| 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-a4 |
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 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 214 a + 705\) , \( -2687 a - 8802\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(214a+705\right){x}-2687a-8802$ |
| 4.1-d4 |
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 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 238 a - 1017\) , \( 6020 a - 25743\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-1017\right){x}+6020a-25743$ |
| 36.1-b4 |
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 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B.1.1, 7B.6.1 |
$1$ |
\( 3 \cdot 7 \) |
$1$ |
$5.406911526$ |
1.671046829 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 4 a - 3\) , \( -4 a + 31\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(4a-3\right){x}-4a+31$ |
| 36.1-f4 |
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 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 7$ |
3B.1.2, 7B.6.1 |
$1$ |
\( 7 \) |
$1$ |
$1.802303842$ |
1.671046829 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 193879 a + 634939\) , \( -82120563 a - 268938047\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(193879a+634939\right){x}-82120563a-268938047$ |
| 128.5-c4 |
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 |
$1$ |
\( 2 \cdot 3 \cdot 7 \) |
$1$ |
$1.103681193$ |
6.139818101 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 8426 a + 27590\) , \( -733568 a - 2402377\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(8426a+27590\right){x}-733568a-2402377$ |
| 128.5-j4 |
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 |
$1$ |
\( 2^{2} \) |
$0.471574970$ |
$3.311043580$ |
1.654505450 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 392 a - 1649\) , \( -11804 a + 50489\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(392a-1649\right){x}-11804a+50489$ |
| 128.6-c4 |
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 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$3.311043580$ |
6.139818101 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 353 a + 1153\) , \( 7663 a + 25095\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(353a+1153\right){x}+7663a+25095$ |
| 128.6-j4 |
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 |
$1$ |
\( 2^{2} \) |
$1.414724910$ |
$1.103681193$ |
1.654505450 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 9410 a - 40226\) , \( 1409836 a - 6026932\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(9410a-40226\right){x}+1409836a-6026932$ |
| 256.1-a4 |
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 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 3825 a - 16342\) , \( -365194 a + 1561180\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(3825a-16342\right){x}-365194a+1561180$ |
| 256.1-r4 |
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 |
\( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 3425 a + 11218\) , \( 186594 a + 611080\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(3425a+11218\right){x}+186594a+611080$ |
*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.
