| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$3.30628$ |
$\textsf{none}$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$54.78414847$ |
$0.317761710$ |
1.411484168 |
\( -371323264041 \) |
\( \bigl[a^{2} - 3 a - 6\) , \( -a^{2} + 4 a + 7\) , \( a\) , \( 136 a^{2} - 631 a - 2026\) , \( 1388 a^{2} - 9741 a - 27946\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+4a+7\right){x}^{2}+\left(136a^{2}-631a-2026\right){x}+1388a^{2}-9741a-27946$ |
| 1.1-a2 |
1.1-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$3.30628$ |
$\textsf{none}$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
✓ |
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$7.826306925$ |
$108.9922668$ |
1.411484168 |
\( 999 \) |
\( \bigl[a^{2} - 3 a - 6\) , \( -a^{2} + 4 a + 7\) , \( a\) , \( a^{2} - a - 1\) , \( a + 1\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+4a+7\right){x}^{2}+\left(a^{2}-a-1\right){x}+a+1$ |
| 8.1-a1 |
8.1-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{6} \) |
$4.67579$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.548828472$ |
$32.31836636$ |
2.876309131 |
\( -\frac{1369}{4} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( a^{2} - 2 a - 6\) , \( a\) , \( 2 a^{2} - 8 a - 13\) , \( -4 a^{2} - 19 a - 17\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-2a-6\right){x}^{2}+\left(2a^{2}-8a-13\right){x}-4a^{2}-19a-17$ |
| 8.1-b1 |
8.1-b |
$2$ |
$3$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{54} \) |
$4.67579$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.572764501$ |
$9.008655559$ |
2.297593568 |
\( \frac{212207543}{262144} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( -a^{2} + 4 a + 7\) , \( 0\) , \( 31 a^{2} - 79 a - 128\) , \( 119 a^{2} - 265 a - 464\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}={x}^{3}+\left(-a^{2}+4a+7\right){x}^{2}+\left(31a^{2}-79a-128\right){x}+119a^{2}-265a-464$ |
| 8.1-b2 |
8.1-b |
$2$ |
$3$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{18} \) |
$4.67579$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$0.524254833$ |
$243.2337001$ |
2.297593568 |
\( -\frac{8398297}{64} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( -a^{2} + 4 a + 7\) , \( 0\) , \( -9 a^{2} + 26 a + 47\) , \( 44 a^{2} - 140 a - 219\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}={x}^{3}+\left(-a^{2}+4a+7\right){x}^{2}+\left(-9a^{2}+26a+47\right){x}+44a^{2}-140a-219$ |
| 11.1-a1 |
11.1-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{5} \) |
$4.93067$ |
$(-a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.809781077$ |
$46.54730820$ |
3.056199681 |
\( -\frac{1252357267456}{161051} a^{2} + \frac{3993799905280}{161051} a + \frac{6288263864320}{161051} \) |
\( \bigl[0\) , \( a^{2} - 4 a - 6\) , \( a^{2} - 2 a - 6\) , \( 2 a^{2} - a - 4\) , \( 3 a^{2} + 9 a + 5\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(a^{2}-4a-6\right){x}^{2}+\left(2a^{2}-a-4\right){x}+3a^{2}+9a+5$ |
| 11.2-a1 |
11.2-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
11.2 |
\( 11 \) |
\( - 11^{5} \) |
$4.93067$ |
$(-a^2+3a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.809781077$ |
$46.54730820$ |
3.056199681 |
\( \frac{1489085370368}{161051} a^{2} - \frac{3214898843648}{161051} a - \frac{1286744715264}{14641} \) |
\( \bigl[0\) , \( a^{2} - 4 a - 8\) , \( a\) , \( 7 a^{2} - 18 a - 40\) , \( 24 a^{2} - 83 a - 107\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a^{2}-4a-8\right){x}^{2}+\left(7a^{2}-18a-40\right){x}+24a^{2}-83a-107$ |
| 11.3-a1 |
11.3-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
11.3 |
\( 11 \) |
\( - 11^{5} \) |
$4.93067$ |
$(a^2-2a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.809781077$ |
$46.54730820$ |
3.056199681 |
\( -\frac{236728102912}{161051} a^{2} - \frac{778901061632}{161051} a - \frac{53128380416}{14641} \) |
\( \bigl[0\) , \( a^{2} - 4 a - 7\) , \( a^{2} - 3 a - 7\) , \( -2 a^{2} + 2 a + 28\) , \( -22 a^{2} + 51 a + 222\bigr] \) |
${y}^2+\left(a^{2}-3a-7\right){y}={x}^{3}+\left(a^{2}-4a-7\right){x}^{2}+\left(-2a^{2}+2a+28\right){x}-22a^{2}+51a+222$ |
| 23.1-a1 |
23.1-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$5.57564$ |
$(a^2-2a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$96.78976410$ |
2.615939570 |
\( \frac{38794}{23} a^{2} - \frac{77118}{23} a - \frac{391937}{23} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( a^{2} - 3 a - 8\) , \( a\) , \( 5 a^{2} - 14 a - 41\) , \( -7 a^{2} + 14 a + 67\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-8\right){x}^{2}+\left(5a^{2}-14a-41\right){x}-7a^{2}+14a+67$ |
| 23.1-b1 |
23.1-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
23.1 |
\( 23 \) |
\( 23 \) |
$5.57564$ |
$(a^2-2a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$67.20189654$ |
1.816267474 |
\( -\frac{32919199}{23} a^{2} + \frac{100068060}{23} a + \frac{185837224}{23} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( -a + 1\) , \( a^{2} - 3 a - 6\) , \( -2 a - 4\) , \( 2 a^{2} - 6 a - 27\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a-4\right){x}+2a^{2}-6a-27$ |
| 23.2-a1 |
23.2-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$5.57564$ |
$(a^2-3a-8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$96.78976410$ |
2.615939570 |
\( -\frac{39264}{23} a^{2} + \frac{117322}{23} a + \frac{193733}{23} \) |
\( \bigl[a^{2} - 3 a - 7\) , \( -a^{2} + 4 a + 7\) , \( a^{2} - 2 a - 6\) , \( -3 a^{2} + 12 a + 17\) , \( 6 a^{2} - 19 a - 31\bigr] \) |
${y}^2+\left(a^{2}-3a-7\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(-a^{2}+4a+7\right){x}^{2}+\left(-3a^{2}+12a+17\right){x}+6a^{2}-19a-31$ |
| 23.2-b1 |
23.2-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( 23 \) |
$5.57564$ |
$(a^2-3a-8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$67.20189654$ |
1.816267474 |
\( -\frac{1310463}{23} a^{2} - \frac{30298273}{23} a - \frac{34113465}{23} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( a\) , \( a + 1\) , \( 3 a^{2} - 6 a - 24\) , \( 2 a - 13\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(3a^{2}-6a-24\right){x}+2a-13$ |
| 23.3-a1 |
23.3-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
23.3 |
\( 23 \) |
\( -23 \) |
$5.57564$ |
$(a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$96.78976410$ |
2.615939570 |
\( \frac{470}{23} a^{2} - 1748 a - \frac{84875}{23} \) |
\( \bigl[1\) , \( -a^{2} + 3 a + 8\) , \( a^{2} - 3 a - 6\) , \( 2\) , \( -a^{2} + 4 a + 2\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(-a^{2}+3a+8\right){x}^{2}+2{x}-a^{2}+4a+2$ |
| 23.3-b1 |
23.3-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
23.3 |
\( 23 \) |
\( 23 \) |
$5.57564$ |
$(a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$67.20189654$ |
1.816267474 |
\( \frac{34229662}{23} a^{2} - 3033469 a - \frac{317124002}{23} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 8\) , \( a^{2} - 2 a - 7\) , \( -3 a^{2} + 7 a + 18\) , \( -3 a^{2} + 6 a + 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a^{2}+2a+8\right){x}^{2}+\left(-3a^{2}+7a+18\right){x}-3a^{2}+6a+11$ |
| 29.1-a1 |
29.1-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
29.1 |
\( 29 \) |
\( 29^{3} \) |
$5.79526$ |
$(a^2-2a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.218029403$ |
$210.0577965$ |
3.713414284 |
\( -\frac{709537158}{24389} a^{2} + \frac{1458012393}{24389} a + \frac{6586762104}{24389} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( -a^{2} + 4 a + 7\) , \( a^{2} - 2 a - 6\) , \( 2 a + 1\) , \( 2 a^{2} - 6 a - 11\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(-a^{2}+4a+7\right){x}^{2}+\left(2a+1\right){x}+2a^{2}-6a-11$ |
| 29.2-a1 |
29.2-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
29.2 |
\( 29 \) |
\( 29^{3} \) |
$5.79526$ |
$(a^2-3a-10)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.218029403$ |
$210.0577965$ |
3.713414284 |
\( \frac{670599081}{24389} a^{2} - \frac{2050735320}{24389} a - \frac{3744790650}{24389} \) |
\( \bigl[a + 1\) , \( -a^{2} + 4 a + 8\) , \( 1\) , \( 3 a^{2} + 2 a - 1\) , \( 3 a^{2} + 14 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+4a+8\right){x}^{2}+\left(3a^{2}+2a-1\right){x}+3a^{2}+14a+12$ |
| 29.3-a1 |
29.3-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
29.3 |
\( 29 \) |
\( 29^{3} \) |
$5.79526$ |
$(a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.218029403$ |
$210.0577965$ |
3.713414284 |
\( \frac{38938077}{24389} a^{2} + \frac{592722927}{24389} a + \frac{637898301}{24389} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( a^{2} - 3 a - 7\) , \( a\) , \( a^{2} - 5 a - 4\) , \( -2 a^{2} + 3 a + 20\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-7\right){x}^{2}+\left(a^{2}-5a-4\right){x}-2a^{2}+3a+20$ |
| 31.1-a1 |
31.1-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
31.1 |
\( 31 \) |
\( 31 \) |
$5.86004$ |
$(a^2-2a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$100.2528089$ |
2.709535376 |
\( \frac{21319}{31} a^{2} + \frac{50062}{31} a + \frac{7513}{31} \) |
\( \bigl[a + 1\) , \( a^{2} - 3 a - 8\) , \( a\) , \( -a + 1\) , \( a^{2} - 3 a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3a-8\right){x}^{2}+\left(-a+1\right){x}+a^{2}-3a-6$ |
| 31.1-b1 |
31.1-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{4} \) |
$5.86004$ |
$(a^2-2a-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.208048551$ |
$102.9638219$ |
3.473752544 |
\( -\frac{677349226}{923521} a^{2} + \frac{687290191}{923521} a + \frac{2180597954}{923521} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( -a + 1\) , \( a\) , \( -a^{2} - 4 a + 3\) , \( a^{2} + 7 a + 11\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a^{2}-4a+3\right){x}+a^{2}+7a+11$ |
| 31.2-a1 |
31.2-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
31.2 |
\( 31 \) |
\( 31 \) |
$5.86004$ |
$(a^2-3a-9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$100.2528089$ |
2.709535376 |
\( -\frac{114019}{31} a^{2} + \frac{249357}{31} a + \frac{1068898}{31} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( -a^{2} + 3 a + 7\) , \( a^{2} - 2 a - 6\) , \( -a^{2} + 2 a + 8\) , \( -a^{2} + 3 a + 8\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-a^{2}+2a+8\right){x}-a^{2}+3a+8$ |
| 31.2-b1 |
31.2-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{4} \) |
$5.86004$ |
$(a^2-3a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.208048551$ |
$102.9638219$ |
3.473752544 |
\( \frac{1344757487}{923521} a^{2} - \frac{3366864200}{923521} a - \frac{13318906524}{923521} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( a\) , \( a^{2} - 2 a - 6\) , \( 5 a^{2} - 10 a - 46\) , \( -11 a^{2} + 23 a + 105\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+a{x}^{2}+\left(5a^{2}-10a-46\right){x}-11a^{2}+23a+105$ |
| 31.3-a1 |
31.3-a |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
31.3 |
\( 31 \) |
\( 31 \) |
$5.86004$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$100.2528089$ |
2.709535376 |
\( \frac{92700}{31} a^{2} - \frac{299419}{31} a - \frac{470835}{31} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( -a^{2} + 2 a + 7\) , \( a^{2} - 3 a - 7\) , \( -a^{2} + 3 a + 8\) , \( -a^{2} + 2 a + 9\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a^{2}-3a-7\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(-a^{2}+3a+8\right){x}-a^{2}+2a+9$ |
| 31.3-b1 |
31.3-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
31.3 |
\( 31 \) |
\( - 31^{4} \) |
$5.86004$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.208048551$ |
$102.9638219$ |
3.473752544 |
\( -\frac{667408261}{923521} a^{2} + \frac{2679574009}{923521} a + \frac{1433661973}{923521} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 8\) , \( a^{2} - 3 a - 7\) , \( -4 a^{2} + 14 a + 26\) , \( 9 a^{2} - 28 a - 44\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3a-7\right){y}={x}^{3}+\left(-a^{2}+2a+8\right){x}^{2}+\left(-4a^{2}+14a+26\right){x}+9a^{2}-28a-44$ |
| 37.1-a1 |
37.1-a |
$3$ |
$9$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$6.03542$ |
$(2a^2-5a-15)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 3 \) |
$1$ |
$0.382153083$ |
2.509816198 |
\( \frac{727057727488000}{37} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -1873\) , \( -31833\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-1873{x}-31833$ |
| 37.1-a2 |
37.1-a |
$3$ |
$9$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{9} \) |
$6.03542$ |
$(2a^2-5a-15)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$9$ |
\( 3^{2} \) |
$1$ |
$10.31813326$ |
2.509816198 |
\( \frac{1404928000}{50653} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -23\) , \( -50\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-23{x}-50$ |
| 37.1-a3 |
37.1-a |
$3$ |
$9$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$6.03542$ |
$(2a^2-5a-15)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 3 \) |
$1$ |
$278.5895980$ |
2.509816198 |
\( \frac{4096000}{37} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-3{x}+1$ |
| 37.1-b1 |
37.1-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$6.03542$ |
$(2a^2-5a-15)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$4$ |
\( 1 \) |
$0.051111408$ |
$214.5901461$ |
3.557190669 |
\( \frac{110592}{37} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}-{x}$ |
| 47.1-a1 |
47.1-a |
$2$ |
$2$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{6} \) |
$6.28092$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$43.83814057$ |
1.777221915 |
\( \frac{395223806856609}{10779215329} a^{2} + \frac{1340990395204473}{10779215329} a + \frac{1044650149435881}{10779215329} \) |
\( \bigl[a^{2} - 3 a - 7\) , \( a^{2} - 2 a - 6\) , \( a^{2} - 3 a - 6\) , \( 66 a^{2} - 143 a - 619\) , \( -206 a^{2} + 443 a + 1966\bigr] \) |
${y}^2+\left(a^{2}-3a-7\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(a^{2}-2a-6\right){x}^{2}+\left(66a^{2}-143a-619\right){x}-206a^{2}+443a+1966$ |
| 47.1-a2 |
47.1-a |
$2$ |
$2$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{3} \) |
$6.28092$ |
$(a^2-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$87.67628115$ |
1.777221915 |
\( -\frac{1347916626}{103823} a^{2} + \frac{2765259567}{103823} a + \frac{13371178734}{103823} \) |
\( \bigl[a^{2} - 3 a - 7\) , \( a^{2} - 2 a - 6\) , \( a^{2} - 3 a - 6\) , \( 36 a^{2} - 78 a - 334\) , \( 171 a^{2} - 370 a - 1617\bigr] \) |
${y}^2+\left(a^{2}-3a-7\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(a^{2}-2a-6\right){x}^{2}+\left(36a^{2}-78a-334\right){x}+171a^{2}-370a-1617$ |
| 47.2-a1 |
47.2-a |
$2$ |
$2$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
47.2 |
\( 47 \) |
\( - 47^{3} \) |
$6.28092$ |
$(a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$87.67628115$ |
1.777221915 |
\( \frac{1278490311}{103823} a^{2} - \frac{3904897248}{103823} a - \frac{6292160136}{103823} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 6\) , \( 1\) , \( -28 a^{2} + 90 a + 143\) , \( -114 a^{2} + 364 a + 574\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+3a+6\right){x}^{2}+\left(-28a^{2}+90a+143\right){x}-114a^{2}+364a+574$ |
| 47.2-a2 |
47.2-a |
$2$ |
$2$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
47.2 |
\( 47 \) |
\( - 47^{6} \) |
$6.28092$ |
$(a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$43.83814057$ |
1.777221915 |
\( -\frac{2526661815774300}{10779215329} a^{2} + \frac{5448547438405209}{10779215329} a + \frac{24024511323626544}{10779215329} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 6\) , \( 1\) , \( -53 a^{2} + 170 a + 268\) , \( 229 a^{2} - 729 a - 1151\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+3a+6\right){x}^{2}+\left(-53a^{2}+170a+268\right){x}+229a^{2}-729a-1151$ |
| 47.3-a1 |
47.3-a |
$2$ |
$2$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
47.3 |
\( 47 \) |
\( - 47^{3} \) |
$6.28092$ |
$(-a^2+a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$87.67628115$ |
1.777221915 |
\( \frac{69426315}{103823} a^{2} + \frac{1139637681}{103823} a + \frac{2101861521}{103823} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 6\) , \( 1\) , \( -6 a^{2} + 20 a + 32\) , \( 47 a^{2} - 149 a - 236\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+3a+6\right){x}^{2}+\left(-6a^{2}+20a+32\right){x}+47a^{2}-149a-236$ |
| 47.3-a2 |
47.3-a |
$2$ |
$2$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
47.3 |
\( 47 \) |
\( - 47^{6} \) |
$6.28092$ |
$(-a^2+a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$43.83814057$ |
1.777221915 |
\( \frac{2131438008917691}{10779215329} a^{2} - \frac{6789537833609682}{10779215329} a - \frac{10713625458135084}{10779215329} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 6\) , \( 1\) , \( -141 a^{2} + 450 a + 712\) , \( 1880 a^{2} - 5991 a - 9455\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+3a+6\right){x}^{2}+\left(-141a^{2}+450a+712\right){x}+1880a^{2}-5991a-9455$ |
| 88.1-a1 |
88.1-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.1 |
\( 2^{3} \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$6.97302$ |
$(-a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 7 \) |
$47.90046731$ |
$0.190808887$ |
5.187473862 |
\( -\frac{97380547627589212972907}{704} a^{2} + \frac{210109395070485307661361}{704} a + \frac{1850683014770341823359889}{1408} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( a^{2} - 4 a - 8\) , \( a^{2} - 3 a - 6\) , \( 526 a^{2} - 1206 a - 5308\) , \( 11598 a^{2} - 25450 a - 113123\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(a^{2}-4a-8\right){x}^{2}+\left(526a^{2}-1206a-5308\right){x}+11598a^{2}-25450a-113123$ |
| 88.1-a2 |
88.1-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.1 |
\( 2^{3} \cdot 11 \) |
\( 2^{3} \cdot 11^{7} \) |
$6.97302$ |
$(-a), (2)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$6.842923901$ |
$65.44744857$ |
5.187473862 |
\( -\frac{27471024869}{38974342} a^{2} + \frac{85546582981}{38974342} a + \frac{105680614975}{19487171} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( a^{2} - 4 a - 8\) , \( a^{2} - 3 a - 6\) , \( 6 a^{2} - 16 a - 28\) , \( 8 a^{2} - 28 a - 43\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(a^{2}-4a-8\right){x}^{2}+\left(6a^{2}-16a-28\right){x}+8a^{2}-28a-43$ |
| 88.1-b1 |
88.1-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.1 |
\( 2^{3} \cdot 11 \) |
\( - 2^{6} \cdot 11^{4} \) |
$6.97302$ |
$(-a), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.188029100$ |
$38.12571116$ |
6.974993347 |
\( -\frac{241528577}{29282} a^{2} - \frac{2206323001}{58564} a - \frac{1235562467}{29282} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( -a^{2} + 2 a + 6\) , \( a + 1\) , \( -3 a + 4\) , \( -3 a^{2} + 8 a + 16\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+6\right){x}^{2}+\left(-3a+4\right){x}-3a^{2}+8a+16$ |
| 88.2-a1 |
88.2-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.2 |
\( 2^{3} \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$6.97302$ |
$(-a^2+3a+7), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 7 \) |
$47.90046731$ |
$0.190808887$ |
5.187473862 |
\( \frac{15348299815306881715547}{704} a^{2} + \frac{25667824090834283913133}{352} a + \frac{7065277755874370161429}{128} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( a + 1\) , \( a + 1\) , \( -149 a^{2} - 71 a - 74\) , \( -2842 a^{2} - 4923 a - 3294\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-149a^{2}-71a-74\right){x}-2842a^{2}-4923a-3294$ |
| 88.2-a2 |
88.2-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.2 |
\( 2^{3} \cdot 11 \) |
\( 2^{3} \cdot 11^{7} \) |
$6.97302$ |
$(-a^2+3a+7), (2)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$6.842923901$ |
$65.44744857$ |
5.187473862 |
\( \frac{30604533243}{38974342} a^{2} - \frac{32171287430}{19487171} a - \frac{20239881973}{3543122} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( a + 1\) , \( a + 1\) , \( a^{2} - a - 4\) , \( a + 2\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a^{2}-a-4\right){x}+a+2$ |
| 88.2-b1 |
88.2-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.2 |
\( 2^{3} \cdot 11 \) |
\( - 2^{6} \cdot 11^{4} \) |
$6.97302$ |
$(-a^2+3a+7), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.188029100$ |
$38.12571116$ |
6.974993347 |
\( -\frac{3172437309}{58564} a^{2} + \frac{10000369081}{58564} a + \frac{1442861727}{5324} \) |
\( \bigl[a + 1\) , \( 1\) , \( a^{2} - 2 a - 7\) , \( -a^{2} + 4 a + 14\) , \( 2 a^{2} - 2 a - 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+{x}^{2}+\left(-a^{2}+4a+14\right){x}+2a^{2}-2a-15$ |
| 88.3-a1 |
88.3-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.3 |
\( 2^{3} \cdot 11 \) |
\( 2^{3} \cdot 11^{7} \) |
$6.97302$ |
$(a^2-2a-8), (2)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$6.842923901$ |
$65.44744857$ |
5.187473862 |
\( -\frac{1566754187}{19487171} a^{2} - \frac{21204008121}{38974342} a + \frac{4012011169}{3543122} \) |
\( \bigl[a + 1\) , \( a^{2} - 4 a - 6\) , \( a^{2} - 3 a - 6\) , \( 3 a^{2} - 10 a - 14\) , \( a^{2} - 3 a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(a^{2}-4a-6\right){x}^{2}+\left(3a^{2}-10a-14\right){x}+a^{2}-3a-5$ |
| 88.3-a2 |
88.3-a |
$2$ |
$7$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.3 |
\( 2^{3} \cdot 11 \) |
\( 2^{21} \cdot 11 \) |
$6.97302$ |
$(a^2-2a-8), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 7 \) |
$47.90046731$ |
$0.190808887$ |
5.187473862 |
\( \frac{5127015488267645703585}{44} a^{2} - \frac{261445043252153875487627}{704} a - \frac{75014601546584041670779}{128} \) |
\( \bigl[a + 1\) , \( a^{2} - 4 a - 6\) , \( a^{2} - 3 a - 6\) , \( -367 a^{2} + 1250 a + 1306\) , \( -10607 a^{2} + 34663 a + 48219\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3a-6\right){y}={x}^{3}+\left(a^{2}-4a-6\right){x}^{2}+\left(-367a^{2}+1250a+1306\right){x}-10607a^{2}+34663a+48219$ |
| 88.3-b1 |
88.3-b |
$1$ |
$1$ |
3.3.1369.1 |
$3$ |
$[3, 0]$ |
88.3 |
\( 2^{3} \cdot 11 \) |
\( - 2^{6} \cdot 11^{4} \) |
$6.97302$ |
$(a^2-2a-8), (2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.188029100$ |
$38.12571116$ |
6.974993347 |
\( \frac{3655494463}{58564} a^{2} - \frac{1948511520}{14641} a - \frac{797647289}{1331} \) |
\( \bigl[a^{2} - 3 a - 6\) , \( 1\) , \( a + 1\) , \( 4 a^{2} - 11 a - 19\) , \( 2 a^{2} - 9 a - 13\bigr] \) |
${y}^2+\left(a^{2}-3a-6\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(4a^{2}-11a-19\right){x}+2a^{2}-9a-13$ |
*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.
