| 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 |
| 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 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 33\) , \( 154 a + 154\bigr] \) |
${y}^2={x}^{3}+\left(-2a-33\right){x}+154a+154$ |
| 256.1-a3 |
256.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{12} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$13.75037163$ |
1.215372628 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2 a + 3\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(2a+3\right){x}$ |
| 256.1-a4 |
256.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{12} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$27.50074327$ |
1.215372628 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 3\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-2a-3\right){x}$ |
| 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 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 22 a - 33\) , \( -70 a + 98\bigr] \) |
${y}^2={x}^{3}+\left(22a-33\right){x}-70a+98$ |
| 256.1-a7 |
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-a8 |
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 |
\( 29071392966 a + 41113158120 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2 a - 33\) , \( -154 a + 154\bigr] \) |
${y}^2={x}^{3}+\left(2a-33\right){x}-154a+154$ |
| 256.1-b1 |
256.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.547952572$ |
1.334302111 |
\( 128 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( -1\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}-1$ |
| 256.1-b2 |
256.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{19} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.547952572$ |
1.334302111 |
\( -36872164 a + 52151080 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 10 a - 22\) , \( 36 a - 46\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(10a-22\right){x}+36a-46$ |
| 256.1-b3 |
256.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{14} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$15.09590514$ |
1.334302111 |
\( 10976 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -2\) , \( -2\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-2{x}-2$ |
| 256.1-b4 |
256.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{19} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.547952572$ |
1.334302111 |
\( 36872164 a + 52151080 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -10 a - 22\) , \( -36 a - 46\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-10a-22\right){x}-36a-46$ |
| 256.1-c1 |
256.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{18} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$6.344993467$ |
1.121646976 |
\( -18473000 a + 26125000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 10 a - 11\) , \( 23 a - 30\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(10a-11\right){x}+23a-30$ |
| 256.1-c2 |
256.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{18} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$12.68998693$ |
1.121646976 |
\( -18473000 a + 26125000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 10 a - 11\) , \( -23 a + 30\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(10a-11\right){x}-23a+30$ |
| 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 |
\( 8000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -1\) , \( -a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}-{x}-a$ |
| 256.1-c5 |
256.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{18} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$6.344993467$ |
1.121646976 |
\( 18473000 a + 26125000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -10 a - 11\) , \( -23 a - 30\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-10a-11\right){x}-23a-30$ |
| 256.1-c6 |
256.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( - 2^{18} \) |
$1.01098$ |
$(a)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$12.68998693$ |
1.121646976 |
\( 18473000 a + 26125000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -10 a - 11\) , \( 23 a + 30\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-10a-11\right){x}+23a+30$ |
| 256.1-d1 |
256.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.01098$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.108082791$ |
$16.29302268$ |
1.245211766 |
\( 128 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 1\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+{x}+1$ |
| 256.1-d2 |
256.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{19} \) |
$1.01098$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.108082791$ |
$16.29302268$ |
1.245211766 |
\( -36872164 a + 52151080 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 10 a - 22\) , \( -36 a + 46\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(10a-22\right){x}-36a+46$ |
| 256.1-d3 |
256.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{14} \) |
$1.01098$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.216165582$ |
$32.58604536$ |
1.245211766 |
\( 10976 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -2\) , \( 2\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-2{x}+2$ |
| 256.1-d4 |
256.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{19} \) |
$1.01098$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.108082791$ |
$16.29302268$ |
1.245211766 |
\( 36872164 a + 52151080 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -10 a - 22\) , \( 36 a + 46\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-10a-22\right){x}+36a+46$ |
*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.
