| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$12.57159272$ |
2.039381637 |
\( -8000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 11\) , \( -a\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+11{x}-a$ |
| 32.1-b1 |
32.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.869038602$ |
$12.57159272$ |
3.544602734 |
\( -8000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -740 a - 4549\) , \( 28481 a + 175560\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-740a-4549\right){x}+28481a+175560$ |
| 32.1-c1 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$7.228619307$ |
$27.50074327$ |
4.031048281 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}$ |
| 32.1-c2 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$14.45723861$ |
$13.75037163$ |
4.031048281 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 444 a + 2737\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(444a+2737\right){x}$ |
| 32.1-c3 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.62028$ |
$(-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.614309653$ |
$13.75037163$ |
4.031048281 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -1221 a - 7509\) , \( -61761 a - 380687\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1221a-7509\right){x}-61761a-380687$ |
| 32.1-c4 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.62028$ |
$(-a-6)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$3.614309653$ |
$55.00148654$ |
4.031048281 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -1221 a - 7490\) , \( 53214 a + 328076\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-1221a-7490\right){x}+53214a+328076$ |
| 32.1-d1 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$13.75037163$ |
0.557651206 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}$ |
| 32.1-d2 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$27.50074327$ |
0.557651206 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -444 a - 2737\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-444a-2737\right){x}$ |
| 32.1-d3 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.62028$ |
$(-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$13.75037163$ |
0.557651206 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 15\) , \( 22\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+15{x}+22$ |
| 32.1-d4 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$2.62028$ |
$(-a-6)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$55.00148654$ |
0.557651206 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 34\) , \( 35\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+34{x}+35$ |
| 32.1-e1 |
32.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$12.57159272$ |
2.039381637 |
\( -8000 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 11\) , \( a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+11{x}+a$ |
| 32.1-f1 |
32.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{38}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$2.62028$ |
$(-a-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.869038602$ |
$12.57159272$ |
3.544602734 |
\( -8000 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 740 a - 4549\) , \( -28481 a + 175560\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(740a-4549\right){x}-28481a+175560$ |
*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.
