| 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 |
| 54684.2-a1 |
54684.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 31^{3} \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.325393587$ |
2.254392903 |
\( -\frac{27687863199645}{14019525436} a - \frac{39320031191761}{28039050872} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -367 a + 554\) , \( 2293 a + 3767\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-367a+554\right){x}+2293a+3767$ |
| 54684.2-a2 |
54684.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{24} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$9$ |
\( 2 \) |
$1$ |
$0.108464529$ |
2.254392903 |
\( \frac{20687422086138241443}{50480821535223919} a + \frac{113695097195902805459}{100961643070447838} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 3053 a - 4156\) , \( 6565 a - 58879\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(3053a-4156\right){x}+6565a-58879$ |
| 54684.2-a3 |
54684.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{8} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.976180762$ |
2.254392903 |
\( -\frac{1566729405}{3038} a + \frac{66816311}{3038} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 83 a + 89\) , \( -527 a + 752\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(83a+89\right){x}-527a+752$ |
| 54684.2-a4 |
54684.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{8} \cdot 31^{9} \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.108464529$ |
2.254392903 |
\( \frac{406635051366709052855}{2591082971745758} a + \frac{44028606239712586643}{1295541485872879} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -7717 a - 3856\) , \( 397303 a - 10303\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-7717a-3856\right){x}+397303a-10303$ |
| 54684.2-a5 |
54684.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{18} \cdot 3^{6} \cdot 7^{8} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$9$ |
\( 2 \) |
$1$ |
$0.108464529$ |
2.254392903 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -32887 a + 47339\) , \( 1679851 a + 2156108\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-32887a+47339\right){x}+1679851a+2156108$ |
| 54684.2-b1 |
54684.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{9} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.014489309$ |
$0.722590618$ |
3.385861220 |
\( \frac{199369}{1488} a + \frac{2804207}{1488} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 8 a - 97\) , \( 20 a - 155\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8a-97\right){x}+20a-155$ |
| 54684.2-b2 |
54684.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{4} \cdot 3^{10} \cdot 7^{9} \cdot 31^{2} \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.507244654$ |
$0.361295309$ |
3.385861220 |
\( \frac{12103686821}{34596} a + \frac{4543215451}{5766} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 68 a - 1237\) , \( 1784 a - 17207\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(68a-1237\right){x}+1784a-17207$ |
| 54684.2-c1 |
54684.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{10} \cdot 3^{6} \cdot 7^{8} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$0.923764880$ |
2.133343609 |
\( \frac{8685387}{48608} a + \frac{17171919}{48608} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -11 a - 30\) , \( 164 a - 94\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-11a-30\right){x}+164a-94$ |
| 54684.2-d1 |
54684.2-d |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{50} \cdot 3^{6} \cdot 7^{6} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.420856605$ |
$0.080398407$ |
3.907067914 |
\( \frac{936087656892551}{1040187392} a - \frac{833285178768245}{1040187392} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 19941 a - 26254\) , \( -1517258 a + 1241735\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(19941a-26254\right){x}-1517258a+1241735$ |
| 54684.2-d2 |
54684.2-d |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{6} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.420856605$ |
$2.009960176$ |
3.907067914 |
\( \frac{24551}{62} a + \frac{66955}{62} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -9 a - 4\) , \( -8 a + 5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a-4\right){x}-8a+5$ |
| 54684.2-d3 |
54684.2-d |
$3$ |
$25$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{10} \cdot 3^{6} \cdot 7^{6} \cdot 31^{5} \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5Cs.4.1 |
$1$ |
\( 2 \cdot 5^{2} \) |
$0.084171321$ |
$0.401992035$ |
3.907067914 |
\( -\frac{511363962461}{916132832} a + \frac{764718499383}{458066416} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -219 a + 311\) , \( 475 a - 1234\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-219a+311\right){x}+475a-1234$ |
| 54684.2-e1 |
54684.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{20} \cdot 3^{6} \cdot 7^{9} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.203114854$ |
$0.434157302$ |
4.073035142 |
\( \frac{10621452329}{10888192} a - \frac{7274546105}{10888192} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -228 a + 108\) , \( 635 a - 1797\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-228a+108\right){x}+635a-1797$ |
| 54684.2-e2 |
54684.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{10} \cdot 3^{6} \cdot 7^{12} \cdot 31^{2} \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.406229709$ |
$0.217078651$ |
4.073035142 |
\( -\frac{6512659898044201}{3617942048} a + \frac{303261539125773}{452242756} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -4068 a + 2508\) , \( 51707 a - 92517\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-4068a+2508\right){x}+51707a-92517$ |
| 54684.2-f1 |
54684.2-f |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
54684.2 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 31 \) |
\( 2^{4} \cdot 3^{6} \cdot 7^{8} \cdot 31 \) |
$2.36681$ |
$(-2a+1), (-3a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$1.258389327$ |
2.906125668 |
\( \frac{179685}{124} a + \frac{238249}{124} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 26 a - 37\) , \( 41 a - 44\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(26a-37\right){x}+41a-44$ |
*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.
