| 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 |
| 54684.6-a1 |
54684.6-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{4} \cdot 3^{8} \cdot 7^{3} \cdot 31^{6} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.157010943$ |
$0.454040668$ |
3.951257046 |
\( \frac{277280928034937}{10650044172} a - \frac{980879090622803}{10650044172} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 361 a + 223\) , \( -2381 a + 5031\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(361a+223\right){x}-2381a+5031$ |
| 54684.6-a2 |
54684.6-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{10} \cdot 7^{3} \cdot 31^{3} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.078505471$ |
$0.908081336$ |
3.951257046 |
\( -\frac{2118903643}{4289904} a + \frac{51419833}{59582} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( a + 43\) , \( 103 a + 99\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+43\right){x}+103a+99$ |
| 54684.6-b1 |
54684.6-b |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{50} \cdot 3^{6} \cdot 7^{6} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 5^{2} \) |
$1$ |
$0.080398407$ |
2.320902097 |
\( \frac{936087656892551}{1040187392} a - \frac{833285178768245}{1040187392} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 22958 a + 1487\) , \( -108899 a + 1456079\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22958a+1487\right){x}-108899a+1456079$ |
| 54684.6-b2 |
54684.6-b |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{6} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$2.009960176$ |
2.320902097 |
\( \frac{24551}{62} a + \frac{66955}{62} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 8 a - 13\) , \( a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a-13\right){x}+a-1$ |
| 54684.6-b3 |
54684.6-b |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{10} \cdot 3^{6} \cdot 7^{6} \cdot 31^{5} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5Cs.4.1 |
$1$ |
\( 5 \) |
$1$ |
$0.401992035$ |
2.320902097 |
\( -\frac{511363962461}{916132832} a + \frac{764718499383}{458066416} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -277 a + 2\) , \( -1046 a - 325\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-277a+2\right){x}-1046a-325$ |
| 54684.6-c1 |
54684.6-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{2} \cdot 3^{7} \cdot 7^{8} \cdot 31^{2} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.374382161$ |
$0.631215303$ |
4.365982776 |
\( -\frac{132282755267}{141267} a - \frac{7519192561}{282534} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 418 a - 359\) , \( -3201 a + 750\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(418a-359\right){x}-3201a+750$ |
| 54684.6-c2 |
54684.6-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{4} \cdot 3^{8} \cdot 7^{7} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.187191080$ |
$1.262430606$ |
4.365982776 |
\( -\frac{492284}{651} a + \frac{1462605}{868} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 28 a - 29\) , \( -27 a - 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(28a-29\right){x}-27a-36$ |
| 54684.6-d1 |
54684.6-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{14} \cdot 7^{9} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.628184799$ |
$0.376770198$ |
4.372736608 |
\( -\frac{6777439}{1674} a - \frac{345359429}{40176} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 413 a + 142\) , \( -2054 a + 5479\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(413a+142\right){x}-2054a+5479$ |
| 54684.6-d2 |
54684.6-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{4} \cdot 3^{22} \cdot 7^{9} \cdot 31^{2} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1.256369599$ |
$0.188385099$ |
4.372736608 |
\( -\frac{3093240911}{4203414} a - \frac{12866283125}{25220484} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -667 a + 1282\) , \( 13198 a + 12475\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-667a+1282\right){x}+13198a+12475$ |
| 54684.6-e1 |
54684.6-e |
$4$ |
$10$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{40} \cdot 3^{8} \cdot 7^{7} \cdot 31^{5} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{5} \cdot 5 \) |
$1$ |
$0.050435338$ |
2.329508499 |
\( -\frac{1677804026195491}{210138884472832} a + \frac{1018430192992530617}{630416653418496} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 154 a + 18103\) , \( -4963 a - 248987\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(154a+18103\right){x}-4963a-248987$ |
| 54684.6-e2 |
54684.6-e |
$4$ |
$10$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{16} \cdot 7^{11} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{5} \) |
$1$ |
$0.252176692$ |
2.329508499 |
\( -\frac{89945456429}{675238032} a + \frac{232000014839}{126607131} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 829 a - 617\) , \( 1364 a + 727\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(829a-617\right){x}+1364a+727$ |
| 54684.6-e3 |
54684.6-e |
$4$ |
$10$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{4} \cdot 3^{11} \cdot 7^{16} \cdot 31^{2} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{6} \) |
$1$ |
$0.126088346$ |
2.329508499 |
\( \frac{2196024357305119847}{29317541143212} a + \frac{2618367099098063807}{29317541143212} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 7849 a - 6557\) , \( -268096 a + 77299\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(7849a-6557\right){x}-268096a+77299$ |
| 54684.6-e4 |
54684.6-e |
$4$ |
$10$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.6 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{20} \cdot 3^{7} \cdot 7^{8} \cdot 31^{10} \) |
$2.36681$ |
$(-2a+1), (3a-2), (6a-5), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{6} \cdot 5 \) |
$1$ |
$0.025217669$ |
2.329508499 |
\( -\frac{31232843268087705559355953}{123377006782646012928} a + \frac{19153800941117482964106493}{61688503391323006464} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -30566 a + 217783\) , \( -40063843 a + 13117285\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-30566a+217783\right){x}-40063843a+13117285$ |
*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.
