| 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 |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.75474$ |
$(2,a)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$27.50074327$ |
2.121728406 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}$ |
| 32.1-a2 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{24} \) |
$2.75474$ |
$(2,a)$ |
$0 \le r \le 2$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$16$ |
\( 2 \) |
$1$ |
$13.75037163$ |
2.121728406 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+4{x}$ |
| 32.1-a3 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{18} \) |
$2.75474$ |
$(2,a)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$1$ |
$13.75037163$ |
2.121728406 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( -14\bigr] \) |
${y}^2={x}^{3}-11{x}-14$ |
| 32.1-a4 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{18} \) |
$2.75474$ |
$(2,a)$ |
$0 \le r \le 2$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$1$ |
$55.00148654$ |
2.121728406 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -11\) , \( 14\bigr] \) |
${y}^2={x}^{3}-11{x}+14$ |
| 32.1-b1 |
32.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$4.268241698$ |
$13.75037163$ |
4.528024828 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -52 a + 337\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-52a+337\right){x}$ |
| 32.1-b2 |
32.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{24} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$2.134120849$ |
$27.50074327$ |
4.528024828 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 208 a - 1348\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(208a-1348\right){x}$ |
| 32.1-b3 |
32.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.067060424$ |
$55.00148654$ |
4.528024828 |
\( 287496 \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 143 a - 890\) , \( -1862 a + 12110\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(143a-890\right){x}-1862a+12110$ |
| 32.1-b4 |
32.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$1.067060424$ |
$13.75037163$ |
4.528024828 |
\( 287496 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 143 a - 911\) , \( 2863 a - 18522\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(143a-911\right){x}+2863a-18522$ |
| 32.1-c1 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.322914441$ |
$13.75037163$ |
3.525160981 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}$ |
| 32.1-c2 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{24} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1.661457220$ |
$27.50074327$ |
3.525160981 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4\) , \( 0\bigr] \) |
${y}^2={x}^{3}-4{x}$ |
| 32.1-c3 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.322914441$ |
$13.75037163$ |
3.525160981 |
\( 287496 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 13\) , \( 21\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+13{x}+21$ |
| 32.1-c4 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.75474$ |
$(2,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$0.830728610$ |
$55.00148654$ |
3.525160981 |
\( 287496 \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 34\) , \( 35\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+34{x}+35$ |
| 32.1-d1 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{24} \) |
$2.75474$ |
$(2,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$2.655997444$ |
$13.75037163$ |
5.635305225 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -208 a + 1348\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-208a+1348\right){x}$ |
| 32.1-d2 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.75474$ |
$(2,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.655997444$ |
$27.50074327$ |
5.635305225 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 52 a - 337\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(52a-337\right){x}$ |
| 32.1-d3 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{18} \) |
$2.75474$ |
$(2,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$2.655997444$ |
$13.75037163$ |
5.635305225 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 572 a - 3707\) , \( 18900 a - 122486\bigr] \) |
${y}^2={x}^{3}+\left(572a-3707\right){x}+18900a-122486$ |
| 32.1-d4 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{42}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{18} \) |
$2.75474$ |
$(2,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.663999361$ |
$55.00148654$ |
5.635305225 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 572 a - 3707\) , \( -18900 a + 122486\bigr] \) |
${y}^2={x}^{3}+\left(572a-3707\right){x}-18900a+122486$ |
*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.
