| 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-a1 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{9} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$3.437592909$ |
0.607686314 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 15 a - 22\) , \( 46 a - 69\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(15a-22\right){x}+46a-69$ |
| 32.1-a2 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{9} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$27.50074327$ |
0.607686314 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 15 a - 23\) , \( -31 a + 46\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(15a-23\right){x}-31a+46$ |
| 256.1-a1 |
256.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{21} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$6.875185818$ |
1.215372628 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 33\) , \( -154 a - 154\bigr] \) |
${y}^2={x}^{3}+\left(-2a-33\right){x}-154a-154$ |
| 256.1-a2 |
256.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{21} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$3.437592909$ |
1.215372628 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 33\) , \( 154 a + 154\bigr] \) |
${y}^2={x}^{3}+\left(-2a-33\right){x}+154a+154$ |
| 1024.1-f1 |
1024.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( - 2^{27} \) |
$1.42974$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$9.722981027$ |
1.718796454 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 a - 66\) , \( 308 a + 616\bigr] \) |
${y}^2={x}^{3}+\left(-4a-66\right){x}+308a+616$ |
| 1024.1-f2 |
1024.1-f |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( - 2^{27} \) |
$1.42974$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$4$ |
\( 2^{2} \) |
$1$ |
$1.215372628$ |
1.718796454 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 a - 66\) , \( -308 a - 616\bigr] \) |
${y}^2={x}^{3}+\left(-4a-66\right){x}-308a-616$ |
| 1024.1-k1 |
1024.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( - 2^{27} \) |
$1.42974$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$2.434836127$ |
$2.430745256$ |
2.092493851 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 120 a - 182\) , \( 924 a - 1232\bigr] \) |
${y}^2={x}^{3}+\left(120a-182\right){x}+924a-1232$ |
| 1024.1-k2 |
1024.1-k |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( - 2^{27} \) |
$1.42974$ |
$(a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$2.434836127$ |
$4.861490513$ |
2.092493851 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 120 a - 182\) , \( -924 a + 1232\bigr] \) |
${y}^2={x}^{3}+\left(120a-182\right){x}-924a+1232$ |
| 1568.2-i1 |
1568.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.2 |
\( 2^{5} \cdot 7^{2} \) |
\( - 2^{9} \cdot 7^{6} \) |
$1.59045$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$4$ |
\( 2 \) |
$1$ |
$2.598575984$ |
1.837470700 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -55 a - 97\) , \( 1397 a + 1934\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-55a-97\right){x}+1397a+1934$ |
| 1568.2-i2 |
1568.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.2 |
\( 2^{5} \cdot 7^{2} \) |
\( - 2^{9} \cdot 7^{6} \) |
$1.59045$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.197151969$ |
1.837470700 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -55 a - 96\) , \( -1452 a - 2031\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-55a-96\right){x}-1452a-2031$ |
| 1568.3-i1 |
1568.3-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.3 |
\( 2^{5} \cdot 7^{2} \) |
\( - 2^{9} \cdot 7^{6} \) |
$1.59045$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.197151969$ |
1.837470700 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 44 a - 84\) , \( -286 a + 208\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(44a-84\right){x}-286a+208$ |
| 1568.3-i2 |
1568.3-i |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1568.3 |
\( 2^{5} \cdot 7^{2} \) |
\( - 2^{9} \cdot 7^{6} \) |
$1.59045$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$2.598575984$ |
1.837470700 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 44 a - 85\) , \( 330 a - 293\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(44a-85\right){x}+330a-293$ |
| 2592.1-d1 |
2592.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2592.1 |
\( 2^{5} \cdot 3^{4} \) |
\( - 2^{9} \cdot 3^{12} \) |
$1.80340$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$3.554503499$ |
$1.145864303$ |
2.880030840 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 135 a - 205\) , \( 1107 a - 1662\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(135a-205\right){x}+1107a-1662$ |
| 2592.1-d2 |
2592.1-d |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2592.1 |
\( 2^{5} \cdot 3^{4} \) |
\( - 2^{9} \cdot 3^{12} \) |
$1.80340$ |
$(a), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.222156468$ |
$9.166914424$ |
2.880030840 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 135 a - 204\) , \( -972 a + 1457\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(135a-204\right){x}-972a+1457$ |
*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.
