| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.29185$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$11$ |
11Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.56070946$ |
1.214699673 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 12880 a - 99542\) , \( -2141323 a + 16549035\bigr] \) |
${y}^2+{y}={x}^{3}+\left(12880a-99542\right){x}-2141323a+16549035$ |
| 1.1-a2 |
1.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.29185$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$11$ |
11Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$17.56070946$ |
1.214699673 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -12880 a - 86662\) , \( 2141323 a + 14407712\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-12880a-86662\right){x}+2141323a+14407712$ |
| 25.2-a1 |
25.2-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$2.88866$ |
$(-4a+31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$7.853388020$ |
0.543230208 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 43199424 a - 333863126\) , \( 415934860788 a - 3214517694381\bigr] \) |
${y}^2+{y}={x}^{3}+\left(43199424a-333863126\right){x}+415934860788a-3214517694381$ |
| 25.2-a2 |
25.2-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$2.88866$ |
$(-4a+31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$7.853388020$ |
0.543230208 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -96 a - 646\) , \( -1378 a - 9272\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-96a-646\right){x}-1378a-9272$ |
| 25.3-a1 |
25.3-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{6} \) |
$2.88866$ |
$(-4a-27)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$7.853388020$ |
0.543230208 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -43199424 a - 290663702\) , \( -415934860788 a - 2798582833593\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-43199424a-290663702\right){x}-415934860788a-2798582833593$ |
| 25.3-a2 |
25.3-a |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{6} \) |
$2.88866$ |
$(-4a-27)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$7.853388020$ |
0.543230208 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 96 a - 742\) , \( 1378 a - 10650\bigr] \) |
${y}^2+{y}={x}^{3}+\left(96a-742\right){x}+1378a-10650$ |
| 64.6-d1 |
64.6-d |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$0.583694721$ |
$54.12574167$ |
4.370654518 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 325670798 a - 2516919454\) , \( -8609473917464 a + 66537597244333\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(325670798a-2516919454\right){x}-8609473917464a+66537597244333$ |
| 64.6-d2 |
64.6-d |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$11.09019970$ |
$2.848723246$ |
4.370654518 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -2 a - 14\) , \( -5 a - 42\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-2a-14\right){x}-5a-42$ |
| 64.7-d1 |
64.7-d |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$0.583694721$ |
$54.12574167$ |
4.370654518 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -325670798 a - 2191248656\) , \( 8609473917463 a + 57928123326870\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-325670798a-2191248656\right){x}+8609473917463a+57928123326870$ |
| 64.7-d2 |
64.7-d |
$2$ |
$19$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11Ns.2.1 |
$1$ |
\( 1 \) |
$11.09019970$ |
$2.848723246$ |
4.370654518 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 2 a - 16\) , \( 4 a - 46\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(2a-16\right){x}+4a-46$ |
*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.
