| 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 |
| 25.2-a1 |
25.2-a |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{9} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$37.43514264$ |
1.538574885 |
\( 7028736 a^{2} - 17440800 a + 4747984 \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( 1\) , \( 58786385 a^{2} + 68785155 a - 27089419\) , \( 1275216929162 a^{2} + 1492114096329 a - 587634150538\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(58786385a^{2}+68785155a-27089419\right){x}+1275216929162a^{2}+1492114096329a-587634150538$ |
| 25.2-a2 |
25.2-a |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{9} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$37.43514264$ |
1.538574885 |
\( 4096 a \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( -3046 a^{2} - 3557 a + 1402\) , \( 147974 a^{2} + 173152 a - 68191\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3046a^{2}-3557a+1402\right){x}+147974a^{2}+173152a-68191$ |
| 25.2-a3 |
25.2-a |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{9} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$37.43514264$ |
1.538574885 |
\( 491380736 a^{2} + 574509056 a - 227098624 \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -4 a^{2} + a\) , \( 95 a^{2} - 265 a + 73\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a^{2}+a\right){x}+95a^{2}-265a+73$ |
| 25.2-a4 |
25.2-a |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{9} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$37.43514264$ |
1.538574885 |
\( -194645795136 a^{2} + 134089931040 a + 625653842512 \) |
\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - a - 1\) , \( -41518 a^{2} - 48578 a + 19134\) , \( -8607796 a^{2} - 10071865 a + 3966568\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-41518a^{2}-48578a+19134\right){x}-8607796a^{2}-10071865a+3966568$ |
| 25.2-b1 |
25.2-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{3} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$271.5184057$ |
0.446373509 |
\( 7028736 a^{2} - 17440800 a + 4747984 \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - a - 3\) , \( a^{2} - a - 1\) , \( 10713 a^{2} + 12536 a - 4937\) , \( 3135860 a^{2} + 3669228 a - 1445040\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(10713a^{2}+12536a-4937\right){x}+3135860a^{2}+3669228a-1445040$ |
| 25.2-b2 |
25.2-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{3} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$10.86073623$ |
0.446373509 |
\( 491380736 a^{2} + 574509056 a - 227098624 \) |
\( \bigl[0\) , \( a^{2} - 2 a - 3\) , \( 1\) , \( -377381 a^{2} - 441567 a + 173905\) , \( -232144315 a^{2} - 271628927 a + 106974682\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-377381a^{2}-441567a+173905\right){x}-232144315a^{2}-271628927a+106974682$ |
| 25.2-b3 |
25.2-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{3} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$271.5184057$ |
0.446373509 |
\( 4096 a \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 2\) , \( a\) , \( -a^{2} - a + 4\) , \( -a^{2} + a + 2\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-a^{2}-a+4\right){x}-a^{2}+a+2$ |
| 25.2-b4 |
25.2-b |
$4$ |
$10$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{3} \) |
$1.85891$ |
$(a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$10.86073623$ |
0.446373509 |
\( -194645795136 a^{2} + 134089931040 a + 625653842512 \) |
\( \bigl[a + 1\) , \( a^{2} - 2 a - 1\) , \( a^{2} - 2\) , \( -5312850765 a^{2} - 6216494888 a + 2448220752\) , \( -394174470602711 a^{2} - 461218221425205 a + 181639982107606\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-2a-1\right){x}^{2}+\left(-5312850765a^{2}-6216494888a+2448220752\right){x}-394174470602711a^{2}-461218221425205a+181639982107606$ |
*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.
