| 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 |
| 32.1-a5 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$13.75037163$ |
0.607686314 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-2{x}-3$ |
| 32.1-a6 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$55.00148654$ |
0.607686314 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -3\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}$ |
| 256.1-a5 |
256.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{18} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$13.75037163$ |
1.215372628 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 22 a - 33\) , \( 70 a - 98\bigr] \) |
${y}^2={x}^{3}+\left(22a-33\right){x}+70a-98$ |
| 256.1-a6 |
256.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{18} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$13.75037163$ |
1.215372628 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 22 a - 33\) , \( -70 a + 98\bigr] \) |
${y}^2={x}^{3}+\left(22a-33\right){x}-70a+98$ |
| 1024.1-f5 |
1024.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.42974$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$4.861490513$ |
1.718796454 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -44 a - 66\) , \( -196 a - 280\bigr] \) |
${y}^2={x}^{3}+\left(-44a-66\right){x}-196a-280$ |
| 1024.1-f6 |
1024.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.42974$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$19.44596205$ |
1.718796454 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -44 a - 66\) , \( 196 a + 280\bigr] \) |
${y}^2={x}^{3}+\left(-44a-66\right){x}+196a+280$ |
| 1024.1-k5 |
1024.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.42974$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.217418063$ |
$9.722981027$ |
2.092493851 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -22\) , \( -28 a\bigr] \) |
${y}^2={x}^{3}-22{x}-28a$ |
| 1024.1-k6 |
1024.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.42974$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.217418063$ |
$9.722981027$ |
2.092493851 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -22\) , \( 28 a\bigr] \) |
${y}^2={x}^{3}-22{x}+28a$ |
| 1568.2-i5 |
1568.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.2 |
\( 2^{5} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$1.59045$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.39430393$ |
1.837470700 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 11 a - 24\) , \( 44 a - 56\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(11a-24\right){x}+44a-56$ |
| 1568.2-i6 |
1568.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.2 |
\( 2^{5} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$1.59045$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.39430393$ |
1.837470700 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 11 a - 25\) , \( -33 a + 31\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(11a-25\right){x}-33a+31$ |
| 1568.3-i5 |
1568.3-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.3 |
\( 2^{5} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$1.59045$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.39430393$ |
1.837470700 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -11 a - 24\) , \( -44 a - 56\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-11a-24\right){x}-44a-56$ |
| 1568.3-i6 |
1568.3-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.3 |
\( 2^{5} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$1.59045$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$10.39430393$ |
1.837470700 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -11 a - 25\) , \( 33 a + 31\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-11a-25\right){x}+33a+31$ |
| 2592.1-d5 |
2592.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2592.1 |
\( 2^{5} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{12} \) |
$1.80340$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.444312937$ |
$18.33382884$ |
2.880030840 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -24\) , \( 35\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-24{x}+35$ |
| 2592.1-d6 |
2592.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2592.1 |
\( 2^{5} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{12} \) |
$1.80340$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.777251749$ |
$4.583457212$ |
2.880030840 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -25\) , \( -60\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-25{x}-60$ |
*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.
