| 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 |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$27.22522842$ |
1.964811619 |
\( -512 a + 512 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-a+2\right){x}$ |
| 256.1-a2 |
256.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$27.22522842$ |
1.964811619 |
\( 249872 a + 434912 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a - 8\) , \( -4 a + 8\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(4a-8\right){x}-4a+8$ |
| 256.1-b1 |
256.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$11.09117728$ |
0.800436773 |
\( -512 a + 512 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-a+2\right){x}$ |
| 256.1-b2 |
256.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$11.09117728$ |
0.800436773 |
\( 249872 a + 434912 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 4 a - 8\) , \( 4 a - 8\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(4a-8\right){x}+4a-8$ |
| 256.1-c1 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 0 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+{x}$ |
| 256.1-c2 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 0 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+{x}$ |
| 256.1-c3 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$4.423757977$ |
1.277028929 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 20 a - 44\) , \( 92 a - 160\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(20a-44\right){x}+92a-160$ |
| 256.1-c4 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( -818626500 a + 1417905000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 20 a - 44\) , \( -92 a + 160\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(20a-44\right){x}-92a+160$ |
| 256.1-c5 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 54000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -4\) , \( 4 a\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}-4{x}+4a$ |
| 256.1-c6 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 54000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -4\) , \( -4 a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}-4{x}-4a$ |
| 256.1-c7 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$4.423757977$ |
1.277028929 |
\( 818626500 a + 1417905000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -20 a - 44\) , \( -92 a - 160\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-20a-44\right){x}-92a-160$ |
| 256.1-c8 |
256.1-c |
$8$ |
$12$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{20} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-48$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$17.69503190$ |
1.277028929 |
\( 818626500 a + 1417905000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -20 a - 44\) , \( 92 a + 160\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-20a-44\right){x}+92a+160$ |
| 256.1-d1 |
256.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$27.22522842$ |
1.964811619 |
\( 512 a + 512 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(a+2\right){x}$ |
| 256.1-d2 |
256.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$27.22522842$ |
1.964811619 |
\( -249872 a + 434912 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4 a - 8\) , \( 4 a + 8\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-4a-8\right){x}+4a+8$ |
| 256.1-e1 |
256.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{8} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$11.09117728$ |
0.800436773 |
\( 512 a + 512 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(a+2\right){x}$ |
| 256.1-e2 |
256.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{16} \) |
$1.23820$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$11.09117728$ |
0.800436773 |
\( -249872 a + 434912 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -4 a - 8\) , \( -4 a - 8\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-4a-8\right){x}-4a-8$ |
| 256.1-f1 |
256.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{12} \) |
$1.23820$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.444312937$ |
$13.75037163$ |
1.763651500 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}$ |
| 256.1-f2 |
256.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{12} \) |
$1.23820$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.888625874$ |
$27.50074327$ |
1.763651500 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4 a - 7\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(4a-7\right){x}$ |
| 256.1-f3 |
256.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{18} \) |
$1.23820$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.444312937$ |
$6.875185818$ |
1.763651500 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 44 a - 77\) , \( 210 a - 364\bigr] \) |
${y}^2={x}^{3}+\left(44a-77\right){x}+210a-364$ |
| 256.1-f4 |
256.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{18} \) |
$1.23820$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.444312937$ |
$27.50074327$ |
1.763651500 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 44 a - 77\) , \( -210 a + 364\bigr] \) |
${y}^2={x}^{3}+\left(44a-77\right){x}-210a+364$ |
*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.
