| 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 |
| 4.1-a3 |
4.1-a |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 1 \) |
$1$ |
$7.662151750$ |
1.014876791 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -215 a + 920\) , \( 2686 a - 11488\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-215a+920\right){x}+2686a-11488$ |
| 4.1-d3 |
4.1-d |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{28} \) |
$0.95409$ |
$(a-4), (a+3)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 7$ |
3B, 7B.1.1 |
$1$ |
\( 3 \cdot 7^{2} \) |
$1$ |
$3.815467665$ |
1.516113114 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -239 a - 778\) , \( -6021 a - 19722\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-239a-778\right){x}-6021a-19722$ |
| 36.1-b3 |
36.1-b |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{6} \) |
$1.65254$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B.1.2, 7B.6.1 |
$1$ |
\( 7 \) |
$1$ |
$1.802303842$ |
1.671046829 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -193880 a + 828819\) , \( 82120562 a - 351058609\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-193880a+828819\right){x}+82120562a-351058609$ |
| 36.1-f3 |
36.1-f |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{6} \) |
$1.65254$ |
$(a-4), (a+3), (4a+13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B.1.1, 7B.6.1 |
$1$ |
\( 3 \cdot 7 \) |
$1$ |
$5.406911526$ |
1.671046829 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -5 a + 1\) , \( 3 a + 27\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5a+1\right){x}+3a+27$ |
| 128.5-c3 |
128.5-c |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{46} \) |
$2.26923$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$3.311043580$ |
6.139818101 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -345 a + 1491\) , \( -6165 a + 26374\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-345a+1491\right){x}-6165a+26374$ |
| 128.5-j3 |
128.5-j |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{46} \) |
$2.26923$ |
$(a-4), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 2^{2} \) |
$1.414724910$ |
$1.103681193$ |
1.654505450 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -9410 a - 30816\) , \( -1409836 a - 4617096\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-9410a-30816\right){x}-1409836a-4617096$ |
| 128.6-c3 |
128.6-c |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{46} \) |
$2.26923$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 2 \cdot 3 \cdot 7 \) |
$1$ |
$1.103681193$ |
6.139818101 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -8428 a + 36016\) , \( 733567 a - 3135945\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-8428a+36016\right){x}+733567a-3135945$ |
| 128.6-j3 |
128.6-j |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{46} \) |
$2.26923$ |
$(a-4), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 2^{2} \) |
$0.471574970$ |
$3.311043580$ |
1.654505450 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -382 a - 1250\) , \( 10156 a + 33260\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-382a-1250\right){x}+10156a+33260$ |
| 256.1-a3 |
256.1-a |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{52} \) |
$2.69858$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 2^{4} \) |
$1$ |
$1.915537937$ |
4.059507167 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -3823 a - 12518\) , \( 369018 a + 1208504\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3823a-12518\right){x}+369018a+1208504$ |
| 256.1-r3 |
256.1-r |
$4$ |
$21$ |
\(\Q(\sqrt{57}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{52} \) |
$2.69858$ |
$(a-4), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B, 7B.6.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.953866916$ |
0.505371038 |
\( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -3423 a + 14642\) , \( -190018 a + 812316\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-3423a+14642\right){x}-190018a+812316$ |
*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.
