| 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 |
| 336.1-a1 |
336.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{12} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$0.533054337$ |
2.093795872 |
\( -\frac{10061824000}{352947} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -113\) , \( -516\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-113{x}-516$ |
| 336.1-a2 |
336.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{8} \cdot 3^{6} \cdot 7^{4} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$4.797489038$ |
2.093795872 |
\( \frac{2048000}{1323} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 7\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+7{x}$ |
| 336.1-a3 |
336.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{16} \cdot 3^{12} \cdot 7^{2} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$4.797489038$ |
2.093795872 |
\( \frac{9826000}{5103} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -28\) , \( -28\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-28{x}-28$ |
| 336.1-a4 |
336.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{6} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$0.533054337$ |
2.093795872 |
\( \frac{2640279346000}{3087} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -1828\) , \( -30700\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-1828{x}-30700$ |
| 336.1-b1 |
336.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{4} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.782729173$ |
2.476377538 |
\( -\frac{16384}{147} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -1\) , \( -2\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-{x}-2$ |
| 336.1-b2 |
336.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{2} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.782729173$ |
2.476377538 |
\( \frac{20720464}{63} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -36\) , \( -72\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-36{x}-72$ |
| 336.1-c1 |
336.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{4} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.097774690$ |
$15.45742511$ |
1.978815036 |
\( -\frac{16384}{147} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 7 a - 17\) , \( -54 a + 150\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(7a-17\right){x}-54a+150$ |
| 336.1-c2 |
336.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{2} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.048887345$ |
$15.45742511$ |
1.978815036 |
\( \frac{20720464}{63} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 182 a - 507\) , \( -1909 a + 5330\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(182a-507\right){x}-1909a+5330$ |
| 336.1-d1 |
336.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{12} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.499476405$ |
$5.711517702$ |
2.490100344 |
\( -\frac{10061824000}{352947} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 567 a - 1585\) , \( -11818 a + 32988\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(567a-1585\right){x}-11818a+32988$ |
| 336.1-d2 |
336.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{8} \cdot 3^{6} \cdot 7^{4} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.166492135$ |
$5.711517702$ |
2.490100344 |
\( \frac{2048000}{1323} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -33 a + 95\) , \( -34 a + 96\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-33a+95\right){x}-34a+96$ |
| 336.1-d3 |
336.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{16} \cdot 3^{12} \cdot 7^{2} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.332984270$ |
$5.711517702$ |
2.490100344 |
\( \frac{9826000}{5103} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 142 a - 395\) , \( -531 a + 1482\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(142a-395\right){x}-531a+1482$ |
| 336.1-d4 |
336.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
336.1 |
\( 2^{4} \cdot 3 \cdot 7 \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{6} \) |
$1.75321$ |
$(-a+2), (a+3), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.998952810$ |
$5.711517702$ |
2.490100344 |
\( \frac{2640279346000}{3087} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 9142 a - 25595\) , \( -727659 a + 2031306\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(9142a-25595\right){x}-727659a+2031306$ |
*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.
