| 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 |
| 1296.1-a1 |
1296.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{12} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$49$ |
\( 1 \) |
$1$ |
$0.364059351$ |
3.892768911 |
\( -5646360960 a - 10114260048 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -765 a - 1386\) , \( -17544 a - 31458\bigr] \) |
${y}^2={x}^{3}+\left(-765a-1386\right){x}-17544a-31458$ |
| 1296.1-a2 |
1296.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{12} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$49$ |
\( 1 \) |
$1$ |
$0.364059351$ |
3.892768911 |
\( 5646360960 a - 15760621008 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 765 a - 2151\) , \( 17544 a - 49002\bigr] \) |
${y}^2={x}^{3}+\left(765a-2151\right){x}+17544a-49002$ |
| 1296.1-b1 |
1296.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$1.604161811$ |
1.050170418 |
\( -5646360960 a - 10114260048 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -48 a - 111\) , \( -336 a - 538\bigr] \) |
${y}^2={x}^{3}+\left(-48a-111\right){x}-336a-538$ |
| 1296.1-b2 |
1296.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$4.812485434$ |
1.050170418 |
\( 5646360960 a - 15760621008 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -123 a - 231\) , \( -1536 a - 2738\bigr] \) |
${y}^2={x}^{3}+\left(-123a-231\right){x}-1536a-2738$ |
| 1296.1-c1 |
1296.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{12} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$3.715723863$ |
0.810837422 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 48 a + 84\bigr] \) |
${y}^2={x}^{3}+48a+84$ |
| 1296.1-c2 |
1296.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{12} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$3.715723863$ |
0.810837422 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -48 a + 132\bigr] \) |
${y}^2={x}^{3}-48a+132$ |
| 1296.1-d1 |
1296.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 7$ |
3B.1.2, 7Ns.3.1 |
$1$ |
\( 3 \) |
$1$ |
$3.715723863$ |
2.432512266 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -96 a - 172\bigr] \) |
${y}^2={x}^{3}-96a-172$ |
| 1296.1-d2 |
1296.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 7$ |
3B.1.1, 7Ns.3.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$11.14717159$ |
2.432512266 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 4\bigr] \) |
${y}^2={x}^{3}+4$ |
| 1296.1-e1 |
1296.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{12} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$7.068449444$ |
1.542462125 |
\( -5646360960 a - 10114260048 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 45 a - 279\) , \( -600 a + 2406\bigr] \) |
${y}^2={x}^{3}+\left(45a-279\right){x}-600a+2406$ |
| 1296.1-e2 |
1296.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{12} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$7.068449444$ |
1.542462125 |
\( 5646360960 a - 15760621008 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -45 a - 234\) , \( 600 a + 1806\bigr] \) |
${y}^2={x}^{3}+\left(-45a-234\right){x}+600a+1806$ |
| 1296.1-f1 |
1296.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$4.812485434$ |
1.050170418 |
\( -5646360960 a - 10114260048 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 123 a - 354\) , \( 1536 a - 4274\bigr] \) |
${y}^2={x}^{3}+\left(123a-354\right){x}+1536a-4274$ |
| 1296.1-f2 |
1296.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{16} \cdot 3^{6} \) |
$2.45697$ |
$(-a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$1.604161811$ |
1.050170418 |
\( 5646360960 a - 15760621008 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 48 a - 159\) , \( 336 a - 874\bigr] \) |
${y}^2={x}^{3}+\left(48a-159\right){x}+336a-874$ |
*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.
