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 |
12.1-a12 |
12.1-a |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$0.95541$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$7.858181321$ |
1.367933784 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -28020 a - 66474\) , \( 4365739 a + 10356761\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-28020a-66474\right){x}+4365739a+10356761$ |
12.1-b12 |
12.1-b |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \) |
$0.95541$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$0.873131257$ |
1.367933784 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 1112 a - 3819\) , \( 27462 a - 92979\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1112a-3819\right){x}+27462a-92979$ |
288.3-e12 |
288.3-e |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{10} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{4} \) |
$1$ |
$0.252051283$ |
1.579553877 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 2258 a - 11752\) , \( 102646 a - 451168\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(2258a-11752\right){x}+102646a-451168$ |
288.3-l12 |
288.3-l |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{10} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$3.098072458$ |
$2.268461550$ |
4.893572364 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -1522975 a - 3612936\) , \( 1732933137 a + 4111004920\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-1522975a-3612936\right){x}+1732933137a+4111004920$ |
288.4-e12 |
288.4-e |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{10} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$2.268461550$ |
1.579553877 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -74213 a - 176219\) , \( 18651419 a + 44246973\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-74213a-176219\right){x}+18651419a+44246973$ |
288.4-l12 |
288.4-l |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{10} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$3.098072458$ |
$0.252051283$ |
4.893572364 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 61058 a - 206116\) , \( 11235272 a - 37889633\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(61058a-206116\right){x}+11235272a-37889633$ |
384.5-c12 |
384.5-c |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.5 |
\( 2^{7} \cdot 3 \) |
\( 2^{28} \cdot 3^{4} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$36$ |
\( 2^{2} \) |
$1$ |
$0.408581278$ |
2.560495363 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 6174 a - 21692\) , \( -366817 a + 1227352\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(6174a-21692\right){x}-366817a+1227352$ |
384.5-r12 |
384.5-r |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.5 |
\( 2^{7} \cdot 3 \) |
\( 2^{28} \cdot 3^{4} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.556041159$ |
$2.099100015$ |
2.274349657 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -318663 a - 755977\) , \( 165397799 a + 392370097\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-318663a-755977\right){x}+165397799a+392370097$ |
384.6-c12 |
384.6-c |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.6 |
\( 2^{7} \cdot 3 \) |
\( 2^{28} \cdot 3^{4} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.408581278$ |
2.560495363 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -157691 a - 374123\) , \( -57969564 a - 137520045\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-157691a-374123\right){x}-57969564a-137520045$ |
384.6-r12 |
384.6-r |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.6 |
\( 2^{7} \cdot 3 \) |
\( 2^{28} \cdot 3^{4} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.556041159$ |
$2.099100015$ |
2.274349657 |
\( \frac{2879604455941411323125}{4608} a + \frac{6831231869232063827875}{4608} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 12711 a - 43281\) , \( 1064067 a - 3586068\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(12711a-43281\right){x}+1064067a-3586068$ |
*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.