| 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 |
| 6.1-a1 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{26} \cdot 3^{14} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( \frac{773243804465483}{78364164096} a - \frac{2224704505657277}{78364164096} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -10 a - 173\) , \( -27 a + 899\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-10a-173\right){x}-27a+899$ |
| 6.1-a2 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{28} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$0.712350106$ |
1.039746854 |
\( \frac{413197302910024471}{2928229434235008} a - \frac{374086572953648785}{2928229434235008} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -90 a - 13\) , \( -683 a + 3107\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-90a-13\right){x}-683a+3107$ |
| 6.1-a3 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{28} \cdot 3^{7} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( -\frac{291660246420647}{587068342272} a + \frac{520440745417985}{587068342272} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 6 a - 22\) , \( 14 a - 31\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(6a-22\right){x}+14a-31$ |
| 6.1-a4 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( \frac{9841}{48} a + \frac{43625}{48} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( a - 2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(a+1\right){x}^2+\left(a-2\right){x}-1$ |
| 6.1-a5 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{14} \cdot 3^{2} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( -\frac{15457}{36} a + \frac{95803}{36} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -3\) , \( -a - 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2-3{x}-a-3$ |
| 6.1-a6 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3 \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 1 \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( \frac{57217}{6} a + \frac{106031}{6} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -a - 1\) , \( a + 4\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-a-1\right){x}+a+4$ |
| 6.1-a7 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3^{4} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 2 \) |
$1$ |
$4.986450742$ |
1.039746854 |
\( -\frac{238419887}{162} a + \frac{349824755}{162} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 10 a + 7\) , \( 11 a - 103\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(10a+7\right){x}+11a-103$ |
| 6.1-a8 |
6.1-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{7} \) |
$0.67072$ |
$(2,a), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.1 |
$1$ |
\( 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( -\frac{133689253600013}{279936} a + \frac{73951848116651}{279936} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -206 a + 609\) , \( 1402 a + 5610\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-206a+609\right){x}+1402a+5610$ |
| 6.4-a1 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{7} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( \frac{133689253600013}{279936} a - \frac{9956234247227}{46656} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 206 a + 403\) , \( -1402 a + 7012\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-1\right){x}^2+\left(206a+403\right){x}-1402a+7012$ |
| 6.4-a2 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{26} \cdot 3^{14} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( -\frac{773243804465483}{78364164096} a - \frac{241910116865299}{13060694016} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 8 a - 178\) , \( -154 a + 999\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(8a-178\right){x}-154a+999$ |
| 6.4-a3 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{19} \cdot 3^{28} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$0.712350106$ |
1.039746854 |
\( -\frac{413197302910024471}{2928229434235008} a + \frac{6518454992729281}{488038239039168} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 88 a - 98\) , \( 582 a + 1991\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(88a-98\right){x}+582a+1991$ |
| 6.4-a4 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{28} \cdot 3^{7} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$1.424700212$ |
1.039746854 |
\( \frac{291660246420647}{587068342272} a + \frac{38130083166223}{97844723712} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -4 a - 17\) , \( -9 a\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(-4a-17\right){x}-9a$ |
| 6.4-a5 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( -\frac{9841}{48} a + \frac{8911}{8} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( a - 2\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(-a-1\right){x}^2+\left(a-2\right){x}+1$ |
| 6.4-a6 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{14} \cdot 3^{2} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2^{2} \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( \frac{15457}{36} a + \frac{13391}{6} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -2 a + 2\) , \( 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(-2a+2\right){x}+3$ |
| 6.4-a7 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3 \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 1 \) |
$1$ |
$9.972901485$ |
1.039746854 |
\( -\frac{57217}{6} a + 27208 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( a - 2\) , \( -a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-1\right){x}^2+\left(a-2\right){x}-a+5$ |
| 6.4-a8 |
6.4-a |
$8$ |
$28$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{13} \cdot 3^{4} \) |
$0.67072$ |
$(2,a+1), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.2.3 |
$1$ |
\( 2 \) |
$1$ |
$4.986450742$ |
1.039746854 |
\( \frac{238419887}{162} a + \frac{18567478}{27} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -12 a + 22\) , \( 8 a - 45\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(-12a+22\right){x}+8a-45$ |
| 26.1-a1 |
26.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.1 |
\( 2 \cdot 13 \) |
\( 2^{3} \cdot 13^{3} \) |
$0.96771$ |
$(2,a), (13,a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$3.545248488$ |
0.492823607 |
\( \frac{16964307415}{17576} a - \frac{7085161553}{17576} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 4 a + 4\) , \( a - 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(4a+4\right){x}+a-30$ |
| 26.1-a2 |
26.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.1 |
\( 2 \cdot 13 \) |
\( 2^{9} \cdot 13^{9} \) |
$0.96771$ |
$(2,a), (13,a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$1.181749496$ |
0.492823607 |
\( \frac{2187264503235295}{5429503678976} a - \frac{3398685237054473}{5429503678976} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 9 a - 21\) , \( 26 a - 58\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(9a-21\right){x}+26a-58$ |
| 26.1-a3 |
26.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.1 |
\( 2 \cdot 13 \) |
\( 2 \cdot 13 \) |
$0.96771$ |
$(2,a), (13,a+4)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$10.63574546$ |
0.492823607 |
\( \frac{703}{26} a - \frac{2345}{26} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-a-1\right){x}$ |
| 26.1-a4 |
26.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.1 |
\( 2 \cdot 13 \) |
\( 2^{27} \cdot 13^{3} \) |
$0.96771$ |
$(2,a), (13,a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$0.393916498$ |
0.492823607 |
\( \frac{305308359952290279703}{294876348416} a + \frac{525859732378041384175}{294876348416} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 824 a - 2026\) , \( 19455 a - 22260\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(824a-2026\right){x}+19455a-22260$ |
| 26.4-a1 |
26.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{9} \cdot 13^{9} \) |
$0.96771$ |
$(2,a+1), (13,a+8)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$1.181749496$ |
0.492823607 |
\( -\frac{2187264503235295}{5429503678976} a - \frac{605710366909589}{2714751839488} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -10 a - 11\) , \( -26 a - 32\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-10a-11\right){x}-26a-32$ |
| 26.4-a2 |
26.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2 \cdot 13 \) |
$0.96771$ |
$(2,a+1), (13,a+8)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$10.63574546$ |
0.492823607 |
\( -\frac{703}{26} a - \frac{821}{13} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2-{x}$ |
| 26.4-a3 |
26.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{3} \cdot 13^{3} \) |
$0.96771$ |
$(2,a+1), (13,a+8)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$3.545248488$ |
0.492823607 |
\( -\frac{16964307415}{17576} a + \frac{4939572931}{8788} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -5 a + 9\) , \( -a - 29\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a+9\right){x}-a-29$ |
| 26.4-a4 |
26.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{27} \cdot 13^{3} \) |
$0.96771$ |
$(2,a+1), (13,a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$0.393916498$ |
0.492823607 |
\( -\frac{305308359952290279703}{294876348416} a + \frac{415584046165165831939}{147438174208} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -825 a - 1201\) , \( -19455 a - 2805\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-825a-1201\right){x}-19455a-2805$ |
| 27.2-a1 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{27256702}{81} a - \frac{57582961}{81} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -2 a + 13\) , \( -11 a - 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(-2a+13\right){x}-11a-20$ |
| 27.2-a2 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{27256702}{81} a - \frac{10108753}{27} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10 a + 60\) , \( 93 a + 58\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-10a+60\right){x}+93a+58$ |
| 27.2-a3 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{2924207}{81} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -16 a + 13\) , \( 34 a - 88\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-16a+13\right){x}+34a-88$ |
| 27.2-a4 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{1950520}{6561} a + \frac{58507}{2187} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 1\) , \( a - 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+{x}+a-2$ |
| 27.2-a5 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{1950520}{6561} a + \frac{2126041}{6561} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -2 a + 12\) , \( -a - 26\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-2a+12\right){x}-a-26$ |
| 27.2-a6 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -17 a - 78\) , \( -130 a - 80\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-17a-78\right){x}-130a-80$ |
| 27.2-a7 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( -\frac{67902559538}{43046721} a + \frac{44765633473}{14348907} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 10 a - 14\) , \( 13 a + 34\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(10a-14\right){x}+13a+34$ |
| 27.2-a8 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{10} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{12214672127}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -241 a + 148\) , \( 1654 a - 4948\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-241a+148\right){x}+1654a-4948$ |
| 27.3-a1 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{27256702}{81} a - \frac{57582961}{81} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 10 a + 47\) , \( -32 a + 94\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(10a+47\right){x}-32a+94$ |
| 27.3-a2 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{27256702}{81} a - \frac{10108753}{27} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 4 a + 11\) , \( 14 a - 20\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(4a+11\right){x}+14a-20$ |
| 27.3-a3 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{2924207}{81} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 14 a - 1\) , \( -35 a - 53\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(14a-1\right){x}-35a-53$ |
| 27.3-a4 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{1950520}{6561} a + \frac{58507}{2187} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 2 a + 7\) , \( 14 a - 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(2a+7\right){x}+14a-36$ |
| 27.3-a5 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{1950520}{6561} a + \frac{2126041}{6561} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 2 a + 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(2a+1\right){x}$ |
| 27.3-a6 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -8 a - 4\) , \( -22 a + 43\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-8a-4\right){x}-22a+43$ |
| 27.3-a7 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( -\frac{67902559538}{43046721} a + \frac{44765633473}{14348907} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 17 a - 98\) , \( 53 a - 309\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(17a-98\right){x}+53a-309$ |
| 27.3-a8 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{10} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{12214672127}{9} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 239 a - 91\) , \( -1655 a - 3293\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(239a-91\right){x}-1655a-3293$ |
| 32.3-a1 |
32.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
32.3 |
\( 2^{5} \) |
\( 2^{20} \) |
$1.01927$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2 \cdot 3 \) |
$0.019902703$ |
$8.580110917$ |
0.854579154 |
\( 512 a + 256 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4\) , \( -a - 1\bigr] \) |
${y}^2={x}^3+\left(a-1\right){x}^2-4{x}-a-1$ |
| 32.4-a1 |
32.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
32.4 |
\( 2^{5} \) |
\( 2^{20} \) |
$1.01927$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 2 \cdot 3 \) |
$0.019902703$ |
$8.580110917$ |
0.854579154 |
\( -512 a + 768 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -4\) , \( a - 2\bigr] \) |
${y}^2={x}^3-a{x}^2-4{x}+a-2$ |
| 36.4-a1 |
36.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.4 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{9} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.159931548$ |
$2.500359038$ |
0.667056446 |
\( \frac{520033}{32} a - \frac{141195}{64} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -26 a + 15\) , \( -25 a + 239\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-26a+15\right){x}-25a+239$ |
| 36.4-a2 |
36.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.4 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{3} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.479794644$ |
$7.501077115$ |
0.667056446 |
\( -\frac{7739}{2} a - \frac{8427}{4} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -a\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2-a{x}$ |
| 36.4-a3 |
36.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.4 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{17} \cdot 3^{3} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.959589288$ |
$7.501077115$ |
0.667056446 |
\( \frac{21797}{16} a - \frac{5757}{8} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -a\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}-a+1$ |
| 36.4-a4 |
36.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.4 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{27} \cdot 3^{9} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.319863096$ |
$2.500359038$ |
0.667056446 |
\( -\frac{1982707}{4096} a + \frac{3848079}{2048} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 4 a + 30\) , \( 21 a - 27\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(4a+30\right){x}+21a-27$ |
| 36.6-a1 |
36.6-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.6 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{3} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.479794644$ |
$7.501077115$ |
0.667056446 |
\( \frac{7739}{2} a - \frac{23905}{4} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -a + 5\) , \( 3 a + 3\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-a+5\right){x}+3a+3$ |
| 36.6-a2 |
36.6-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.6 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{17} \cdot 3^{3} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.959589288$ |
$7.501077115$ |
0.667056446 |
\( -\frac{21797}{16} a + \frac{10283}{16} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-a-1\right){x}$ |
| 36.6-a3 |
36.6-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.6 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{27} \cdot 3^{9} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.319863096$ |
$2.500359038$ |
0.667056446 |
\( \frac{1982707}{4096} a + \frac{5713451}{4096} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -6 a + 34\) , \( -22 a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-6a+34\right){x}-22a-6$ |
| 36.6-a4 |
36.6-a |
$4$ |
$6$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
36.6 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{9} \) |
$1.04973$ |
$(2,a), (2,a+1), (3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.159931548$ |
$2.500359038$ |
0.667056446 |
\( -\frac{520033}{32} a + \frac{898871}{64} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 24 a - 5\) , \( 43 a + 67\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(24a-5\right){x}+43a+67$ |
*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.
