| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{25} \cdot 13^{3} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.981452668$ |
0.942818116 |
\( -\frac{141266017}{1384448} a + \frac{873216369}{692224} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -34 a + 109\) , \( 44 a - 139\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-34a+109\right){x}+44a-139$ |
| 338.1-b1 |
338.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{14} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \cdot 7^{2} \) |
$1$ |
$0.385597965$ |
5.974902374 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -855\) , \( -10907\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-855{x}-10907$ |
| 338.1-b2 |
338.1-b |
$2$ |
$7$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{26} \cdot 13^{2} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$1$ |
$18.89430030$ |
5.974902374 |
\( -\frac{2146689}{1664} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -15\) , \( 13\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-15{x}+13$ |
| 338.1-c1 |
338.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{30} \cdot 13^{2} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$2.977948821$ |
$0.265819283$ |
4.505844684 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -1839\) , \( -32477\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-1839{x}-32477$ |
| 338.1-c2 |
338.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{6} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.992649607$ |
$2.392373550$ |
4.505844684 |
\( -\frac{10218313}{17576} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -19\) , \( -81\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-19{x}-81$ |
| 338.1-c3 |
338.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.330883202$ |
$21.53136195$ |
4.505844684 |
\( \frac{12167}{26} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 1\) , \( 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+{x}+3$ |
| 338.1-d1 |
338.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{25} \cdot 13^{3} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.981452668$ |
0.942818116 |
\( \frac{141266017}{1384448} a + \frac{873216369}{692224} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -217 a + 703\) , \( 3030 a - 9566\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-217a+703\right){x}+3030a-9566$ |
| 338.1-e1 |
338.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{37} \cdot 13^{3} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 5^{2} \) |
$0.108452395$ |
$2.981452668$ |
5.112544171 |
\( \frac{141266017}{1384448} a + \frac{873216369}{692224} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 136 a + 436\) , \( -352 a - 1112\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(136a+436\right){x}-352a-1112$ |
| 338.1-f1 |
338.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$0.265819283$ |
0.756534943 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-460{x}-3830$ |
| 338.1-f2 |
338.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.392373550$ |
0.756534943 |
\( -\frac{10218313}{17576} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -5\) , \( -8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-5{x}-8$ |
| 338.1-f3 |
338.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$21.53136195$ |
0.756534943 |
\( \frac{12167}{26} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}$ |
| 338.1-g1 |
338.1-g |
$2$ |
$7$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 2 \) |
$7.165071991$ |
$0.385597965$ |
1.747371659 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -213\) , \( -1257\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$ |
| 338.1-g2 |
338.1-g |
$2$ |
$7$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 2 \cdot 7 \) |
$1.023581713$ |
$18.89430030$ |
1.747371659 |
\( -\frac{2146689}{1664} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 338.1-h1 |
338.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
338.1 |
\( 2 \cdot 13^{2} \) |
\( 2^{37} \cdot 13^{3} \) |
$2.42325$ |
$(2,a), (13,a+6), (13,a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 5^{2} \) |
$0.108452395$ |
$2.981452668$ |
5.112544171 |
\( -\frac{141266017}{1384448} a + \frac{873216369}{692224} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 877 a + 2771\) , \( -22323 a - 70593\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(877a+2771\right){x}-22323a-70593$ |
*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.
