| 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-a4 |
1.1-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$4.57742$ |
$\textsf{none}$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$869.9086354$ |
0.471725361 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 1\) , \( a + 1\) , \( a\) , \( a + 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}$ |
| 49.3-a4 |
49.3-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$7.44552$ |
$(-a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.019181815$ |
$937.6524294$ |
1.404460983 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 1\) , \( -a\) , \( a^{2} - 2 a - 1\) , \( -a^{3} + 4 a^{2} - 2 a - 3\) , \( a^{3} - 3 a^{2} + 1\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+1\right){x}{y}+\left(a^{2}-2a-1\right){y}={x}^{3}-a{x}^{2}+\left(-a^{3}+4a^{2}-2a-3\right){x}+a^{3}-3a^{2}+1$ |
| 49.4-a2 |
49.4-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( 7^{6} \) |
$7.44552$ |
$(-a^3+3a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.038363631$ |
$937.6524294$ |
1.404460983 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a\) , \( 3 a^{3} - a^{2} - 9 a - 1\) , \( 3 a^{3} - 7 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(3a^{3}-a^{2}-9a-1\right){x}+3a^{3}-7a-2$ |
| 256.1-e4 |
256.1-e |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$9.15483$ |
$(a^3-2a^2-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$316.8820943$ |
1.546520898 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[0\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( 0\) , \( -2 a^{3} + 5 a^{2} + 4 a - 6\) , \( -5 a^{3} + 12 a^{2} + 11 a - 14\bigr] \) |
${y}^2={x}^{3}+\left(a^{3}-2a^{2}-2a+2\right){x}^{2}+\left(-2a^{3}+5a^{2}+4a-6\right){x}-5a^{3}+12a^{2}+11a-14$ |
| 256.1-j1 |
256.1-j |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$9.15483$ |
$(a^3-2a^2-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$316.8820943$ |
1.546520898 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 2 a - 2\) , \( 0\) , \( -2 a^{3} + 5 a^{2} + 4 a - 6\) , \( 5 a^{3} - 12 a^{2} - 11 a + 14\bigr] \) |
${y}^2={x}^{3}+\left(-a^{3}+2a^{2}+2a-2\right){x}^{2}+\left(-2a^{3}+5a^{2}+4a-6\right){x}+5a^{3}-12a^{2}-11a+14$ |
| 256.1-n2 |
256.1-n |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$9.15483$ |
$(a^3-2a^2-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$461.7232551$ |
2.253408053 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( a^{2} - 2 a - 1\) , \( a^{3} - 2 a^{2} - a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(a^{2}-2a-1\right){x}+a^{3}-2a^{2}-a$ |
| 289.3-a2 |
289.3-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
289.3 |
\( 17^{2} \) |
\( 17^{6} \) |
$9.29464$ |
$(a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$307.4207872$ |
3.000691301 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 4 a - 1\) , \( a\) , \( -3 a^{3} + 9 a^{2} + 7 a - 7\) , \( 4 a^{3} - 7 a^{2} - 6 a + 7\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-4a-1\right){x}^{2}+\left(-3a^{3}+9a^{2}+7a-7\right){x}+4a^{3}-7a^{2}-6a+7$ |
| 289.4-a4 |
289.4-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
289.4 |
\( 17^{2} \) |
\( 17^{6} \) |
$9.29464$ |
$(-a^3+3a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$307.4207872$ |
3.000691301 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 1\) , \( -a - 1\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} - 6 a - 4\) , \( a^{3} - a^{2} - 4 a - 2\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+1\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a^{3}-6a-4\right){x}+a^{3}-a^{2}-4a-2$ |
| 625.2-a4 |
625.2-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
625.2 |
\( 5^{4} \) |
\( 5^{12} \) |
$10.23542$ |
$(a^3-a^2-5a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$96.42847788$ |
0.941224884 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 1\) , \( -a^{2} + 3 a + 1\) , \( a\) , \( 5 a^{3} - 2 a^{2} - 27 a - 17\) , \( 25 a^{3} - 27 a^{2} - 103 a - 42\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+3a+1\right){x}^{2}+\left(5a^{3}-2a^{2}-27a-17\right){x}+25a^{3}-27a^{2}-103a-42$ |
| 625.3-a3 |
625.3-a |
$4$ |
$6$ |
4.4.2624.1 |
$4$ |
$[4, 0]$ |
625.3 |
\( 5^{4} \) |
\( 5^{12} \) |
$10.23542$ |
$(a^3-a^2-3a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$96.42847788$ |
0.941224884 |
\( 67584 a^{3} - 160832 a^{2} - 143360 a + 192256 \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + a - 1\) , \( a\) , \( -4 a^{3} + 11 a^{2} - a - 2\) , \( 6 a^{3} - 20 a^{2} + 6 a + 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+a-1\right){x}^{2}+\left(-4a^{3}+11a^{2}-a-2\right){x}+6a^{3}-20a^{2}+6a+5$ |
*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.
