| 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 |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{13} \) |
$10.24424$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$117.9540467$ |
1.258180972 |
\( \frac{544633931021}{1220703125} a^{3} - \frac{1105452496523}{1220703125} a^{2} - \frac{112445163749}{48828125} a + \frac{835667587658}{1220703125} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( -2 a^{3} + 3 a^{2} + 11 a\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( -3 a^{3} + 5 a^{2} + 16 a - 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+11a\right){x}^{2}+\left(a^{3}-2a^{2}-3a+5\right){x}-3a^{3}+5a^{2}+16a-3$ |
| 5.1-b1 |
5.1-b |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5 \) |
$10.24424$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$100.9717091$ |
1.077035393 |
\( \frac{1316}{5} a^{3} - \frac{92313}{5} a^{2} - 11606 a + \frac{8698}{5} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( a + 1\) , \( -a^{2} + a + 3\) , \( a^{3} - 2 a^{2} - 4 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-1\right){x}^{2}+\left(-a^{2}+a+3\right){x}+a^{3}-2a^{2}-4a-2$ |
| 7.1-a1 |
7.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$10.68429$ |
$(-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.285024227$ |
$320.9155264$ |
1.951339205 |
\( \frac{290072363351}{49} a^{3} - \frac{70940688524}{7} a^{2} - \frac{1386705191304}{49} a + \frac{407403205719}{49} \) |
\( \bigl[a\) , \( -a^{3} + 2 a^{2} + 3 a\) , \( 0\) , \( -72 a^{3} + 192 a^{2} + 112 a - 44\) , \( -422 a^{3} + 1125 a^{2} + 658 a - 254\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a\right){x}^{2}+\left(-72a^{3}+192a^{2}+112a-44\right){x}-422a^{3}+1125a^{2}+658a-254$ |
| 7.1-a2 |
7.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$10.68429$ |
$(-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.142512113$ |
$1283.662105$ |
1.951339205 |
\( \frac{281343}{7} a^{3} - 69213 a^{2} - \frac{1348621}{7} a + \frac{411118}{7} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 5 a - 1\) , \( a + 1\) , \( -11 a^{3} + 20 a^{2} + 52 a - 17\) , \( 32 a^{3} - 54 a^{2} - 153 a + 44\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-1\right){x}^{2}+\left(-11a^{3}+20a^{2}+52a-17\right){x}+32a^{3}-54a^{2}-153a+44$ |
| 13.1-a1 |
13.1-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
13.1 |
\( 13 \) |
\( 13^{3} \) |
$11.54386$ |
$(-a^3+a^2+4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$202.6393083$ |
2.161493641 |
\( -\frac{41120614}{2197} a^{3} + \frac{86825361}{2197} a^{2} + \frac{174272011}{2197} a - \frac{143293803}{2197} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 1\) , \( a^{3} - a^{2} - 5 a - 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( -2 a^{3} + 4 a^{2} + 9 a + 1\) , \( a^{2} - 3\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+1\right){x}{y}+\left(a^{3}-a^{2}-6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-2\right){x}^{2}+\left(-2a^{3}+4a^{2}+9a+1\right){x}+a^{2}-3$ |
| 17.2-a1 |
17.2-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( - 17^{6} \) |
$11.93753$ |
$(a^3-a^2-5a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$424.6923656$ |
2.265034003 |
\( -\frac{10829536429963169633}{24137569} a^{3} + \frac{28863622406304923890}{24137569} a^{2} + \frac{16911537140223424137}{24137569} a - \frac{6503177294240882283}{24137569} \) |
\( \bigl[a + 1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -4 a^{3} - 9 a^{2} + 4\) , \( -6 a^{3} - 12 a^{2} - 3 a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-4a^{3}-9a^{2}+4\right){x}-6a^{3}-12a^{2}-3a+1$ |
| 17.2-a2 |
17.2-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( 17^{3} \) |
$11.93753$ |
$(a^3-a^2-5a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$849.3847312$ |
2.265034003 |
\( \frac{1137990911}{4913} a^{3} - \frac{2904519268}{4913} a^{2} - \frac{1728835193}{4913} a + \frac{667833748}{4913} \) |
\( \bigl[a + 1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -4 a^{3} - 4 a^{2} + 5 a + 4\) , \( -16 a^{3} - 34 a^{2} - 11 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-4a^{3}-4a^{2}+5a+4\right){x}-16a^{3}-34a^{2}-11a+3$ |
| 17.2-b1 |
17.2-b |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
17.2 |
\( 17 \) |
\( - 17^{3} \) |
$11.93753$ |
$(a^3-a^2-5a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.042328471$ |
$456.8325930$ |
2.475148073 |
\( -\frac{55525}{4913} a^{3} + \frac{786325}{4913} a^{2} + \frac{1592750}{4913} a + \frac{28239}{4913} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( a^{3} - a^{2} - 6 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( -2 a^{3} - 5 a^{2} - a + 1\) , \( 3 a^{3} + 5 a^{2} + 2 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a-2\right){x}^{2}+\left(-2a^{3}-5a^{2}-a+1\right){x}+3a^{3}+5a^{2}+2a-1$ |
| 17.3-a1 |
17.3-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
17.3 |
\( 17 \) |
\( -17 \) |
$11.93753$ |
$(-a^3+a^2+5a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.048141706$ |
$1183.553452$ |
2.431082063 |
\( -\frac{1257553}{17} a^{3} - \frac{2541725}{17} a^{2} - \frac{423630}{17} a + \frac{533078}{17} \) |
\( \bigl[a\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 2 a^{3} - 3 a^{2} - 9 a - 1\) , \( 4 a^{3} - 5 a^{2} - 20 a - 3\) , \( 2 a^{3} - 2 a^{2} - 10 a - 5\bigr] \) |
${y}^2+a{x}{y}+\left(2a^{3}-3a^{2}-9a-1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-9a\right){x}^{2}+\left(4a^{3}-5a^{2}-20a-3\right){x}+2a^{3}-2a^{2}-10a-5$ |
| 19.1-a1 |
19.1-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$12.10466$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.058770250$ |
$363.4061378$ |
2.733765840 |
\( \frac{547144747}{6859} a^{3} - \frac{532241339}{6859} a^{2} - \frac{3864802156}{6859} a - \frac{2279687744}{6859} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( -2 a^{3} + 3 a^{2} + 11 a\) , \( 1\) , \( 2 a^{3} - 5 a^{2} - 6 a + 10\) , \( -6 a^{3} + 10 a^{2} + 29 a - 6\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+3a^{2}+11a\right){x}^{2}+\left(2a^{3}-5a^{2}-6a+10\right){x}-6a^{3}+10a^{2}+29a-6$ |
| 25.1-a1 |
25.1-a |
$2$ |
$5$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{3} \) |
$12.52711$ |
$(a^2-2a-3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$1.473161916$ |
$601.2979291$ |
3.023572853 |
\( -124 a^{3} + 90 a^{2} + 767 a + 465 \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( a^{2} - 3 a - 3\) , \( 0\) , \( a^{2} - 3 a + 2\) , \( -13 a^{3} + 23 a^{2} + 60 a - 18\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}={x}^{3}+\left(a^{2}-3a-3\right){x}^{2}+\left(a^{2}-3a+2\right){x}-13a^{3}+23a^{2}+60a-18$ |
| 25.1-a2 |
25.1-a |
$2$ |
$5$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{3} \) |
$12.52711$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$7.365809584$ |
$4.810383433$ |
3.023572853 |
\( -30599407213 a^{3} + 52487536154 a^{2} + 146149189772 a - 43484265248 \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( 1\) , \( -16 a^{3} + 29 a^{2} + 74 a - 32\) , \( -43 a^{3} + 76 a^{2} + 205 a - 82\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-1\right){x}^{2}+\left(-16a^{3}+29a^{2}+74a-32\right){x}-43a^{3}+76a^{2}+205a-82$ |
| 25.1-b1 |
25.1-b |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{7} \) |
$12.52711$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$89.24985604$ |
1.904003698 |
\( \frac{1316}{5} a^{3} - \frac{92313}{5} a^{2} - 11606 a + \frac{8698}{5} \) |
\( \bigl[1\) , \( -2 a^{3} + 3 a^{2} + 10 a\) , \( 1\) , \( 5 a^{3} - 9 a^{2} - 24 a + 8\) , \( -13 a^{3} + 22 a^{2} + 62 a - 18\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-2a^{3}+3a^{2}+10a\right){x}^{2}+\left(5a^{3}-9a^{2}-24a+8\right){x}-13a^{3}+22a^{2}+62a-18$ |
| 25.1-c1 |
25.1-c |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{19} \) |
$12.52711$ |
$(a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$85.38971925$ |
1.821653821 |
\( \frac{544633931021}{1220703125} a^{3} - \frac{1105452496523}{1220703125} a^{2} - \frac{112445163749}{48828125} a + \frac{835667587658}{1220703125} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 7 a - 1\) , \( a^{3} - a^{2} - 6 a - 2\) , \( 2 a^{3} - 4 a^{2} - 9 a + 8\) , \( 2 a^{3} - 7 a^{2} - 2 a + 14\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-6a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-7a-1\right){x}^{2}+\left(2a^{3}-4a^{2}-9a+8\right){x}+2a^{3}-7a^{2}-2a+14$ |
| 25.1-d1 |
25.1-d |
$2$ |
$5$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{9} \) |
$12.52711$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.051549612$ |
$557.2882590$ |
2.451464207 |
\( -124 a^{3} + 90 a^{2} + 767 a + 465 \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 1\) , \( -a^{2} + 2 a + 4\) , \( a\) , \( -a^{2} + 3 a + 6\) , \( 11 a^{3} + 24 a^{2} + 9 a - 1\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-a^{2}+3a+6\right){x}+11a^{3}+24a^{2}+9a-1$ |
| 25.1-d2 |
25.1-d |
$2$ |
$5$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{9} \) |
$12.52711$ |
$(a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$0.257748064$ |
$111.4576518$ |
2.451464207 |
\( -30599407213 a^{3} + 52487536154 a^{2} + 146149189772 a - 43484265248 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( 2 a^{3} - 3 a^{2} - 10 a\) , \( 11 a^{3} - 65 a - 123\) , \( -18 a^{3} - 92 a^{2} + 310 a + 462\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(2a^{3}-3a^{2}-10a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(11a^{3}-65a-123\right){x}-18a^{3}-92a^{2}+310a+462$ |
| 29.2-a1 |
29.2-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29 \) |
$12.76169$ |
$(a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.494654696$ |
$144.3607983$ |
3.046784031 |
\( \frac{87128764}{29} a^{3} - \frac{63491433}{29} a^{2} - \frac{539960553}{29} a - \frac{320840758}{29} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 1\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( a\) , \( -a^{3} + 4 a^{2} - 2 a - 3\) , \( -a^{3} + 4 a^{2} - 2 a - 4\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+4a\right){x}^{2}+\left(-a^{3}+4a^{2}-2a-3\right){x}-a^{3}+4a^{2}-2a-4$ |
| 31.1-a1 |
31.1-a |
$2$ |
$3$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31 \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$2.427779910$ |
$3.844472540$ |
3.584089492 |
\( \frac{10077453354340891857}{31} a^{3} - \frac{17253366899431201762}{31} a^{2} - \frac{48179010153890934085}{31} a + \frac{14152214278679135626}{31} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a\) , \( 37 a^{3} - 88 a^{2} - 96 a + 12\) , \( 300 a^{3} - 579 a^{2} - 1016 a - 309\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(37a^{3}-88a^{2}-96a+12\right){x}+300a^{3}-579a^{2}-1016a-309$ |
| 31.1-a2 |
31.1-a |
$2$ |
$3$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.809259970$ |
$311.4022757$ |
3.584089492 |
\( \frac{8753568431}{29791} a^{3} - \frac{14981602414}{29791} a^{2} - \frac{41870763685}{29791} a + \frac{12298529441}{29791} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a\) , \( 7 a^{3} - 13 a^{2} - 31 a + 12\) , \( 2 a^{3} - 8 a^{2} - 3 a + 24\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(7a^{3}-13a^{2}-31a+12\right){x}+2a^{3}-8a^{2}-3a+24$ |
| 31.1-b1 |
31.1-b |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$167.6906818$ |
1.788706965 |
\( \frac{6213409058807}{29791} a^{3} - \frac{8633170361643}{29791} a^{2} - \frac{36475930058120}{29791} a + \frac{10416320461402}{29791} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( 6 a^{3} - 15 a^{2} - 10 a + 3\) , \( -15 a^{3} + 41 a^{2} + 24 a - 11\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-2a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+1\right){x}^{2}+\left(6a^{3}-15a^{2}-10a+3\right){x}-15a^{3}+41a^{2}+24a-11$ |
| 31.1-c1 |
31.1-c |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$4$ |
\( 1 \) |
$1$ |
$42.42759390$ |
1.810250442 |
\( \frac{41710021098}{29791} a^{3} - \frac{91407542611}{29791} a^{2} - \frac{228254055529}{29791} a + \frac{67866582299}{29791} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( -a^{3} + a^{2} + 5 a + 3\) , \( a + 1\) , \( -a^{3} - 3 a^{2} + 14 a + 8\) , \( 4 a^{3} - 13 a^{2} - a + 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+3\right){x}^{2}+\left(-a^{3}-3a^{2}+14a+8\right){x}+4a^{3}-13a^{2}-a+4$ |
| 31.1-d1 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{6} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$492.1641382$ |
0.656221184 |
\( \frac{25290147472452905}{887503681} a^{3} + \frac{53256018737476624}{887503681} a^{2} + \frac{13664823639496413}{887503681} a - \frac{8138544755072017}{887503681} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a - 1\) , \( 0\) , \( 48 a^{3} - 128 a^{2} - 76 a + 28\) , \( 46 a^{3} - 121 a^{2} - 73 a + 26\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(48a^{3}-128a^{2}-76a+28\right){x}+46a^{3}-121a^{2}-73a+26$ |
| 31.1-d2 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$246.0820691$ |
0.656221184 |
\( \frac{1248919832943643893409}{29791} a^{3} + \frac{2629997049374852711936}{29791} a^{2} + \frac{674771461425427867735}{29791} a - \frac{402122756071306404146}{29791} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a - 1\) , \( 0\) , \( 508 a^{3} - 1353 a^{2} - 821 a + 283\) , \( -12729 a^{3} + 33998 a^{2} + 19826 a - 7716\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(508a^{3}-1353a^{2}-821a+283\right){x}-12729a^{3}+33998a^{2}+19826a-7716$ |
| 31.1-d3 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{12} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$30.76025864$ |
0.656221184 |
\( -\frac{28701736354825279403647}{787662783788549761} a^{3} + \frac{83371699993555035031040}{787662783788549761} a^{2} + \frac{32340922878974239852151}{787662783788549761} a - \frac{13979020961410432424738}{787662783788549761} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( 2 a^{3} - 3 a^{2} - 10 a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( -242 a^{3} + 404 a^{2} + 1171 a - 287\) , \( -3175 a^{3} + 5490 a^{2} + 15103 a - 4738\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+\left(a^{3}-2a^{2}-4a+1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-10a-2\right){x}^{2}+\left(-242a^{3}+404a^{2}+1171a-287\right){x}-3175a^{3}+5490a^{2}+15103a-4738$ |
| 31.1-d4 |
31.1-d |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{3} \) |
$12.86852$ |
$(-2a^3+3a^2+10a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$984.3282764$ |
0.656221184 |
\( -\frac{95127527}{29791} a^{3} + \frac{13279296}{29791} a^{2} + \frac{424673785}{29791} a + \frac{282223088}{29791} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 7 a - 2\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a^{3} - 4 a^{2} - 7 a + 8\) , \( 8 a^{3} - 7 a^{2} - 31 a + 5\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-7a-2\right){x}^{2}+\left(a^{3}-4a^{2}-7a+8\right){x}+8a^{3}-7a^{2}-31a+5$ |
| 35.1-a1 |
35.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{4} \cdot 7^{6} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.483287911$ |
$50.15538214$ |
3.102665735 |
\( -\frac{4043474154791713}{73530625} a^{3} + \frac{2929092948444217}{10504375} a^{2} + \frac{423993467945597}{2941225} a - \frac{4308833732210674}{73530625} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 9 a - 1\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( a^{3} - a^{2} - 5 a - 2\) , \( -82 a^{3} + 133 a^{2} + 401 a - 81\) , \( -546 a^{3} + 940 a^{2} + 2601 a - 797\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-9a-1\right){x}{y}+\left(a^{3}-a^{2}-5a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(-82a^{3}+133a^{2}+401a-81\right){x}-546a^{3}+940a^{2}+2601a-797$ |
| 35.1-a2 |
35.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{3} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.241643955$ |
$200.6215285$ |
3.102665735 |
\( \frac{76517152162}{8575} a^{3} - \frac{18715485758}{1225} a^{2} - \frac{14637107969}{343} a + \frac{107527231676}{8575} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 9 a\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( 2 a^{3} - 3 a^{2} - 10 a - 1\) , \( 21 a^{3} - 51 a^{2} - 47 a + 12\) , \( -69 a^{3} + 189 a^{2} + 92 a - 42\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-9a\right){x}{y}+\left(2a^{3}-3a^{2}-10a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(21a^{3}-51a^{2}-47a+12\right){x}-69a^{3}+189a^{2}+92a-42$ |
| 35.1-b1 |
35.1-b |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{3} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1301.210971$ |
1.734954130 |
\( \frac{29223662}{8575} a^{3} - \frac{7824783}{1225} a^{2} - \frac{5450293}{343} a + \frac{76026126}{8575} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 2\) , \( -2 a^{3} + 3 a^{2} + 10 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 5 a^{3} - 15 a^{2} - 5 a + 11\) , \( 3 a^{3} - 12 a^{2} + 5 a + 7\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+10a\right){x}^{2}+\left(5a^{3}-15a^{2}-5a+11\right){x}+3a^{3}-12a^{2}+5a+7$ |
| 35.1-b2 |
35.1-b |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{8} \cdot 7^{3} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$20.33142143$ |
1.734954130 |
\( -\frac{3102531937073995838684}{133984375} a^{3} + \frac{322877548911664612181}{19140625} a^{2} + \frac{769158047830228689271}{5359375} a + \frac{11426265833850473892943}{133984375} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 2\) , \( -2 a^{3} + 3 a^{2} + 10 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 50 a^{3} - 135 a^{2} - 100 a + 21\) , \( 968 a^{3} - 2518 a^{2} - 1568 a + 505\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+10a\right){x}^{2}+\left(50a^{3}-135a^{2}-100a+21\right){x}+968a^{3}-2518a^{2}-1568a+505$ |
| 35.1-b3 |
35.1-b |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{4} \cdot 7^{6} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$162.6513714$ |
1.734954130 |
\( -\frac{158540855005413}{73530625} a^{3} + \frac{18308241073767}{10504375} a^{2} + \frac{38140123735772}{2941225} a + \frac{557071530887976}{73530625} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 2\) , \( -2 a^{3} + 3 a^{2} + 10 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 65 a^{3} - 175 a^{2} - 100 a + 46\) , \( 619 a^{3} - 1652 a^{2} - 960 a + 374\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+10a\right){x}^{2}+\left(65a^{3}-175a^{2}-100a+46\right){x}+619a^{3}-1652a^{2}-960a+374$ |
| 35.1-b4 |
35.1-b |
$4$ |
$4$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{12} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$10.16571071$ |
1.734954130 |
\( -\frac{23370143732176223008284}{346032180025} a^{3} + \frac{8898423457377306029381}{49433168575} a^{2} + \frac{1459634478451919488455}{13841287201} a - \frac{14032800992833495763857}{346032180025} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( 2 a^{3} - 3 a^{2} - 9 a - 1\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( -1072 a^{3} + 1839 a^{2} + 5118 a - 1526\) , \( -26889 a^{3} + 46041 a^{2} + 128543 a - 37794\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-9a-1\right){x}^{2}+\left(-1072a^{3}+1839a^{2}+5118a-1526\right){x}-26889a^{3}+46041a^{2}+128543a-37794$ |
| 35.1-c1 |
35.1-c |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{8} \cdot 7^{6} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.091391428$ |
$248.2712327$ |
2.904312747 |
\( \frac{759594198684002462}{45956640625} a^{3} + \frac{219644333649069767}{6565234375} a^{2} + \frac{12958127929814397}{1838265625} a - \frac{216434594184658399}{45956640625} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -a^{3} + 4 a^{2} + a - 19\) , \( 4 a^{3} - 14 a^{2} - 2 a + 18\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a\right){x}^{2}+\left(-a^{3}+4a^{2}+a-19\right){x}+4a^{3}-14a^{2}-2a+18$ |
| 35.1-c2 |
35.1-c |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{4} \cdot 7^{3} \) |
$13.06522$ |
$(a^2-2a-3), (-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.045695714$ |
$993.0849308$ |
2.904312747 |
\( -\frac{42753481787}{214375} a^{3} + \frac{5449538533}{30625} a^{2} + \frac{9860780828}{8575} a + \frac{141203683924}{214375} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 4 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 4 a^{3} - 6 a^{2} - 19 a - 9\) , \( -4 a^{3} + 31 a + 20\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a\right){x}^{2}+\left(4a^{3}-6a^{2}-19a-9\right){x}-4a^{3}+31a+20$ |
| 43.1-a1 |
43.1-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
43.1 |
\( 43 \) |
\( 43 \) |
$13.40577$ |
$(a^3-8a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.023494439$ |
$2536.480550$ |
2.542651762 |
\( \frac{3523879}{43} a^{3} - \frac{9725276}{43} a^{2} - \frac{4416083}{43} a + \frac{2914642}{43} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( 2 a^{3} - 3 a^{2} - 10 a - 1\) , \( 2 a^{3} - 3 a^{2} - 10 a\) , \( -a^{3} + 3 a^{2} + 3 a - 8\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(2a^{3}-3a^{2}-10a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-10a-1\right){x}^{2}+\left(-a^{3}+3a^{2}+3a-8\right){x}$ |
| 47.1-a1 |
47.1-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{3} \) |
$13.55566$ |
$(2a^3-3a^2-9a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$176.5561126$ |
1.883271897 |
\( \frac{10777207}{2209} a^{3} - \frac{32559610}{2209} a^{2} - \frac{121850431}{2209} a - \frac{65027825}{2209} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 1\) , \( 2 a^{3} - 3 a^{2} - 10 a\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 5 a^{3} - 7 a^{2} - 26 a - 1\) , \( 4 a^{3} - 6 a^{2} - 21 a\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-10a\right){x}^{2}+\left(5a^{3}-7a^{2}-26a-1\right){x}+4a^{3}-6a^{2}-21a$ |
| 47.1-b1 |
47.1-b |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{3} \) |
$13.55566$ |
$(2a^3-3a^2-9a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.049190634$ |
$438.5094193$ |
2.761041077 |
\( \frac{105677571}{2209} a^{3} - \frac{257038429}{2209} a^{2} - \frac{213723900}{2209} a + \frac{20644148}{2209} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( -a^{3} + 2 a^{2} + 5 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a\right){x}^{2}+\left(-a^{3}+a^{2}+2a-1\right){x}-a^{3}+2a^{2}+5a-2$ |
| 49.1-a1 |
49.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{8} \) |
$13.62645$ |
$(-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.282971866$ |
$318.8223853$ |
3.849305013 |
\( \frac{290072363351}{49} a^{3} - \frac{70940688524}{7} a^{2} - \frac{1386705191304}{49} a + \frac{407403205719}{49} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 6 a - 2\) , \( -45 a^{3} + 163 a^{2} - 25 a - 124\) , \( -219 a^{3} + 749 a^{2} - 50 a - 489\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-6a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-45a^{3}+163a^{2}-25a-124\right){x}-219a^{3}+749a^{2}-50a-489$ |
| 49.1-a2 |
49.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{7} \) |
$13.62645$ |
$(-a^3+a^2+6a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.141485933$ |
$1275.289541$ |
3.849305013 |
\( \frac{281343}{7} a^{3} - 69213 a^{2} - \frac{1348621}{7} a + \frac{411118}{7} \) |
\( \bigl[a\) , \( -2 a^{3} + 3 a^{2} + 11 a + 2\) , \( a + 1\) , \( -33 a^{3} + 56 a^{2} + 161 a - 39\) , \( 173 a^{3} - 296 a^{2} - 825 a + 246\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+11a+2\right){x}^{2}+\left(-33a^{3}+56a^{2}+161a-39\right){x}+173a^{3}-296a^{2}-825a+246$ |
| 55.1-a1 |
55.1-a |
$2$ |
$3$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{3} \cdot 11^{3} \) |
$13.82463$ |
$(a^2-2a-3), (a^3-2a^2-4a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.641102597$ |
$366.3234411$ |
3.340113703 |
\( \frac{23733306}{15125} a^{3} - \frac{160592728}{15125} a^{2} - \frac{5620689}{605} a + \frac{7938838}{15125} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( a^{3} - a^{2} - 6 a - 2\) , \( -a^{2} - 5 a - 2\) , \( -2 a^{3} - 5 a^{2} - 4 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(a^{3}-a^{2}-6a-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+1\right){x}^{2}+\left(-a^{2}-5a-2\right){x}-2a^{3}-5a^{2}-4a-1$ |
| 55.1-a2 |
55.1-a |
$2$ |
$3$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5 \cdot 11^{9} \) |
$13.82463$ |
$(a^2-2a-3), (a^3-2a^2-4a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1.923307793$ |
$4.522511618$ |
3.340113703 |
\( -\frac{9404087719190371726}{805255} a^{3} - \frac{19803291049550068002}{805255} a^{2} - \frac{1016175712480865650}{161051} a + \frac{3027894646330022817}{805255} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 9 a - 1\) , \( a - 1\) , \( 1\) , \( -28 a^{3} + 24 a^{2} + 162 a + 77\) , \( 26 a^{3} - 15 a^{2} - 186 a - 148\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-9a-1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-28a^{3}+24a^{2}+162a+77\right){x}+26a^{3}-15a^{2}-186a-148$ |
| 55.1-b1 |
55.1-b |
$2$ |
$5$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{5} \cdot 11 \) |
$13.82463$ |
$(a^2-2a-3), (a^3-2a^2-4a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$359.2578903$ |
0.766419557 |
\( \frac{136773632}{34375} a^{3} - \frac{237961216}{34375} a^{2} - \frac{27230208}{1375} a + \frac{215515136}{34375} \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 5 a - 2\) , \( 1\) , \( -2 a^{3} + 4 a^{2} + 7 a + 2\) , \( a^{3} - 2 a^{2} - 3 a - 1\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{3}-a^{2}-5a-2\right){x}^{2}+\left(-2a^{3}+4a^{2}+7a+2\right){x}+a^{3}-2a^{2}-3a-1$ |
| 55.1-b2 |
55.1-b |
$2$ |
$5$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{5} \) |
$13.82463$ |
$(a^2-2a-3), (a^3-2a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 5 \) |
$1$ |
$0.574812624$ |
0.766419557 |
\( -\frac{67609207331617046528}{6655} a^{3} - \frac{142372647799593271296}{6655} a^{2} - \frac{7305636936107753472}{1331} a + \frac{21768574389959950336}{6655} \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 5 a - 2\) , \( 1\) , \( 128 a^{3} - 356 a^{2} - 183 a + 52\) , \( 1775 a^{3} - 4812 a^{2} - 2703 a + 1031\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{3}-a^{2}-5a-2\right){x}^{2}+\left(128a^{3}-356a^{2}-183a+52\right){x}+1775a^{3}-4812a^{2}-2703a+1031$ |
| 65.1-a1 |
65.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
65.1 |
\( 5 \cdot 13 \) |
\( - 5^{18} \cdot 13^{2} \) |
$14.11635$ |
$(a^2-2a-3), (-a^3+a^2+4a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{2} \) |
$1$ |
$82.84784127$ |
3.596511325 |
\( -\frac{579299210672725708868}{644683837890625} a^{3} + \frac{421754974678080995659}{644683837890625} a^{2} + \frac{143639125591663658292}{25787353515625} a + \frac{2135150320184614345761}{644683837890625} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a - 1\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a + 1\) , \( 3 a^{3} - 7 a^{2} - 14 a + 7\) , \( 2 a^{3} - 18 a - 9\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(3a^{3}-7a^{2}-14a+7\right){x}+2a^{3}-18a-9$ |
| 65.1-a2 |
65.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
65.1 |
\( 5 \cdot 13 \) |
\( 5^{9} \cdot 13 \) |
$14.11635$ |
$(a^2-2a-3), (-a^3+a^2+4a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 1 \) |
$1$ |
$165.6956825$ |
3.596511325 |
\( \frac{1481584151153}{25390625} a^{3} - \frac{2068470015364}{25390625} a^{2} - \frac{345401759582}{1015625} a + \frac{2428284828544}{25390625} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a - 1\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a + 1\) , \( 3 a^{3} - 7 a^{2} - 9 a + 12\) , \( -2 a + 3\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(3a^{3}-7a^{2}-9a+12\right){x}-2a+3$ |
| 73.1-a1 |
73.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
73.1 |
\( 73 \) |
\( 73^{3} \) |
$14.32266$ |
$(a^3-2a^2-3a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.205710628$ |
$510.8297662$ |
3.362671542 |
\( -\frac{158206294}{389017} a^{3} + \frac{536693350}{389017} a^{2} + \frac{97320569}{389017} a - \frac{63394252}{389017} \) |
\( \bigl[a^{3} - a^{2} - 6 a - 1\) , \( -2 a^{3} + 3 a^{2} + 10 a + 1\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 17 a^{3} - 30 a^{2} - 81 a + 31\) , \( -117 a^{3} + 199 a^{2} + 561 a - 159\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-6a-1\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+10a+1\right){x}^{2}+\left(17a^{3}-30a^{2}-81a+31\right){x}-117a^{3}+199a^{2}+561a-159$ |
| 73.1-a2 |
73.1-a |
$2$ |
$2$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
73.1 |
\( 73 \) |
\( - 73^{6} \) |
$14.32266$ |
$(a^3-2a^2-3a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.411421256$ |
$127.7074415$ |
3.362671542 |
\( -\frac{974610777306767}{151334226289} a^{3} + \frac{3606188865202951}{151334226289} a^{2} + \frac{13625361963351787}{151334226289} a + \frac{8260027442959916}{151334226289} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 10 a\) , \( -2 a^{3} + 3 a^{2} + 10 a\) , \( 2 a^{3} - 3 a^{2} - 9 a - 1\) , \( -63 a^{3} + 165 a^{2} + 107 a - 38\) , \( 857 a^{3} - 2285 a^{2} - 1336 a + 514\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-10a\right){x}{y}+\left(2a^{3}-3a^{2}-9a-1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+10a\right){x}^{2}+\left(-63a^{3}+165a^{2}+107a-38\right){x}+857a^{3}-2285a^{2}-1336a+514$ |
| 77.1-a1 |
77.1-a |
$1$ |
$1$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
77.1 |
\( 7 \cdot 11 \) |
\( - 7 \cdot 11 \) |
$14.41848$ |
$(-a^3+a^2+6a+1), (a^3-2a^2-4a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.037997482$ |
$2594.254859$ |
4.205888218 |
\( \frac{51491666}{77} a^{3} - \frac{5802957}{11} a^{2} - \frac{311733833}{77} a - \frac{182620734}{77} \) |
\( \bigl[2 a^{3} - 3 a^{2} - 9 a\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( a + 1\) , \( a^{3} - 6 a^{2} + 5 a + 7\) , \( -5 a^{3} + 12 a^{2} + 10 a - 1\bigr] \) |
${y}^2+\left(2a^{3}-3a^{2}-9a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(a^{3}-6a^{2}+5a+7\right){x}-5a^{3}+12a^{2}+10a-1$ |
| 77.1-b1 |
77.1-b |
$4$ |
$6$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
77.1 |
\( 7 \cdot 11 \) |
\( - 7^{3} \cdot 11^{6} \) |
$14.41848$ |
$(-a^3+a^2+6a+1), (a^3-2a^2-4a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$2.318432182$ |
$26.30598209$ |
3.903286548 |
\( \frac{1709915870999650}{456533} a^{3} - \frac{419172481574001}{65219} a^{2} - \frac{8165256928043223}{456533} a + \frac{2436039253827782}{456533} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 45 a^{3} - 110 a^{2} - 91 a\) , \( 270 a^{3} - 707 a^{2} - 442 a + 107\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-9a\right){x}^{2}+\left(45a^{3}-110a^{2}-91a\right){x}+270a^{3}-707a^{2}-442a+107$ |
| 77.1-b2 |
77.1-b |
$4$ |
$6$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
77.1 |
\( 7 \cdot 11 \) |
\( - 7 \cdot 11^{2} \) |
$14.41848$ |
$(-a^3+a^2+6a+1), (a^3-2a^2-4a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$0.772810727$ |
$2130.784549$ |
3.903286548 |
\( -\frac{1447868}{77} a^{3} - \frac{371947}{11} a^{2} - \frac{699687}{77} a + \frac{546303}{77} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 5 a^{3} - 10 a^{2} - 16 a\) , \( -7 a^{3} + 19 a^{2} + 11 a - 6\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-9a\right){x}^{2}+\left(5a^{3}-10a^{2}-16a\right){x}-7a^{3}+19a^{2}+11a-6$ |
| 77.1-b3 |
77.1-b |
$4$ |
$6$ |
4.4.8789.1 |
$4$ |
$[4, 0]$ |
77.1 |
\( 7 \cdot 11 \) |
\( - 7^{2} \cdot 11^{4} \) |
$14.41848$ |
$(-a^3+a^2+6a+1), (a^3-2a^2-4a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.545621454$ |
$532.6961373$ |
3.903286548 |
\( \frac{95386699288689}{5929} a^{3} + \frac{28695259780905}{847} a^{2} + \frac{51535697367200}{5929} a - \frac{2792017297693}{539} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( 10 a^{3} - 25 a^{2} - 21 a + 5\) , \( 16 a^{3} - 43 a^{2} - 24 a + 8\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(2a^{3}-3a^{2}-9a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-9a\right){x}^{2}+\left(10a^{3}-25a^{2}-21a+5\right){x}+16a^{3}-43a^{2}-24a+8$ |
*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.
