| 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 |
| 124.1-a1 |
124.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
124.1 |
\( 2^{2} \cdot 31 \) |
\( 2^{50} \cdot 31 \) |
$0.51648$ |
$(-6a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$0.368431786$ |
0.425428381 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 1300 a - 550\) , \( -9800 a - 7280\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(1300a-550\right){x}-9800a-7280$ |
| 7936.1-c1 |
7936.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
7936.1 |
\( 2^{8} \cdot 31 \) |
\( 2^{74} \cdot 31 \) |
$1.46083$ |
$(-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$2.882361124$ |
$0.092107946$ |
2.452476457 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 20800 a - 8792\) , \( 651200 a + 424324\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(20800a-8792\right){x}+651200a+424324$ |
| 20956.1-c3 |
20956.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.1 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{50} \cdot 13^{6} \cdot 31 \) |
$1.86220$ |
$(-4a+1), (-6a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 5^{2} \) |
$1$ |
$0.102184592$ |
2.949815085 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 15654 a - 13496\) , \( -770131 a + 246841\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15654a-13496\right){x}-770131a+246841$ |
| 20956.5-b3 |
20956.5-b |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
20956.5 |
\( 2^{2} \cdot 13^{2} \cdot 31 \) |
\( 2^{50} \cdot 13^{6} \cdot 31 \) |
$1.86220$ |
$(4a-3), (-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.278764670$ |
$0.102184592$ |
3.289216924 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -15105 a + 14246\) , \( 65278 a - 710141\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-15105a+14246\right){x}+65278a-710141$ |
| 34596.1-d3 |
34596.1-d |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
34596.1 |
\( 2^{2} \cdot 3^{2} \cdot 31^{2} \) |
\( 2^{50} \cdot 3^{6} \cdot 31^{7} \) |
$2.11084$ |
$(-2a+1), (-6a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$0.038204596$ |
2.205743407 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 35903 a - 118365\) , \( -6646970 a + 15108733\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(35903a-118365\right){x}-6646970a+15108733$ |
| 54684.1-d3 |
54684.1-d |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.1 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{50} \cdot 3^{6} \cdot 7^{6} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (-6a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 5^{2} \) |
$1$ |
$0.080398407$ |
2.320902097 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 24444 a - 1488\) , \( 133343 a + 1345693\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(24444a-1488\right){x}+133343a+1345693$ |
| 54684.5-d3 |
54684.5-d |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.5 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{50} \cdot 3^{6} \cdot 7^{6} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (3a-2), (-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.420856605$ |
$0.080398407$ |
3.907067914 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 26253 a - 19941\) , \( 1517257 a - 275523\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(26253a-19941\right){x}+1517257a-275523$ |
| 71424.1-a3 |
71424.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.1 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{74} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$7.009610537$ |
$0.053178547$ |
3.443417772 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 26376 a + 36024\) , \( -4499544 a + 5180172\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(26376a+36024\right){x}-4499544a+5180172$ |
| 71424.1-h3 |
71424.1-h |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
71424.1 |
\( 2^{8} \cdot 3^{2} \cdot 31 \) |
\( 2^{74} \cdot 3^{6} \cdot 31 \) |
$2.53023$ |
$(-2a+1), (-6a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$25$ |
\( 2 \) |
$1$ |
$0.053178547$ |
3.070264883 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 36025 a - 62399\) , \( 4473169 a - 5216196\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(36025a-62399\right){x}+4473169a-5216196$ |
| 77500.1-b3 |
77500.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
77500.1 |
\( 2^{2} \cdot 5^{4} \cdot 31 \) |
\( 2^{50} \cdot 5^{12} \cdot 31 \) |
$2.58241$ |
$(-6a+1), (2), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.1.3 |
$25$ |
\( 1 \) |
$1$ |
$0.073686357$ |
2.127141908 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -13738 a - 18762\) , \( -1281257 a - 812508\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-13738a-18762\right){x}-1281257a-812508$ |
| 126976.1-c3 |
126976.1-c |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
126976.1 |
\( 2^{12} \cdot 31 \) |
\( 2^{86} \cdot 31 \) |
$2.92166$ |
$(-6a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$8.763271822$ |
$0.046053973$ |
3.728144549 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 83199 a - 35167\) , \( 5257632 a + 3311393\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(83199a-35167\right){x}+5257632a+3311393$ |
| 126976.1-h3 |
126976.1-h |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
126976.1 |
\( 2^{12} \cdot 31 \) |
\( 2^{86} \cdot 31 \) |
$2.92166$ |
$(-6a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$25$ |
\( 2 \) |
$1$ |
$0.046053973$ |
2.658927385 |
\( -\frac{936087656892551}{1040187392} a + \frac{51401239062153}{520093696} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -48032 a + 83199\) , \( -5257632 a - 3311393\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-48032a+83199\right){x}-5257632a-3311393$ |
*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.
