| 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 |
| 784.1-a1 |
784.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 1 \) |
$1$ |
$9.180288820$ |
2.003303258 |
\( -5784929280 a + 16147406592 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 456 a - 1279\) , \( -8181 a + 22834\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(456a-1279\right){x}-8181a+22834$ |
| 784.1-a2 |
784.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$1$ |
\( 1 \) |
$1$ |
$9.180288820$ |
2.003303258 |
\( 5784929280 a + 10362477312 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -454 a - 824\) , \( 8636 a + 15477\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-454a-824\right){x}+8636a+15477$ |
| 784.1-b1 |
784.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$4$ |
\( 1 \) |
$1$ |
$1.887498018$ |
1.647543341 |
\( -5784929280 a + 16147406592 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -34 a - 152\) , \( -100 a - 539\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-34a-152\right){x}-100a-539$ |
| 784.1-b2 |
784.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{8} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3$ |
2Cn, 3B |
$4$ |
\( 1 \) |
$1$ |
$1.887498018$ |
1.647543341 |
\( 5784929280 a + 10362477312 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 36 a - 187\) , \( 135 a - 826\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(36a-187\right){x}+135a-826$ |
| 784.1-c1 |
784.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$17.19445586$ |
3.752137883 |
\( 1792 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 12 a - 31\) , \( 13 a - 37\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(12a-31\right){x}+13a-37$ |
| 784.1-c2 |
784.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$1.910495096$ |
3.752137883 |
\( 406749952 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -712 a - 1279\) , \( -16113 a - 28864\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-712a-1279\right){x}-16113a-28864$ |
| 784.1-d1 |
784.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$2.16684$ |
$(a+3), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 3^{2} \) |
$0.019390460$ |
$19.01761041$ |
2.896922752 |
\( 1792 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -2\) , \( 1\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-2{x}+1$ |
| 784.1-d2 |
784.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$2.16684$ |
$(a+3), (2)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.174514141$ |
$19.01761041$ |
2.896922752 |
\( 406749952 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -142\) , \( 701\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-142{x}+701$ |
| 784.1-e1 |
784.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{16} \cdot 7^{4} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$5.827272664$ |
1.271615146 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 2\) , \( 12 a + 21\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}+12a+21$ |
| 784.1-e2 |
784.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
784.1 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{16} \cdot 7^{4} \) |
$2.16684$ |
$(a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$5.827272664$ |
1.271615146 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 2\) , \( -12 a + 35\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}-12a+35$ |
*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.
