |
$2.876843090$ |
2.140535770 |
\( -\frac{331776}{5} a - \frac{221184}{5} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -6 a + 15\) , \( -18 a - 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-6a+15\right){x}-18a-2$ |
| 81225.1-b2 |
81225.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{14} \cdot 19^{2} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \) |
$1.127651741$ |
$0.410977584$ |
2.140535770 |
\( -\frac{285311102976}{78125} a + \frac{319450300416}{78125} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 1044 a - 1185\) , \( -16461 a + 10582\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(1044a-1185\right){x}-16461a+10582$ |
| 81225.1-c1 |
81225.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{24} \cdot 5^{2} \cdot 19^{8} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$4.699222060$ |
$0.166671616$ |
3.617570269 |
\( \frac{2953216}{32805} a - \frac{8101888}{98415} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( -517 a + 725\) , \( 27004 a + 3248\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-517a+725\right){x}+27004a+3248$ |
| 81225.1-d1 |
81225.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{4} \cdot 19^{3} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.264147118$ |
$1.348009321$ |
3.289259426 |
\( \frac{15794551}{75} a - \frac{61746}{5} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -30 a + 77\) , \( -184 a - 9\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-30a+77\right){x}-184a-9$ |
| 81225.1-d2 |
81225.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{7} \cdot 5^{2} \cdot 19^{3} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.528294236$ |
$2.696018643$ |
3.289259426 |
\( -\frac{1621}{15} a + \frac{1349}{15} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 2\) , \( -4 a - 3\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+2{x}-4a-3$ |
| 81225.1-e1 |
81225.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{8} \cdot 19^{8} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.790705359$ |
$0.192047993$ |
4.249507442 |
\( \frac{1361807016381}{225625} a - \frac{2510490224016}{225625} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -3663 a + 6648\) , \( 125748 a + 67671\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-3663a+6648\right){x}+125748a+67671$ |
| 81225.1-e2 |
81225.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 19^{8} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.197676339$ |
$0.768191972$ |
4.249507442 |
\( \frac{19435059}{1805} a - \frac{19486224}{1805} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 42 a + 93\) , \( -492 a + 516\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(42a+93\right){x}-492a+516$ |
| 81225.1-e3 |
81225.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 19^{14} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.790705359$ |
$0.192047993$ |
4.249507442 |
\( \frac{147104989379271}{84917815205} a - \frac{100481829971616}{84917815205} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -573 a - 792\) , \( 19482 a + 969\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-573a-792\right){x}+19482a+969$ |
| 81225.1-e4 |
81225.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.1 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{4} \cdot 19^{10} \) |
$2.61289$ |
$(-2a+1), (-5a+3), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.395352679$ |
$0.384095986$ |
4.249507442 |
\( -\frac{10985870511}{3258025} a + \frac{998993331}{651605} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -198 a + 408\) , \( 1737 a + 1584\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-198a+408\right){x}+1737a+1584$ |
*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.
