| 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-a7 |
12.1-a |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{15} \cdot 3^{3} \) |
$0.95541$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$7.858181321$ |
1.367933784 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -3 a + 7\) , \( a - 7\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-3a+7\right){x}+a-7$ |
| 12.1-b7 |
12.1-b |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{15} \cdot 3^{3} \) |
$0.95541$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$15.71636264$ |
1.367933784 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -533 a - 1264\) , \( 10417 a + 24712\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-533a-1264\right){x}+10417a+24712$ |
| 288.3-e7 |
288.3-e |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{27} \cdot 3^{9} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.536923101$ |
1.579553877 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -12587 a + 42448\) , \( -661475 a + 2230680\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(-12587a+42448\right){x}-661475a+2230680$ |
| 288.3-l7 |
288.3-l |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.3 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{27} \cdot 3^{9} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$1.032690819$ |
$2.268461550$ |
4.893572364 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -20 a - 16\) , \( -49 a - 152\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-20a-16\right){x}-49a-152$ |
| 288.4-e7 |
288.4-e |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{27} \cdot 3^{9} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.268461550$ |
1.579553877 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -123 a + 411\) , \( 555 a - 1875\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-123a+411\right){x}+555a-1875$ |
| 288.4-l7 |
288.4-l |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
288.4 |
\( 2^{5} \cdot 3^{2} \) |
\( - 2^{27} \cdot 3^{9} \) |
$2.11468$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{6} \cdot 3 \) |
$0.258172704$ |
$4.536923101$ |
4.893572364 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -1417 a - 3361\) , \( 48278 a + 114529\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1417a-3361\right){x}+48278a+114529$ |
| 384.5-c7 |
384.5-c |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.5 |
\( 2^{7} \cdot 3 \) |
\( - 2^{33} \cdot 3^{3} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.902975340$ |
2.560495363 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -6076 a - 14412\) , \( 432107 a + 1025080\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-6076a-14412\right){x}+432107a+1025080$ |
| 384.5-r7 |
384.5-r |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.5 |
\( 2^{7} \cdot 3 \) |
\( - 2^{33} \cdot 3^{3} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \) |
$2.074721545$ |
$3.148650022$ |
2.274349657 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -15 a + 39\) , \( -39 a + 87\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-15a+39\right){x}-39a+87$ |
| 384.6-c7 |
384.6-c |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.6 |
\( 2^{7} \cdot 3 \) |
\( - 2^{33} \cdot 3^{3} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$4.902975340$ |
2.560495363 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -29 a + 82\) , \( 27 a - 93\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-29a+82\right){x}+27a-93$ |
| 384.6-r7 |
384.6-r |
$12$ |
$36$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
384.6 |
\( 2^{7} \cdot 3 \) |
\( - 2^{33} \cdot 3^{3} \) |
$2.27237$ |
$(-a-2), (-a+3), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \) |
$2.074721545$ |
$3.148650022$ |
2.274349657 |
\( \frac{415636375}{36864} a + \frac{1036704625}{36864} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -3004 a - 7126\) , \( -158217 a - 375336\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-3004a-7126\right){x}-158217a-375336$ |
*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.
