| 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 |
| 36.2-a1 |
36.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{13} \) |
$1.33142$ |
$(a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$7.578921928$ |
2.491934179 |
\( \frac{13801}{4374} a + \frac{503422}{2187} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -19 a + 67\) , \( -200 a + 708\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-19a+67\right){x}-200a+708$ |
| 36.2-b1 |
36.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{7} \) |
$1.33142$ |
$(a-3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.430705504$ |
$2.776255887$ |
1.572638992 |
\( \frac{1293877682358550987489}{6} a - \frac{2291057091849561691514}{3} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -11528 a - 29307\) , \( -2399463 a - 6097932\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-11528a-29307\right){x}-2399463a-6097932$ |
| 36.2-b2 |
36.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{11} \) |
$1.33142$ |
$(a-3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.086141100$ |
$13.88127943$ |
1.572638992 |
\( \frac{19793521}{7776} a - \frac{74116909}{7776} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 66 a - 219\) , \( -525 a + 1870\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(66a-219\right){x}-525a+1870$ |
| 36.2-c1 |
36.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{9} \) |
$1.33142$ |
$(a-3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$0.451986050$ |
$2.448071380$ |
2.182878143 |
\( -\frac{3696743}{4} a - \frac{18785529}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -63 a - 159\) , \( -658 a - 1675\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-63a-159\right){x}-658a-1675$ |
| 36.2-c2 |
36.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{3} \) |
$1.33142$ |
$(a-3), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.355958152$ |
$22.03264242$ |
2.182878143 |
\( 109 a - \frac{765}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 2 a + 6\) , \( a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a+6\right){x}+a$ |
| 36.2-d1 |
36.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{3} \) |
$1.33142$ |
$(a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$5.597662836$ |
1.840500203 |
\( -\frac{3696743}{4} a - \frac{18785529}{8} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -20 a + 66\) , \( -135 a + 474\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-20a+66\right){x}-135a+474$ |
| 36.2-d2 |
36.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
36.2 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{9} \) |
$1.33142$ |
$(a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$5.597662836$ |
1.840500203 |
\( 109 a - \frac{765}{2} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 3\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+3{x}-1$ |
*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.
