| 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 |
| 81225.3-a1 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{38} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$8.055414916$ |
$0.074031481$ |
2.754442525 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 5281 a + 1650\) , \( 142859 a - 409439\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(5281a+1650\right){x}+142859a-409439$ |
| 81225.3-a2 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.503463432$ |
$1.184503703$ |
2.754442525 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( a\) , \( -31 a + 91\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+a{x}-31a+91$ |
| 81225.3-a3 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{10} \cdot 5^{16} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.027707458$ |
$0.148062962$ |
2.754442525 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -1679 a - 525\) , \( 6248 a - 20216\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-1679a-525\right){x}+6248a-20216$ |
| 81225.3-a4 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{14} \cdot 5^{8} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.013853729$ |
$0.296125925$ |
2.754442525 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 481 a + 150\) , \( 1199 a - 2909\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(481a+150\right){x}+1199a-2909$ |
| 81225.3-a5 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{10} \cdot 5^{4} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.006926864$ |
$0.592251851$ |
2.754442525 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 241 a + 75\) , \( -592 a + 2014\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(241a+75\right){x}-592a+2014$ |
| 81225.3-a6 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{22} \cdot 5^{4} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.027707458$ |
$0.148062962$ |
2.754442525 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 6481 a + 2025\) , \( 104774 a - 297134\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(6481a+2025\right){x}+104774a-297134$ |
| 81225.3-a7 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.013853729$ |
$0.296125925$ |
2.754442525 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 3841 a + 1200\) , \( -44287 a + 133549\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(3841a+1200\right){x}-44287a+133549$ |
| 81225.3-a8 |
81225.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{14} \cdot 5^{2} \cdot 19^{6} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$8.055414916$ |
$0.074031481$ |
2.754442525 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 103681 a + 32400\) , \( 6545489 a - 18926729\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(103681a+32400\right){x}+6545489a-18926729$ |
| 81225.3-b1 |
81225.3-b |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 19^{2} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \) |
$0.161093105$ |
$2.876843090$ |
2.140535770 |
\( \frac{331776}{5} a - 110592 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -15 a + 6\) , \( 17 a - 19\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-15a+6\right){x}+17a-19$ |
| 81225.3-b2 |
81225.3-b |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{14} \cdot 19^{2} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \) |
$1.127651741$ |
$0.410977584$ |
2.140535770 |
\( \frac{285311102976}{78125} a + \frac{6827839488}{15625} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 1185 a - 1044\) , \( 16460 a - 5878\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(1185a-1044\right){x}+16460a-5878$ |
| 81225.3-c1 |
81225.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{24} \cdot 5^{2} \cdot 19^{8} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$4.699222060$ |
$0.166671616$ |
3.617570269 |
\( -\frac{2953216}{32805} a + \frac{151552}{19683} \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( 208 a - 725\) , \( -27005 a + 30253\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(208a-725\right){x}-27005a+30253$ |
| 81225.3-d1 |
81225.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{7} \cdot 5^{2} \cdot 19^{3} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.528294236$ |
$2.696018643$ |
3.289259426 |
\( \frac{1621}{15} a - \frac{272}{15} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -2 a - 1\) , \( 4 a - 7\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-2a-1\right){x}+4a-7$ |
| 81225.3-d2 |
81225.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{4} \cdot 19^{3} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.264147118$ |
$1.348009321$ |
3.289259426 |
\( -\frac{15794551}{75} a + \frac{14868361}{75} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -77 a + 29\) , \( 184 a - 193\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-77a+29\right){x}+184a-193$ |
| 81225.3-e1 |
81225.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{8} \cdot 19^{8} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.790705359$ |
$0.192047993$ |
4.249507442 |
\( -\frac{1361807016381}{225625} a - \frac{229736641527}{45125} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 2985 a - 6648\) , \( -125748 a + 193419\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(2985a-6648\right){x}-125748a+193419$ |
| 81225.3-e2 |
81225.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{4} \cdot 19^{10} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.395352679$ |
$0.384095986$ |
4.249507442 |
\( \frac{10985870511}{3258025} a - \frac{5990903856}{3258025} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 210 a - 408\) , \( -1737 a + 3321\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(210a-408\right){x}-1737a+3321$ |
| 81225.3-e3 |
81225.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 19^{8} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.197676339$ |
$0.768191972$ |
4.249507442 |
\( -\frac{19435059}{1805} a - \frac{10233}{361} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 135 a - 93\) , \( 492 a + 24\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(135a-93\right){x}+492a+24$ |
| 81225.3-e4 |
81225.3-e |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
81225.3 |
\( 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 19^{14} \) |
$2.61289$ |
$(-2a+1), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.790705359$ |
$0.192047993$ |
4.249507442 |
\( -\frac{147104989379271}{84917815205} a + \frac{9324631881531}{16983563041} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1365 a + 792\) , \( -19482 a + 20451\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-1365a+792\right){x}-19482a+20451$ |
*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.
