| 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 |
| 338.1-a1 |
338.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$2.54152$ |
$(a+3), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \cdot 7 \) |
$0.170407109$ |
$3.254622356$ |
2.341100226 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -208\) , \( 1046\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-208{x}+1046$ |
| 338.1-a2 |
338.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.54152$ |
$(a+3), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$1.192849768$ |
$3.254622356$ |
2.341100226 |
\( -\frac{2146689}{1664} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 2\) , \( -4\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+2{x}-4$ |
| 338.1-b1 |
338.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$2.54152$ |
$(a+3), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.3 |
$4$ |
\( 2 \cdot 7 \) |
$1$ |
$0.385597965$ |
3.255340507 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -213\) , \( -1257\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$ |
| 338.1-b2 |
338.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.54152$ |
$(a+3), (13)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.1 |
$4$ |
\( 2 \cdot 7 \) |
$1$ |
$18.89430030$ |
3.255340507 |
\( -\frac{2146689}{1664} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 338.1-c1 |
338.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$2.54152$ |
$(a+3), (13)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$8.323604404$ |
$0.265819283$ |
1.334232659 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-460{x}-3830$ |
| 338.1-c2 |
338.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$2.54152$ |
$(a+3), (13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.774534801$ |
$2.392373550$ |
1.334232659 |
\( -\frac{10218313}{17576} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -5\) , \( -8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-5{x}-8$ |
| 338.1-c3 |
338.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$2.54152$ |
$(a+3), (13)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$0.924844933$ |
$21.53136195$ |
1.334232659 |
\( \frac{12167}{26} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}$ |
| 338.1-d1 |
338.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$2.54152$ |
$(a+3), (13)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.043225854$ |
$12.10583107$ |
5.679947919 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -457\) , \( 3371\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-457{x}+3371$ |
| 338.1-d2 |
338.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$2.54152$ |
$(a+3), (13)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.043225854$ |
$12.10583107$ |
5.679947919 |
\( -\frac{10218313}{17576} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -2\) , \( 4\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-2{x}+4$ |
| 338.1-d3 |
338.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{11}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$2.54152$ |
$(a+3), (13)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.389032689$ |
$12.10583107$ |
5.679947919 |
\( \frac{12167}{26} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 3\) , \( 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+3{x}+1$ |
*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.
