| 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 |
| 61731.3-a1 |
61731.3-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$1.182529221$ |
$0.123884704$ |
2.706567867 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 36929 a + 11540\) , \( 1351376 a - 4007862\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(36929a+11540\right){x}+1351376a-4007862$ |
| 61731.3-a2 |
61731.3-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$1.182529221$ |
$0.123884704$ |
2.706567867 |
\( \frac{14306161739497472}{322687697779} a - \frac{27123251845038080}{322687697779} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 7731 a - 4308\) , \( -195516 a - 34450\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(7731a-4308\right){x}-195516a-34450$ |
| 61731.3-a3 |
61731.3-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$1.182529221$ |
$0.123884704$ |
2.706567867 |
\( -\frac{14306161739497472}{322687697779} a - \frac{12817090105540608}{322687697779} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -6699 a - 18\) , \( 251154 a - 123514\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6699a-18\right){x}+251154a-123514$ |
| 61731.3-a4 |
61731.3-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{12} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2^{3} \) |
$0.394176407$ |
$0.371654113$ |
2.706567867 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 141 a - 588\) , \( 2064 a - 5467\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(141a-588\right){x}+2064a-5467$ |
| 61731.3-a5 |
61731.3-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$0.131392135$ |
$1.114962340$ |
2.706567867 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 43 a - 32\) , \( -16 a + 20\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(43a-32\right){x}-16a+20$ |
| 61731.3-b1 |
61731.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{7} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.108104655$ |
0.998628029 |
\( \frac{7240152655469734}{50950689123} a - \frac{5512832666599067}{50950689123} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -9143 a + 10240\) , \( 68164 a + 329450\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-9143a+10240\right){x}+68164a+329450$ |
| 61731.3-b2 |
61731.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{8} \cdot 19^{14} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.216209310$ |
0.998628029 |
\( \frac{67419143}{390963} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -128 a + 535\) , \( -4787 a + 13769\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-128a+535\right){x}-4787a+13769$ |
| 61731.3-b3 |
61731.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{8} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.864837242$ |
0.998628029 |
\( \frac{389017}{57} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 22 a - 95\) , \( 103 a - 271\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(22a-95\right){x}+103a-271$ |
| 61731.3-b4 |
61731.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{10} \cdot 19^{10} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.432418621$ |
0.998628029 |
\( \frac{30664297}{3249} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 97 a - 410\) , \( -980 a + 2978\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(97a-410\right){x}-980a+2978$ |
| 61731.3-b5 |
61731.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{7} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.108104655$ |
0.998628029 |
\( -\frac{7240152655469734}{50950689123} a + \frac{1727319988870667}{50950689123} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 5287 a + 5950\) , \( -322106 a + 404732\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5287a+5950\right){x}-322106a+404732$ |
| 61731.3-b6 |
61731.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{14} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.216209310$ |
0.998628029 |
\( \frac{115714886617}{1539} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 1522 a - 6395\) , \( -66245 a + 194783\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1522a-6395\right){x}-66245a+194783$ |
| 61731.3-c1 |
61731.3-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{9} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$1.078091533$ |
1.244872874 |
\( \frac{29840721}{6859} a - \frac{35267232}{6859} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -16 a - 40\) , \( -81 a - 75\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-16a-40\right){x}-81a-75$ |
| 61731.3-c2 |
61731.3-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{12} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.539045766$ |
1.244872874 |
\( -\frac{36038181633}{47045881} a - \frac{39546962313}{47045881} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -146 a + 145\) , \( 222 a - 1131\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-146a+145\right){x}+222a-1131$ |
| 61731.3-c3 |
61731.3-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{7} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$1.078091533$ |
1.244872874 |
\( -\frac{9153}{19} a + \frac{36801}{19} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 18 a + 27\) , \( -45 a + 20\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(18a+27\right){x}-45a+20$ |
| 61731.3-c4 |
61731.3-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.539045766$ |
1.244872874 |
\( -\frac{363527109}{361} a + \frac{287391186}{361} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 183 a + 417\) , \( -4773 a + 4499\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(183a+417\right){x}-4773a+4499$ |
| 61731.3-d1 |
61731.3-d |
$2$ |
$5$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{10} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.035996263$ |
0.831298097 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 65856 a - 276591\) , \( 18858975 a - 54638371\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(65856a-276591\right){x}+18858975a-54638371$ |
| 61731.3-d2 |
61731.3-d |
$2$ |
$5$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{26} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.179981317$ |
0.831298097 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -294 a + 1239\) , \( 6225 a - 19261\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-294a+1239\right){x}+6225a-19261$ |
| 61731.3-e1 |
61731.3-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{12} \cdot 19^{10} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.371111235$ |
1.714089374 |
\( -\frac{1024000}{513} a - \frac{512000}{171} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 366 a - 450\) , \( -4386 a + 3212\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(366a-450\right){x}-4386a+3212$ |
| 61731.3-f1 |
61731.3-f |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{10} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.705796155$ |
3.259932801 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 36 a - 147\) , \( -279 a + 677\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(36a-147\right){x}-279a+677$ |
| 61731.3-g1 |
61731.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{13} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.769040107$ |
$0.356166928$ |
4.555249792 |
\( \frac{5845799184}{130321} a - \frac{4133064549}{130321} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 238 a - 817\) , \( 3795 a - 8972\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-817\right){x}+3795a-8972$ |
| 61731.3-g2 |
61731.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{11} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.384520053$ |
$0.712333856$ |
4.555249792 |
\( \frac{61536}{361} a - \frac{40335}{361} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -42 a - 2\) , \( 343 a - 394\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-42a-2\right){x}+343a-394$ |
| 61731.3-h1 |
61731.3-h |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{7} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.896333780$ |
4.139988395 |
\( \frac{5845799184}{130321} a - \frac{4133064549}{130321} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 7 a - 118\) , \( -7 a + 466\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7a-118\right){x}-7a+466$ |
| 61731.3-h2 |
61731.3-h |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.3 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{5} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.792667560$ |
4.139988395 |
\( \frac{61536}{361} a - \frac{40335}{361} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -8 a + 2\) , \( -16 a + 25\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a+2\right){x}-16a+25$ |
*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.
