| 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 |
| 3969.3-a1 |
3969.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{12} \cdot 7^{4} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.383631390$ |
$2.439445643$ |
2.646977654 |
\( -\frac{1728}{49} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -2 a + 1\) , \( -6 a + 3\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2a+1\right){x}-6a+3$ |
| 3969.3-a2 |
3969.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{12} \cdot 7^{2} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.767262781$ |
$2.439445643$ |
2.646977654 |
\( \frac{1259712}{7} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -14 a - 5\) , \( 30 a - 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-14a-5\right){x}+30a-6$ |
| 3969.3-b1 |
3969.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{12} \cdot 7^{4} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.383631390$ |
$2.439445643$ |
2.646977654 |
\( -\frac{1728}{49} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( a + 1\) , \( 5 a + 3\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(a+1\right){x}+5a+3$ |
| 3969.3-b2 |
3969.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{12} \cdot 7^{2} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.767262781$ |
$2.439445643$ |
2.646977654 |
\( \frac{1259712}{7} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 13 a - 5\) , \( -31 a - 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(13a-5\right){x}-31a-6$ |
| 3969.3-c1 |
3969.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{14} \cdot 7^{16} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.287358976$ |
1.625547847 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -306\) , \( 5859\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-306{x}+5859$ |
| 3969.3-c2 |
3969.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{2} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.298871812$ |
1.625547847 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 9\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+9{x}$ |
| 3969.3-c3 |
3969.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{20} \cdot 7^{4} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.149435906$ |
1.625547847 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -36\) , \( 27\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-36{x}+27$ |
| 3969.3-c4 |
3969.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{28} \cdot 7^{2} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.574717953$ |
1.625547847 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -351\) , \( -2430\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-351{x}-2430$ |
| 3969.3-c5 |
3969.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{8} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.574717953$ |
1.625547847 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -441\) , \( 3672\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-441{x}+3672$ |
| 3969.3-c6 |
3969.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
3969.3 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{14} \cdot 7^{4} \) |
$2.00611$ |
$(-a-1), (a-1), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$0.287358976$ |
1.625547847 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -7056\) , \( 229905\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-7056{x}+229905$ |
*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.
