| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$11$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.65012$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$31$ |
31Ns.7.1 |
$1$ |
\( 1 \) |
$1$ |
$23.06325055$ |
1.248945040 |
\( -32768 \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 10 a - 97\) , \( 57 a - 555\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(10a-97\right){x}+57a-555$ |
| 1.1-a2 |
1.1-a |
$2$ |
$11$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.65012$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$31$ |
31Ns.7.1 |
$1$ |
\( 1 \) |
$1$ |
$23.06325055$ |
1.248945040 |
\( -32768 \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -10 a - 87\) , \( -57 a - 498\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-10a-87\right){x}-57a-498$ |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$2.46751$ |
$(a-10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.371768946$ |
$48.62641779$ |
3.915869319 |
\( -\frac{77824}{25} a - \frac{679936}{25} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( 6 a - 20\) , \( -236 a + 2314\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-20\right){x}-236a+2314$ |
| 5.1-b1 |
5.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$2.46751$ |
$(a-10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$2.129507344$ |
$7.470843813$ |
3.446129560 |
\( -\frac{77824}{25} a - \frac{679936}{25} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( -4 a - 15\) , \( -34 a - 329\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-15\right){x}-34a-329$ |
| 5.2-a1 |
5.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$2.46751$ |
$(a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.371768946$ |
$48.62641779$ |
3.915869319 |
\( \frac{77824}{25} a - \frac{151552}{5} \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( -4 a - 15\) , \( 240 a + 2094\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-15\right){x}+240a+2094$ |
| 5.2-b1 |
5.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 5^{2} \) |
$2.46751$ |
$(a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$2.129507344$ |
$7.470843813$ |
3.446129560 |
\( \frac{77824}{25} a - \frac{151552}{5} \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( 6 a - 20\) , \( 28 a - 342\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a-20\right){x}+28a-342$ |
| 11.1-a1 |
11.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$3.00513$ |
$(-3a+29)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
|
\( 2 \) |
$1$ |
$0.064435690$ |
4.998604420 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -7820\) , \( -263580\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-7820{x}-263580$ |
| 11.1-a2 |
11.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{10} \) |
$3.00513$ |
$(-3a+29)$ |
$0 \le r \le 1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.1.1 |
|
\( 2 \cdot 5 \) |
$1$ |
$1.610892258$ |
4.998604420 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20$ |
| 11.1-a3 |
11.1-a |
$3$ |
$25$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$3.00513$ |
$(-3a+29)$ |
$0 \le r \le 1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
|
\( 2 \) |
$1$ |
$40.27230645$ |
4.998604420 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}$ |
| 11.1-b1 |
11.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$3.00513$ |
$(-3a+29)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$8.512583687$ |
0.921964503 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 32493485 a - 316262100\) , \( -306370187085 a + 2981929418029\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(32493485a-316262100\right){x}-306370187085a+2981929418029$ |
| 11.1-b2 |
11.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{10} \) |
$3.00513$ |
$(-3a+29)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.512583687$ |
0.921964503 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -42935 a - 374955\) , \( 26666635 a + 232882194\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-42935a-374955\right){x}+26666635a+232882194$ |
| 11.1-b3 |
11.1-b |
$3$ |
$25$ |
\(\Q(\sqrt{341}) \) |
$2$ |
$[2, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$3.00513$ |
$(-3a+29)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$8.512583687$ |
0.921964503 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -1385 a - 12095\) , \( -202015 a - 1764216\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-1385a-12095\right){x}-202015a-1764216$ |
*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.
