| 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 |
| 37.1-a1 |
37.1-a |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$61.22512179$ |
1.722091094 |
\( \frac{231143481}{1369} a^{2} - \frac{650356337}{1369} a + \frac{254921595}{1369} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -8602 a^{2} + 4553 a + 36552\) , \( 7282823 a^{2} - 3854918 a - 30945737\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-8602a^{2}+4553a+36552\right){x}+7282823a^{2}-3854918a-30945737$ |
| 37.1-a2 |
37.1-a |
$2$ |
$2$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$122.4502435$ |
1.722091094 |
\( \frac{58180}{37} a^{2} + \frac{489688}{37} a + \frac{672155}{37} \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( 0\) , \( a^{2} - a - 4\) , \( 4 a^{2} - 2 a - 17\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(a^{2}-a-4\right){x}+4a^{2}-2a-17$ |
| 37.1-b1 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$142.6743765$ |
0.501628796 |
\( \frac{574631192853100701}{37} a^{2} - \frac{1616786066804313023}{37} a + \frac{633688955893848379}{37} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - a - 4\) , \( 0\) , \( -31 a^{2} - 107 a - 94\) , \( 323 a^{2} + 821 a + 427\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-31a^{2}-107a-94\right){x}+323a^{2}+821a+427$ |
| 37.1-b2 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$285.3487530$ |
0.501628796 |
\( \frac{75545001782}{1369} a^{2} - \frac{128675750816}{1369} a + \frac{43853909475}{1369} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - a - 4\) , \( 0\) , \( -66 a^{2} - 92 a + 51\) , \( 494 a^{2} + 672 a - 406\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-66a^{2}-92a+51\right){x}+494a^{2}+672a-406$ |
| 37.1-b3 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$8.917148531$ |
0.501628796 |
\( -\frac{969015990419780}{37} a^{2} + \frac{512916449133472}{37} a + \frac{4117485524184467}{37} \) |
\( \bigl[a^{2} + a - 3\) , \( -a + 1\) , \( a + 1\) , \( 272629937 a^{2} - 144307547 a - 1158442916\) , \( 3583317227543 a^{2} - 1896709220400 a - 15226018492530\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(272629937a^{2}-144307547a-1158442916\right){x}+3583317227543a^{2}-1896709220400a-15226018492530$ |
| 37.1-b4 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$71.33718825$ |
0.501628796 |
\( -\frac{3386438280}{1369} a^{2} + \frac{1793150944}{1369} a + \frac{14395786521}{1369} \) |
\( \bigl[a^{2} + a - 3\) , \( -a + 1\) , \( a + 1\) , \( 17039757 a^{2} - 9019427 a - 72404321\) , \( 55994592296 a^{2} - 29638866100 a - 237928892040\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(17039757a^{2}-9019427a-72404321\right){x}+55994592296a^{2}-29638866100a-237928892040$ |
| 37.1-b5 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$142.6743765$ |
0.501628796 |
\( \frac{62582775088455315}{37} a^{2} + \frac{84043853250844831}{37} a - \frac{53422816607739723}{37} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} + 3\) , \( a + 1\) , \( -19221 a^{2} + 2696 a + 68110\) , \( -9776007 a^{2} + 5673613 a + 42444636\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-19221a^{2}+2696a+68110\right){x}-9776007a^{2}+5673613a+42444636$ |
| 37.1-b6 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{4} \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$142.6743765$ |
0.501628796 |
\( \frac{5001063400}{1874161} a^{2} - \frac{6649464920}{1874161} a + \frac{9630513457}{1874161} \) |
\( \bigl[1\) , \( a^{2} - a - 4\) , \( 0\) , \( 2754 a^{2} - 1458 a - 11702\) , \( 83572 a^{2} - 44236 a - 355109\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(2754a^{2}-1458a-11702\right){x}+83572a^{2}-44236a-355109$ |
| 37.1-b7 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( - 37^{8} \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$17.83429706$ |
0.501628796 |
\( -\frac{470570565437518}{3512479453921} a^{2} + \frac{325402034253328}{3512479453921} a + \frac{2230582325275909}{3512479453921} \) |
\( \bigl[1\) , \( a^{2} - a - 4\) , \( 0\) , \( -6616 a^{2} + 3502 a + 28113\) , \( 561018 a^{2} - 296956 a - 2383844\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-6616a^{2}+3502a+28113\right){x}+561018a^{2}-296956a-2383844$ |
| 37.1-b8 |
37.1-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$2.89967$ |
$(-2a^2+2a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$35.66859412$ |
0.501628796 |
\( \frac{33908012}{37} a^{2} - \frac{95345192}{37} a + \frac{37270317}{37} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{2} - 3\) , \( 6552923945 a^{2} - 3468571292 a - 27844294777\) , \( 546134280974000 a^{2} - 289077929896281 a - 2320601312548858\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6552923945a^{2}-3468571292a-27844294777\right){x}+546134280974000a^{2}-289077929896281a-2320601312548858$ |
*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.
