| 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 |
| 64.1-a3 |
64.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \) |
$0.71487$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$17.94635162$ |
0.793124183 |
\( 8000 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4 a - 4\) , \( -4 a - 6\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-4\right){x}-4a-6$ |
| 64.1-a4 |
64.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \) |
$0.71487$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$35.89270325$ |
0.793124183 |
\( 8000 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a - 4\) , \( 4 a + 6\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-4\right){x}+4a+6$ |
| 256.1-c3 |
256.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{12} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$25.37997386$ |
1.121646976 |
\( 8000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -1\) , \( a\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}-{x}+a$ |
| 256.1-c4 |
256.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{12} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$25.37997386$ |
1.121646976 |
\( 8000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -1\) , \( -a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}-{x}-a$ |
| 1024.1-d3 |
1024.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{18} \) |
$1.42974$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.960251595$ |
$12.68998693$ |
2.154126597 |
\( 8000 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -3\) , \( -1\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-3{x}-1$ |
| 1024.1-d4 |
1024.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{18} \) |
$1.42974$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.480125797$ |
$25.37997386$ |
2.154126597 |
\( 8000 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-3{x}+1$ |
| 1024.1-m3 |
1024.1-m |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{18} \) |
$1.42974$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.94635162$ |
1.586248366 |
\( 8000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a - 9\) , \( -5 a - 7\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a-9\right){x}-5a-7$ |
| 1024.1-m4 |
1024.1-m |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{18} \) |
$1.42974$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.94635162$ |
1.586248366 |
\( 8000 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -6 a - 9\) , \( 5 a + 7\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a-9\right){x}+5a+7$ |
| 3136.2-a3 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.868344182$ |
$9.592728448$ |
2.945025478 |
\( 8000 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -10 a - 18\) , \( -20 a - 24\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-10a-18\right){x}-20a-24$ |
| 3136.2-a4 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.434172091$ |
$9.592728448$ |
2.945025478 |
\( 8000 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -10 a - 18\) , \( 20 a + 24\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-10a-18\right){x}+20a+24$ |
| 3136.3-a3 |
3136.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.3 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$1.89137$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.868344182$ |
$9.592728448$ |
2.945025478 |
\( 8000 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 10 a - 18\) , \( 20 a - 24\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(10a-18\right){x}+20a-24$ |
| 3136.3-a4 |
3136.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.3 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$1.89137$ |
$(a), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.434172091$ |
$9.592728448$ |
2.945025478 |
\( 8000 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 10 a - 18\) , \( -20 a + 24\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(10a-18\right){x}-20a+24$ |
*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.
