| 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 |
| 676.1-a1 |
676.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$12.10583107$ |
2.641708916 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 2297 a - 6434\) , \( -89617 a + 250159\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(2297a-6434\right){x}-89617a+250159$ |
| 676.1-a2 |
676.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$12.10583107$ |
2.641708916 |
\( -\frac{10218313}{17576} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 22 a - 64\) , \( -164 a + 455\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(22a-64\right){x}-164a+455$ |
| 676.1-a3 |
676.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$12.10583107$ |
2.641708916 |
\( \frac{12167}{26} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -3 a + 6\) , \( 3 a - 11\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-3a+6\right){x}+3a-11$ |
| 676.1-b1 |
676.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.246468705$ |
$11.25615983$ |
3.061695412 |
\( -\frac{132651}{208} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -3 a - 2\) , \( -3 a - 4\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a-2\right){x}-3a-4$ |
| 676.1-b2 |
676.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{4} \) |
$2.08802$ |
$(2), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.623234352$ |
$11.25615983$ |
3.061695412 |
\( \frac{1033364331}{676} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -23 a - 42\) , \( -87 a - 156\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-23a-42\right){x}-87a-156$ |
| 676.1-c1 |
676.1-c |
$2$ |
$7$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.4 |
$1$ |
\( 7 \) |
$1$ |
$3.254622356$ |
4.971517769 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 1062 a - 2975\) , \( -30162 a + 84202\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1062a-2975\right){x}-30162a+84202$ |
| 676.1-c2 |
676.1-c |
$2$ |
$7$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.6 |
$1$ |
\( 7 \) |
$1$ |
$3.254622356$ |
4.971517769 |
\( -\frac{2146689}{1664} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 12 a - 35\) , \( 78 a - 218\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(12a-35\right){x}+78a-218$ |
| 676.1-d1 |
676.1-d |
$2$ |
$7$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{14} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 7 \) |
$1$ |
$0.385597965$ |
0.589010621 |
\( -\frac{1064019559329}{125497034} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -213\) , \( -1257\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$ |
| 676.1-d2 |
676.1-d |
$2$ |
$7$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{14} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$1$ |
$18.89430030$ |
0.589010621 |
\( -\frac{2146689}{1664} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 676.1-e1 |
676.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.246468705$ |
$11.25615983$ |
3.061695412 |
\( -\frac{132651}{208} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( a - 4\) , \( 2 a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-4\right){x}+2a-6$ |
| 676.1-e2 |
676.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 13^{4} \) |
$2.08802$ |
$(2), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.623234352$ |
$11.25615983$ |
3.061695412 |
\( \frac{1033364331}{676} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 21 a - 64\) , \( 86 a - 242\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(21a-64\right){x}+86a-242$ |
| 676.1-f1 |
676.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{18} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$0.265819283$ |
0.522058708 |
\( -\frac{10730978619193}{6656} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-460{x}-3830$ |
| 676.1-f2 |
676.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$2.392373550$ |
0.522058708 |
\( -\frac{10218313}{17576} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -5\) , \( -8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-5{x}-8$ |
| 676.1-f3 |
676.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
676.1 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 13^{2} \) |
$2.08802$ |
$(2), (13)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$21.53136195$ |
0.522058708 |
\( \frac{12167}{26} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}$ |
*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.
