| 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 |
| 33124.5-a1 |
33124.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{40} \cdot 7^{6} \cdot 13^{4} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.220089901$ |
1.270689638 |
\( \frac{71903073502287}{60782804992} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -867 a\) , \( 6445\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}-867a{x}+6445$ |
| 33124.5-a2 |
33124.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{20} \cdot 7^{12} \cdot 13^{8} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \cdot 5 \) |
$1$ |
$0.110044950$ |
1.270689638 |
\( \frac{8511781274893233}{3440817243136} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 4253 a\) , \( 59693\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+4253a{x}+59693$ |
| 33124.5-a3 |
33124.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{10} \cdot 7^{24} \cdot 13^{4} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.055022475$ |
1.270689638 |
\( \frac{3389174547561866673}{74853681183008} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 31293 a\) , \( -2081875\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+31293a{x}-2081875$ |
| 33124.5-a4 |
33124.5-a |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{10} \cdot 7^{6} \cdot 13^{16} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.055022475$ |
1.270689638 |
\( \frac{22868021811807457713}{8953460393696} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 59133 a\) , \( 5547693\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+59133a{x}+5547693$ |
| 33124.5-b1 |
33124.5-b |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{22} \cdot 7^{14} \cdot 13^{2} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 11 \) |
$1$ |
$0.175111233$ |
2.224211397 |
\( -\frac{10824513276632329}{21926008832} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -4609 a + 4608\) , \( 120244\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-4609a+4608\right){x}+120244$ |
| 33124.5-c1 |
33124.5-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1[2] |
$81$ |
\( 1 \) |
$1$ |
$0.256426207$ |
2.664859317 |
\( -\frac{424962187484640625}{182} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -15663 a + 15663\) , \( -755809\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-15663a+15663\right){x}-755809$ |
| 33124.5-c2 |
33124.5-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{6} \cdot 7^{6} \cdot 13^{6} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/3\Z\oplus\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1[2] |
$1$ |
\( 3^{5} \) |
$1$ |
$0.769278622$ |
2.664859317 |
\( -\frac{795309684625}{6028568} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -193 a + 193\) , \( -1055\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-193a+193\right){x}-1055$ |
| 33124.5-c3 |
33124.5-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 7^{10} \cdot 13^{10} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3^{4} \) |
$1$ |
$0.256426207$ |
2.664859317 |
\( \frac{6994863867180068874625}{855859600259576822} a - \frac{5471444874746322606000}{427929800129788411} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -23 a + 1093\) , \( 16670 a - 8609\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-23a+1093\right){x}+16670a-8609$ |
| 33124.5-c4 |
33124.5-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 7^{10} \cdot 13^{10} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3^{4} \) |
$1$ |
$0.256426207$ |
2.664859317 |
\( -\frac{6994863867180068874625}{855859600259576822} a - \frac{3948025882312576337375}{855859600259576822} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -1093 a + 23\) , \( -16670 a + 8061\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-1093a+23\right){x}-16670a+8061$ |
| 33124.5-c5 |
33124.5-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{18} \cdot 7^{2} \cdot 13^{2} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3^{2} \) |
$1$ |
$2.307835866$ |
2.664859317 |
\( \frac{37595375}{46592} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 7 a - 7\) , \( -7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(7a-7\right){x}-7$ |
| 33124.5-d1 |
33124.5-d |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{2} \cdot 7^{6} \cdot 13^{2} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.051746300$ |
3.523853096 |
\( \frac{4019679}{8918} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -4 a\) , \( -5\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}-4a{x}-5$ |
| 33124.5-e1 |
33124.5-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
33124.5 |
\( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( 2^{14} \cdot 7^{2} \cdot 13^{10} \) |
$2.08802$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 7 \) |
$1$ |
$0.533192753$ |
4.309745717 |
\( -\frac{1207949625}{332678528} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -22 a + 22\) , \( 884\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-22a+22\right){x}+884$ |
*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.
