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 |
61731.2-a1 |
61731.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$1.182529221$ |
$0.123884704$ |
2.706567867 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -36927 a + 48468\) , \( -1388304 a - 2608018\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-36927a+48468\right){x}-1388304a-2608018$ |
61731.2-a2 |
61731.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$1.182529221$ |
$0.123884704$ |
2.706567867 |
\( \frac{14306161739497472}{322687697779} a - \frac{27123251845038080}{322687697779} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 19 a + 6700\) , \( -244436 a + 127622\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(19a+6700\right){x}-244436a+127622$ |
61731.2-a3 |
61731.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$1.182529221$ |
$0.123884704$ |
2.706567867 |
\( -\frac{14306161739497472}{322687697779} a - \frac{12817090105540608}{322687697779} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 4309 a - 7730\) , \( 192094 a - 234274\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4309a-7730\right){x}+192094a-234274$ |
61731.2-a4 |
61731.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{12} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2^{3} \) |
$0.394176407$ |
$0.371654113$ |
2.706567867 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 589 a - 140\) , \( -1616 a - 3991\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(589a-140\right){x}-1616a-3991$ |
61731.2-a5 |
61731.2-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{6} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{3} \) |
$0.131392135$ |
$1.114962340$ |
2.706567867 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 11 a + 32\) , \( 16 a + 4\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(11a+32\right){x}+16a+4$ |
61731.2-b1 |
61731.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{7} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.108104655$ |
0.998628029 |
\( \frac{7240152655469734}{50950689123} a - \frac{5512832666599067}{50950689123} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 11237 a - 5950\) , \( 322105 a + 82626\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(11237a-5950\right){x}+322105a+82626$ |
61731.2-b2 |
61731.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{8} \cdot 19^{14} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.216209310$ |
0.998628029 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 407 a - 535\) , \( 4786 a + 8982\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(407a-535\right){x}+4786a+8982$ |
61731.2-b3 |
61731.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{8} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.864837242$ |
0.998628029 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -73 a + 95\) , \( -104 a - 168\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-73a+95\right){x}-104a-168$ |
61731.2-b4 |
61731.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{10} \cdot 19^{10} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.432418621$ |
0.998628029 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -313 a + 410\) , \( 979 a + 1998\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-313a+410\right){x}+979a+1998$ |
61731.2-b5 |
61731.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{7} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.108104655$ |
0.998628029 |
\( -\frac{7240152655469734}{50950689123} a + \frac{1727319988870667}{50950689123} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 1097 a - 10240\) , \( -68165 a + 397614\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(1097a-10240\right){x}-68165a+397614$ |
61731.2-b6 |
61731.2-b |
$6$ |
$8$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{14} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.216209310$ |
0.998628029 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -4873 a + 6395\) , \( 66244 a + 128538\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-4873a+6395\right){x}+66244a+128538$ |
61731.2-c1 |
61731.2-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.539045766$ |
1.244872874 |
\( \frac{363527109}{361} a - \frac{76135923}{361} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 601 a - 417\) , \( 4772 a - 273\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(601a-417\right){x}+4772a-273$ |
61731.2-c2 |
61731.2-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{12} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{3} \) |
$1$ |
$0.539045766$ |
1.244872874 |
\( \frac{36038181633}{47045881} a - \frac{75585143946}{47045881} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -a - 145\) , \( -222 a - 909\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-a-145\right){x}-222a-909$ |
61731.2-c3 |
61731.2-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{9} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$1.078091533$ |
1.244872874 |
\( -\frac{29840721}{6859} a - \frac{5426511}{6859} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -56 a + 40\) , \( 81 a - 156\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-56a+40\right){x}+81a-156$ |
61731.2-c4 |
61731.2-c |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{7} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B[2] |
$1$ |
\( 2^{2} \) |
$1$ |
$1.078091533$ |
1.244872874 |
\( \frac{9153}{19} a + \frac{27648}{19} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 46 a - 27\) , \( 44 a - 24\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(46a-27\right){x}+44a-24$ |
61731.2-d1 |
61731.2-d |
$2$ |
$5$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{10} \cdot 19^{16} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.035996263$ |
0.831298097 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( 276592 a - 65855\) , \( -18648239 a - 36055987\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(276592a-65855\right){x}-18648239a-36055987$ |
61731.2-d2 |
61731.2-d |
$2$ |
$5$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{26} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.179981317$ |
0.831298097 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -1238 a + 295\) , \( -7169 a - 11797\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1238a+295\right){x}-7169a-11797$ |
61731.2-e1 |
61731.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{12} \cdot 19^{10} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.371111235$ |
1.714089374 |
\( \frac{1024000}{513} a - \frac{2560000}{513} \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -84 a + 450\) , \( 4386 a - 1174\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-84a+450\right){x}+4386a-1174$ |
61731.2-f1 |
61731.2-f |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{10} \cdot 19^{8} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.705796155$ |
3.259932801 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( 148 a - 35\) , \( 167 a + 545\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(148a-35\right){x}+167a+545$ |
61731.2-g1 |
61731.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{11} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.384520053$ |
$0.712333856$ |
4.555249792 |
\( -\frac{61536}{361} a + \frac{21201}{361} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 42 a - 44\) , \( -343 a - 51\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(42a-44\right){x}-343a-51$ |
61731.2-g2 |
61731.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{3} \cdot 19^{13} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.769040107$ |
$0.356166928$ |
4.555249792 |
\( -\frac{5845799184}{130321} a + \frac{1712734635}{130321} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -238 a - 579\) , \( -3795 a - 5177\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-238a-579\right){x}-3795a-5177$ |
61731.2-h1 |
61731.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{5} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.792667560$ |
4.139988395 |
\( -\frac{61536}{361} a + \frac{21201}{361} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -7 a - 1\) , \( 15 a + 9\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-1\right){x}+15a+9$ |
61731.2-h2 |
61731.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
61731.2 |
\( 3^{2} \cdot 19^{3} \) |
\( 3^{9} \cdot 19^{7} \) |
$2.43964$ |
$(-2a+1), (-5a+3), (-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.896333780$ |
4.139988395 |
\( -\frac{5845799184}{130321} a + \frac{1712734635}{130321} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -112 a + 119\) , \( 6 a + 459\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-112a+119\right){x}+6a+459$ |
*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.