| 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 |
| 25.3-a1 |
25.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$10.14732862$ |
2.071314782 |
\( -1835626496 a - 4496347136 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -19 a - 12\) , \( 30 a + 117\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-19a-12\right){x}+30a+117$ |
| 25.3-a2 |
25.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{2} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$10.14732862$ |
2.071314782 |
\( 118784 a - 290816 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( a - 2\) , \( -3\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a-2\right){x}-3$ |
| 25.3-b1 |
25.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.99242927$ |
2.447944375 |
\( \frac{213248}{625} a + \frac{84032}{625} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( a + 2\) , \( 5\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a+2\right){x}+5$ |
| 25.3-b2 |
25.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$23.98485855$ |
2.447944375 |
\( -\frac{8704256}{25} a + \frac{21394496}{25} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -76 a - 188\) , \( 346 a + 847\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-76a-188\right){x}+346a+847$ |
| 25.3-c1 |
25.3-c |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$1.175470154$ |
0.239941840 |
\( -1835626496 a - 4496347136 \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -685 a + 1678\) , \( 11772 a - 28836\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-685a+1678\right){x}+11772a-28836$ |
| 25.3-c2 |
25.3-c |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{2} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$29.38675386$ |
0.239941840 |
\( 118784 a - 290816 \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 3 a + 8\) , \( -13 a - 32\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3a+8\right){x}-13a-32$ |
| 25.3-d1 |
25.3-d |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$7.346204439$ |
1.499537701 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 2\) , \( 192 a - 470\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+2{x}+192a-470$ |
| 25.3-d2 |
25.3-d |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$7.346204439$ |
1.499537701 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 2\) , \( -2 a - 5\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+2{x}-2a-5$ |
| 25.3-e1 |
25.3-e |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.313341678$ |
0.880457183 |
\( \frac{213248}{625} a + \frac{84032}{625} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 55 a + 136\) , \( -302 a - 740\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(55a+136\right){x}-302a-740$ |
| 25.3-e2 |
25.3-e |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{8} \) |
$0.97888$ |
$(-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.626683357$ |
0.880457183 |
\( -\frac{8704256}{25} a + \frac{21394496}{25} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 2 a - 10\) , \( a - 10\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a-10\right){x}+a-10$ |
*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.
