| 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 |
| 25.1-a1 |
25.1-a |
$1$ |
$1$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5Nn |
$1$ |
\( 1 \) |
$1$ |
$260.0488896$ |
2.612402493 |
\( 256753581255500 a^{3} + 130257445927750 a^{2} - 1474438598926750 a - 1518279740390875 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a + 1\) , \( a\) , \( -4 a^{3} - a^{2} + 2 a - 2\) , \( -17 a^{3} - 13 a^{2} + 14 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a^{3}-a^{2}+2a-2\right){x}-17a^{3}-13a^{2}+14a-1$ |
| 25.1-b1 |
25.1-b |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.070938334$ |
$2779.685780$ |
2.641195305 |
\( 4036500 a^{3} - 7580142 a^{2} - 9865638 a + 6254145 \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 4 a - 1\) , \( a^{2} - a - 2\) , \( 5 a^{3} - 2 a^{2} - 20 a - 6\) , \( -a^{3} - a^{2} + 7 a + 12\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(5a^{3}-2a^{2}-20a-6\right){x}-a^{3}-a^{2}+7a+12$ |
| 25.1-b2 |
25.1-b |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.212815003$ |
$308.8539756$ |
2.641195305 |
\( 906201 a^{3} + 1751706 a^{2} - 179496 a - 477441 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a - 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 2 a^{3} - a\) , \( -58 a^{3} + 114 a^{2} + 145 a - 95\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a^{3}-a\right){x}-58a^{3}+114a^{2}+145a-95$ |
| 25.1-c1 |
25.1-c |
$1$ |
$1$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5Nn |
$4$ |
\( 1 \) |
$1$ |
$73.05431961$ |
2.935560108 |
\( 256753581255500 a^{3} + 130257445927750 a^{2} - 1474438598926750 a - 1518279740390875 \) |
\( \bigl[1\) , \( -a + 1\) , \( a^{2} - a - 2\) , \( -360 a^{3} + 434 a^{2} + 1633 a - 894\) , \( 3368 a^{3} - 4065 a^{2} - 15295 a + 8363\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-360a^{3}+434a^{2}+1633a-894\right){x}+3368a^{3}-4065a^{2}-15295a+8363$ |
| 25.1-d1 |
25.1-d |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.280250781$ |
$769.6351248$ |
2.889053175 |
\( 906201 a^{3} + 1751706 a^{2} - 179496 a - 477441 \) |
\( \bigl[a^{3} - 4 a - 2\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 3\) , \( -5 a^{3} + 3 a^{2} + 25 a + 4\) , \( 5 a^{3} - 10 a^{2} - 19 a + 31\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-5a^{3}+3a^{2}+25a+4\right){x}+5a^{3}-10a^{2}-19a+31$ |
| 25.1-d2 |
25.1-d |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.840752345$ |
$85.51501386$ |
2.889053175 |
\( 4036500 a^{3} - 7580142 a^{2} - 9865638 a + 6254145 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 2 a^{3} + 4 a^{2} - 6 a - 13\) , \( 2 a^{3} + 7 a^{2} - 6 a - 23\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(2a^{3}+4a^{2}-6a-13\right){x}+2a^{3}+7a^{2}-6a-23$ |
| 25.1-e1 |
25.1-e |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.446692615$ |
$89.94453277$ |
1.614464947 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a\) , \( a + 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( -5 a^{3} + 6 a^{2} + 21 a - 12\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-a^{3}+2a^{2}+3a-1\right){x}-5a^{3}+6a^{2}+21a-12$ |
| 25.1-e2 |
25.1-e |
$2$ |
$3$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$13.30136$ |
$(-a^3+2a^2+3a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.148897538$ |
$809.5007950$ |
1.614464947 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( -2 a^{3} + 4 a^{2} + 5 a - 6\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-a^{3}+2a^{2}+3a-1\right){x}-2a^{3}+4a^{2}+5a-6$ |
*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.
