| 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 |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{16} \) |
$10.08921$ |
$(-a^2+2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$260.7831942$ |
1.412220173 |
\( \frac{22110897043}{390625} a^{3} - \frac{73795458033}{390625} a^{2} - \frac{74816746524}{390625} a + \frac{300892942736}{390625} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} - 5 a - 4\) , \( -37 a^{3} + 126 a^{2} + 118 a - 501\) , \( 210 a^{3} - 727 a^{2} - 669 a + 2892\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(-37a^{3}+126a^{2}+118a-501\right){x}+210a^{3}-727a^{2}-669a+2892$ |
| 5.1-a2 |
5.1-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{8} \) |
$10.08921$ |
$(-a^2+2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$260.7831942$ |
1.412220173 |
\( -\frac{1101127}{625} a^{3} - \frac{938751}{625} a^{2} + \frac{8246283}{625} a + \frac{11499669}{625} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} - 5 a - 4\) , \( -2 a^{3} + 11 a^{2} + 3 a - 46\) , \( -2 a^{3} + 8 a^{2} + 6 a - 34\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(-2a^{3}+11a^{2}+3a-46\right){x}-2a^{3}+8a^{2}+6a-34$ |
| 5.1-b1 |
5.1-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{8} \) |
$10.08921$ |
$(-a^2+2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.031864064$ |
$634.2218560$ |
1.750996052 |
\( -\frac{1101127}{625} a^{3} - \frac{938751}{625} a^{2} + \frac{8246283}{625} a + \frac{11499669}{625} \) |
\( \bigl[1\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{3} - a^{2} - 4 a\) , \( 4 a^{3} + 7 a^{2} - 24 a - 38\) , \( 43 a^{3} + 39 a^{2} - 227 a - 279\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(4a^{3}+7a^{2}-24a-38\right){x}+43a^{3}+39a^{2}-227a-279$ |
| 5.1-b2 |
5.1-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{16} \) |
$10.08921$ |
$(-a^2+2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.015932032$ |
$634.2218560$ |
1.750996052 |
\( \frac{22110897043}{390625} a^{3} - \frac{73795458033}{390625} a^{2} - \frac{74816746524}{390625} a + \frac{300892942736}{390625} \) |
\( \bigl[1\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{3} - a^{2} - 4 a\) , \( -61 a^{3} - 33 a^{2} + 311 a + 292\) , \( 410 a^{3} + 341 a^{2} - 2156 a - 2496\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(-61a^{3}-33a^{2}+311a+292\right){x}+410a^{3}+341a^{2}-2156a-2496$ |
| 5.2-a1 |
5.2-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{16} \) |
$10.08921$ |
$(-a^3+5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$260.7831942$ |
1.412220173 |
\( -\frac{22110897043}{390625} a^{3} - \frac{7462766904}{390625} a^{2} + \frac{156074971461}{390625} a + \frac{174391635222}{390625} \) |
\( \bigl[a^{3} - 6 a - 4\) , \( -a^{3} + 2 a^{2} + 5 a - 4\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 35 a^{3} + 10 a^{2} - 239 a - 257\) , \( -127 a^{3} - 55 a^{2} + 898 a + 1014\bigr] \) |
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-4\right){x}^{2}+\left(35a^{3}+10a^{2}-239a-257\right){x}-127a^{3}-55a^{2}+898a+1014$ |
| 5.2-a2 |
5.2-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{8} \) |
$10.08921$ |
$(-a^3+5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$260.7831942$ |
1.412220173 |
\( \frac{1101127}{625} a^{3} - \frac{4242132}{625} a^{2} - \frac{122616}{25} a + \frac{17706074}{625} \) |
\( \bigl[a^{3} - 6 a - 4\) , \( -a^{3} + 2 a^{2} + 5 a - 4\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a + 3\) , \( 5 a^{3} + 4 a^{2} - 36 a - 49\bigr] \) |
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-4\right){x}^{2}+\left(a+3\right){x}+5a^{3}+4a^{2}-36a-49$ |
| 5.2-b1 |
5.2-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{8} \) |
$10.08921$ |
$(-a^3+5a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.031864064$ |
$634.2218560$ |
1.750996052 |
\( \frac{1101127}{625} a^{3} - \frac{4242132}{625} a^{2} - \frac{122616}{25} a + \frac{17706074}{625} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a + 1\) , \( a^{3} - 5 a - 4\) , \( -5 a^{3} + 20 a^{2} + 2 a - 51\) , \( -44 a^{3} + 169 a^{2} + 23 a - 424\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a+1\right){x}^{2}+\left(-5a^{3}+20a^{2}+2a-51\right){x}-44a^{3}+169a^{2}+23a-424$ |
| 5.2-b2 |
5.2-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( 5^{16} \) |
$10.08921$ |
$(-a^3+5a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.015932032$ |
$634.2218560$ |
1.750996052 |
\( -\frac{22110897043}{390625} a^{3} - \frac{7462766904}{390625} a^{2} + \frac{156074971461}{390625} a + \frac{174391635222}{390625} \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a + 1\) , \( a^{3} - 5 a - 4\) , \( 60 a^{3} - 215 a^{2} - 58 a + 509\) , \( -411 a^{3} + 1572 a^{2} + 247 a - 3901\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a+1\right){x}^{2}+\left(60a^{3}-215a^{2}-58a+509\right){x}-411a^{3}+1572a^{2}+247a-3901$ |
| 25.1-a1 |
25.1-a |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{2} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$187.1078804$ |
2.026491960 |
\( -1144513001 a^{3} + 3885584496 a^{2} + 3688844885 a - 15399442982 \) |
\( \bigl[a^{2} - 4\) , \( a^{3} - 6 a - 4\) , \( a^{3} - a^{2} - 4 a\) , \( 9 a^{3} - 32 a^{2} - 5 a + 77\) , \( -33 a^{3} + 137 a^{2} + 11 a - 361\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-6a-4\right){x}^{2}+\left(9a^{3}-32a^{2}-5a+77\right){x}-33a^{3}+137a^{2}+11a-361$ |
| 25.1-a2 |
25.1-a |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{10} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$187.1078804$ |
2.026491960 |
\( -12049 a^{3} - 5029 a^{2} + 84404 a + 95254 \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a + 1\) , \( a\) , \( 4 a^{3} - 18 a - 8\) , \( 4 a^{3} + 10 a^{2} - 24 a - 56\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a^{3}-18a-8\right){x}+4a^{3}+10a^{2}-24a-56$ |
| 25.1-b1 |
25.1-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{14} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$261.0106538$ |
1.413451936 |
\( -\frac{1101127}{625} a^{3} - \frac{938751}{625} a^{2} + \frac{8246283}{625} a + \frac{11499669}{625} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{2} + 2 a + 6\) , \( a^{3} - a^{2} - 5 a + 1\) , \( -a^{3} + 8 a^{2} + 2 a - 29\) , \( 12 a^{3} - 38 a^{2} - 12 a + 86\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{2}+2a+6\right){x}^{2}+\left(-a^{3}+8a^{2}+2a-29\right){x}+12a^{3}-38a^{2}-12a+86$ |
| 25.1-b2 |
25.1-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{22} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$130.5053269$ |
1.413451936 |
\( \frac{22110897043}{390625} a^{3} - \frac{73795458033}{390625} a^{2} - \frac{74816746524}{390625} a + \frac{300892942736}{390625} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( a + 1\) , \( -116 a^{3} + 386 a^{2} + 391 a - 1549\) , \( 1643 a^{3} - 5606 a^{2} - 5244 a + 22192\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-1\right){x}^{2}+\left(-116a^{3}+386a^{2}+391a-1549\right){x}+1643a^{3}-5606a^{2}-5244a+22192$ |
| 25.1-c1 |
25.1-c |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{4} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 3 \) |
$0.064141910$ |
$309.1954019$ |
2.577561096 |
\( -12049 a^{3} - 5029 a^{2} + 84404 a + 95254 \) |
\( \bigl[a^{3} - 5 a - 4\) , \( a^{2} - 2 a - 6\) , \( a^{3} - 6 a - 4\) , \( -a^{3} - a^{2} + 5 a + 8\) , \( -5 a^{3} - 3 a^{2} + 24 a + 22\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{3}-6a-4\right){y}={x}^{3}+\left(a^{2}-2a-6\right){x}^{2}+\left(-a^{3}-a^{2}+5a+8\right){x}-5a^{3}-3a^{2}+24a+22$ |
| 25.1-c2 |
25.1-c |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{8} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 3 \) |
$0.320709550$ |
$61.83908038$ |
2.577561096 |
\( -1144513001 a^{3} + 3885584496 a^{2} + 3688844885 a - 15399442982 \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( -a^{3} + 2 a^{2} + 4 a - 6\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 19 a^{3} + 8 a^{2} - 140 a - 164\) , \( 163 a^{3} + 78 a^{2} - 1111 a - 1266\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-6\right){x}^{2}+\left(19a^{3}+8a^{2}-140a-164\right){x}+163a^{3}+78a^{2}-1111a-1266$ |
| 25.1-d1 |
25.1-d |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{8} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.4[2] |
$1$ |
\( 1 \) |
$6.922862578$ |
$9.866227646$ |
2.959031852 |
\( -1144513001 a^{3} + 3885584496 a^{2} + 3688844885 a - 15399442982 \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a^{3} + 2 a^{2} + 4 a - 6\) , \( a^{3} - 5 a - 5\) , \( 7 a^{3} - 12 a^{2} - 27 a + 10\) , \( 17 a^{3} - 26 a^{2} - 67 a + 3\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{3}-5a-5\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-6\right){x}^{2}+\left(7a^{3}-12a^{2}-27a+10\right){x}+17a^{3}-26a^{2}-67a+3$ |
| 25.1-d2 |
25.1-d |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{4} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1[2] |
$1$ |
\( 1 \) |
$1.384572515$ |
$1233.278455$ |
2.959031852 |
\( -12049 a^{3} - 5029 a^{2} + 84404 a + 95254 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 7 a - 4\) , \( 1\) , \( -2 a^{2} + 2 a + 21\) , \( -3 a^{3} + 18 a^{2} + 2 a - 77\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-7a-4\right){x}^{2}+\left(-2a^{2}+2a+21\right){x}-3a^{3}+18a^{2}+2a-77$ |
| 25.1-e1 |
25.1-e |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{14} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$301.4463466$ |
1.632423489 |
\( -\frac{1101127}{625} a^{3} - \frac{938751}{625} a^{2} + \frac{8246283}{625} a + \frac{11499669}{625} \) |
\( \bigl[a^{3} - 5 a - 4\) , \( a^{2} - 6\) , \( 0\) , \( 8 a^{3} + 6 a^{2} - 42 a - 45\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}={x}^{3}+\left(a^{2}-6\right){x}^{2}+\left(8a^{3}+6a^{2}-42a-45\right){x}$ |
| 25.1-e2 |
25.1-e |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{22} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$150.7231733$ |
1.632423489 |
\( \frac{22110897043}{390625} a^{3} - \frac{73795458033}{390625} a^{2} - \frac{74816746524}{390625} a + \frac{300892942736}{390625} \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( a^{3} - 5 a - 4\) , \( -25 a^{3} + 47 a^{2} + 107 a - 147\) , \( 60 a^{3} - 319 a^{2} - 132 a + 1351\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(-25a^{3}+47a^{2}+107a-147\right){x}+60a^{3}-319a^{2}-132a+1351$ |
| 25.1-f1 |
25.1-f |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{2} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$121.2559617$ |
1.313275694 |
\( -1144513001 a^{3} + 3885584496 a^{2} + 3688844885 a - 15399442982 \) |
\( \bigl[a^{2} - a - 4\) , \( a^{3} - 5 a - 5\) , \( a^{2} - a - 5\) , \( a^{3} + a^{2} - 6 a - 9\) , \( a^{2} + 2 a\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(a^{3}+a^{2}-6a-9\right){x}+a^{2}+2a$ |
| 25.1-f2 |
25.1-f |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{10} \) |
$12.33753$ |
$(-a^2+2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$121.2559617$ |
1.313275694 |
\( -12049 a^{3} - 5029 a^{2} + 84404 a + 95254 \) |
\( \bigl[a\) , \( -a^{3} + 7 a + 4\) , \( a^{3} - a^{2} - 5 a\) , \( -11 a^{3} - 11 a^{2} + 65 a + 87\) , \( -49 a^{3} - 43 a^{2} + 262 a + 314\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(-a^{3}+7a+4\right){x}^{2}+\left(-11a^{3}-11a^{2}+65a+87\right){x}-49a^{3}-43a^{2}+262a+314$ |
| 25.2-a1 |
25.2-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$12.33753$ |
$(-a^2+2a+3), (-a^3+5a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.032299529$ |
$1018.877028$ |
2.851417145 |
\( -\frac{17760381}{25} a^{2} + \frac{17760381}{25} a + \frac{105598196}{25} \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a\) , \( a^{3} - 5 a - 4\) , \( -6 a - 11\) , \( -16 a^{3} - 12 a^{2} + 96 a + 115\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}-a{x}^{2}+\left(-6a-11\right){x}-16a^{3}-12a^{2}+96a+115$ |
| 25.2-a2 |
25.2-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.33753$ |
$(-a^2+2a+3), (-a^3+5a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.064599058$ |
$1018.877028$ |
2.851417145 |
\( -\frac{5171}{5} a^{2} + \frac{5171}{5} a + 5749 \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a\) , \( a^{3} - 5 a - 4\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}-a{x}^{2}+\left(-a-1\right){x}$ |
| 25.2-b1 |
25.2-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$12.33753$ |
$(-a^2+2a+3), (-a^3+5a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.032299529$ |
$1018.877028$ |
2.851417145 |
\( -\frac{17760381}{25} a^{2} + \frac{17760381}{25} a + \frac{105598196}{25} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{3} - a^{2} - 4 a\) , \( -2 a^{3} + 4 a^{2} + 10 a - 23\) , \( 15 a^{3} - 58 a^{2} - 22 a + 180\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(-2a^{3}+4a^{2}+10a-23\right){x}+15a^{3}-58a^{2}-22a+180$ |
| 25.2-b2 |
25.2-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.33753$ |
$(-a^2+2a+3), (-a^3+5a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.064599058$ |
$1018.877028$ |
2.851417145 |
\( -\frac{5171}{5} a^{2} + \frac{5171}{5} a + 5749 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{3} - a^{2} - 4 a\) , \( -2 a^{3} + 4 a^{2} + 5 a - 8\) , \( -a^{3} + 2 a^{2} + 2 a - 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(-2a^{3}+4a^{2}+5a-8\right){x}-a^{3}+2a^{2}+2a-3$ |
| 25.3-a1 |
25.3-a |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{10} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$187.1078804$ |
2.026491960 |
\( 12049 a^{3} - 41176 a^{2} - 38199 a + 162580 \) |
\( \bigl[a^{3} - 5 a - 5\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} - 3 a^{2} + 9 a + 21\) , \( a^{2} - a - 6\bigr] \) |
${y}^2+\left(a^{3}-5a-5\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(-a^{3}-3a^{2}+9a+21\right){x}+a^{2}-a-6$ |
| 25.3-a2 |
25.3-a |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{2} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$187.1078804$ |
2.026491960 |
\( 1144513001 a^{3} + 452045493 a^{2} - 8026474874 a - 8969526602 \) |
\( \bigl[a^{2} - 5\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a^{2} - 4\) , \( -10 a^{3} - 11 a^{2} + 56 a + 77\) , \( 32 a^{3} + 28 a^{2} - 170 a - 203\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(-10a^{3}-11a^{2}+56a+77\right){x}+32a^{3}+28a^{2}-170a-203$ |
| 25.3-b1 |
25.3-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{22} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$130.5053269$ |
1.413451936 |
\( -\frac{22110897043}{390625} a^{3} - \frac{7462766904}{390625} a^{2} + \frac{156074971461}{390625} a + \frac{174391635222}{390625} \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{2} - 4\) , \( 116 a^{3} + 46 a^{2} - 822 a - 923\) , \( -1369 a^{3} - 567 a^{2} + 9576 a + 10778\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(116a^{3}+46a^{2}-822a-923\right){x}-1369a^{3}-567a^{2}+9576a+10778$ |
| 25.3-b2 |
25.3-b |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{14} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$261.0106538$ |
1.413451936 |
\( \frac{1101127}{625} a^{3} - \frac{4242132}{625} a^{2} - \frac{122616}{25} a + \frac{17706074}{625} \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{2} - 4\) , \( 11 a^{3} + 6 a^{2} - 77 a - 88\) , \( 5 a^{3} + 2 a^{2} - 33 a - 37\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(11a^{3}+6a^{2}-77a-88\right){x}+5a^{3}+2a^{2}-33a-37$ |
| 25.3-c1 |
25.3-c |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{8} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 3 \) |
$0.320709550$ |
$61.83908038$ |
2.577561096 |
\( 1144513001 a^{3} + 452045493 a^{2} - 8026474874 a - 8969526602 \) |
\( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - a^{2} - 6 a - 1\) , \( a^{3} - 6 a - 4\) , \( -19 a^{3} + 65 a^{2} + 65 a - 277\) , \( -163 a^{3} + 567 a^{2} + 465 a - 2136\bigr] \) |
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{3}-6a-4\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(-19a^{3}+65a^{2}+65a-277\right){x}-163a^{3}+567a^{2}+465a-2136$ |
| 25.3-c2 |
25.3-c |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{4} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 3 \) |
$0.064141910$ |
$309.1954019$ |
2.577561096 |
\( 12049 a^{3} - 41176 a^{2} - 38199 a + 162580 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 5 a + 5\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} - 5 a^{2} + 11 a + 22\) , \( 4 a^{3} - 20 a^{2} + 3 a + 54\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+5a+5\right){x}^{2}+\left(-a^{3}-5a^{2}+11a+22\right){x}+4a^{3}-20a^{2}+3a+54$ |
| 25.3-d1 |
25.3-d |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{4} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1[2] |
$1$ |
\( 1 \) |
$1.384572515$ |
$1233.278455$ |
2.959031852 |
\( 12049 a^{3} - 41176 a^{2} - 38199 a + 162580 \) |
\( \bigl[a^{3} - 5 a - 4\) , \( a^{3} - 2 a^{2} - 5 a + 5\) , \( a^{2} - 4\) , \( -4 a^{2} + 6 a + 26\) , \( 10 a^{3} + 3 a^{2} - 65 a - 65\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+5\right){x}^{2}+\left(-4a^{2}+6a+26\right){x}+10a^{3}+3a^{2}-65a-65$ |
| 25.3-d2 |
25.3-d |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{8} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.4[2] |
$1$ |
\( 1 \) |
$6.922862578$ |
$9.866227646$ |
2.959031852 |
\( 1144513001 a^{3} + 452045493 a^{2} - 8026474874 a - 8969526602 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{2} - 4\) , \( a^{2} - a - 4\) , \( -3 a^{3} + 11 a^{2} + 8 a - 43\) , \( -16 a^{3} + 2 a^{2} + 76 a + 24\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-3a^{3}+11a^{2}+8a-43\right){x}-16a^{3}+2a^{2}+76a+24$ |
| 25.3-e1 |
25.3-e |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{22} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$150.7231733$ |
1.632423489 |
\( -\frac{22110897043}{390625} a^{3} - \frac{7462766904}{390625} a^{2} + \frac{156074971461}{390625} a + \frac{174391635222}{390625} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{3} - 5 a - 4\) , \( 21 a^{3} - 13 a^{2} - 114 a - 77\) , \( -56 a^{3} - 59 a^{2} + 449 a + 587\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(21a^{3}-13a^{2}-114a-77\right){x}-56a^{3}-59a^{2}+449a+587$ |
| 25.3-e2 |
25.3-e |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{14} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$301.4463466$ |
1.632423489 |
\( \frac{1101127}{625} a^{3} - \frac{4242132}{625} a^{2} - \frac{122616}{25} a + \frac{17706074}{625} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{3} - 5 a - 4\) , \( a^{3} + 2 a^{2} - 9 a - 12\) , \( 2 a^{3} - 6 a^{2} - 4 a + 12\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(a^{3}+2a^{2}-9a-12\right){x}+2a^{3}-6a^{2}-4a+12$ |
| 25.3-f1 |
25.3-f |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{2} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$121.2559617$ |
1.313275694 |
\( 1144513001 a^{3} + 452045493 a^{2} - 8026474874 a - 8969526602 \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{3} + 5 a + 6\) , \( a^{2} - a - 4\) , \( -a^{3} + 2 a^{2} + 5 a - 5\) , \( -a^{3} + 3 a^{2} + a - 5\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{3}+5a+6\right){x}^{2}+\left(-a^{3}+2a^{2}+5a-5\right){x}-a^{3}+3a^{2}+a-5$ |
| 25.3-f2 |
25.3-f |
$2$ |
$5$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{10} \) |
$12.33753$ |
$(-a^3+5a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$1$ |
$121.2559617$ |
1.313275694 |
\( 12049 a^{3} - 41176 a^{2} - 38199 a + 162580 \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 5\) , \( a^{2} - a - 4\) , \( 10 a^{3} - 29 a^{2} - 14 a + 64\) , \( 39 a^{3} - 140 a^{2} - 31 a + 341\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(10a^{3}-29a^{2}-14a+64\right){x}+39a^{3}-140a^{2}-31a+341$ |
| 31.2-a1 |
31.2-a |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$12.67377$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$7.096706804$ |
$5.756461161$ |
3.539608545 |
\( \frac{6130703730739448}{31} a^{2} - \frac{6130703730739448}{31} a - \frac{20733831642404195}{31} \) |
\( \bigl[a^{3} - 5 a - 5\) , \( a^{3} - 6 a - 5\) , \( a^{3} - a^{2} - 5 a\) , \( 97 a^{3} - 358 a^{2} - 116 a + 1001\) , \( -91 a^{3} - 83 a^{2} + 2499 a - 4292\bigr] \) |
${y}^2+\left(a^{3}-5a-5\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(a^{3}-6a-5\right){x}^{2}+\left(97a^{3}-358a^{2}-116a+1001\right){x}-91a^{3}-83a^{2}+2499a-4292$ |
| 31.2-a2 |
31.2-a |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$12.67377$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$14.19341360$ |
$1.439115290$ |
3.539608545 |
\( -\frac{61725871986044215714}{961} a^{2} + \frac{61725871986044215714}{961} a + \frac{346778046802821801379}{961} \) |
\( \bigl[a^{2} - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( a\) , \( -638 a^{3} + 2416 a^{2} + 806 a - 7071\) , \( 17182 a^{3} - 68857 a^{2} + 9 a + 159201\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(-638a^{3}+2416a^{2}+806a-7071\right){x}+17182a^{3}-68857a^{2}+9a+159201$ |
| 31.2-a3 |
31.2-a |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$12.67377$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.774176701$ |
$368.4135143$ |
3.539608545 |
\( -\frac{106208}{31} a^{2} + \frac{106208}{31} a + \frac{476287}{31} \) |
\( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 2 a^{2} - 5 a + 6\) , \( a^{2} - a - 4\) , \( a^{3} - 8 a^{2} - 3 a + 31\) , \( -4 a^{3} - 10 a^{2} + 25 a + 56\bigr] \) |
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+6\right){x}^{2}+\left(a^{3}-8a^{2}-3a+31\right){x}-4a^{3}-10a^{2}+25a+56$ |
| 31.2-a4 |
31.2-a |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$12.67377$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.548353402$ |
$92.10337857$ |
3.539608545 |
\( \frac{9029272560}{961} a^{2} - \frac{9029272560}{961} a - \frac{30517260007}{961} \) |
\( \bigl[a^{3} - 6 a - 4\) , \( a^{3} - 2 a^{2} - 5 a + 6\) , \( a^{2} - a - 4\) , \( -24 a^{3} - 33 a^{2} + 122 a + 186\) , \( -226 a^{3} - 222 a^{2} + 1175 a + 1473\bigr] \) |
${y}^2+\left(a^{3}-6a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+6\right){x}^{2}+\left(-24a^{3}-33a^{2}+122a+186\right){x}-226a^{3}-222a^{2}+1175a+1473$ |
| 31.2-a5 |
31.2-a |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{16} \) |
$12.67377$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$3.548353402$ |
$5.756461161$ |
3.539608545 |
\( \frac{11889611722383394}{852891037441} a^{2} - \frac{11889611722383394}{852891037441} a - \frac{56187831951653267}{852891037441} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{2} - a - 4\) , \( 74 a^{3} - 335 a^{2} + 195 a + 442\) , \( -3104 a^{3} + 11720 a^{2} + 3919 a - 34076\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(74a^{3}-335a^{2}+195a+442\right){x}-3104a^{3}+11720a^{2}+3919a-34076$ |
| 31.2-a6 |
31.2-a |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{8} \) |
$12.67377$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$7.096706804$ |
$23.02584464$ |
3.539608545 |
\( -\frac{156520379364360}{923521} a^{2} + \frac{156520379364360}{923521} a + \frac{879341773920653}{923521} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} - a - 6\) , \( a + 1\) , \( -213 a^{3} - 202 a^{2} + 1109 a + 1364\) , \( -3678 a^{3} - 3412 a^{2} + 19444 a + 23858\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-213a^{3}-202a^{2}+1109a+1364\right){x}-3678a^{3}-3412a^{2}+19444a+23858$ |
| 31.2-b1 |
31.2-b |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$12.67377$ |
$(a^3-4a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$165.8225355$ |
0.897979375 |
\( \frac{6130703730739448}{31} a^{2} - \frac{6130703730739448}{31} a - \frac{20733831642404195}{31} \) |
\( \bigl[a^{2} - a - 5\) , \( 1\) , \( a^{2} - a - 4\) , \( -17 a^{2} + 17 a + 83\) , \( 171 a^{2} - 171 a - 949\bigr] \) |
${y}^2+\left(a^{2}-a-5\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+{x}^{2}+\left(-17a^{2}+17a+83\right){x}+171a^{2}-171a-949$ |
| 31.2-b2 |
31.2-b |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$12.67377$ |
$(a^3-4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1$ |
$2.590977117$ |
0.897979375 |
\( -\frac{61725871986044215714}{961} a^{2} + \frac{61725871986044215714}{961} a + \frac{346778046802821801379}{961} \) |
\( \bigl[a^{2} - a - 4\) , \( -1\) , \( a^{2} - a - 5\) , \( 134 a^{2} - 134 a - 677\) , \( 738 a^{2} - 738 a - 3716\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}-{x}^{2}+\left(134a^{2}-134a-677\right){x}+738a^{2}-738a-3716$ |
| 31.2-b3 |
31.2-b |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{8} \) |
$12.67377$ |
$(a^3-4a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2 \) |
$1$ |
$41.45563388$ |
0.897979375 |
\( -\frac{156520379364360}{923521} a^{2} + \frac{156520379364360}{923521} a + \frac{879341773920653}{923521} \) |
\( \bigl[a^{2} - a - 4\) , \( -1\) , \( a^{2} - a - 5\) , \( -11 a^{2} + 11 a + 23\) , \( 43 a^{2} - 43 a - 162\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}-{x}^{2}+\left(-11a^{2}+11a+23\right){x}+43a^{2}-43a-162$ |
| 31.2-b4 |
31.2-b |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{16} \) |
$12.67377$ |
$(a^3-4a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1$ |
$2.590977117$ |
0.897979375 |
\( \frac{11889611722383394}{852891037441} a^{2} - \frac{11889611722383394}{852891037441} a - \frac{56187831951653267}{852891037441} \) |
\( \bigl[1\) , \( a^{2} - a - 6\) , \( a^{2} - a - 5\) , \( 29 a^{2} - 29 a - 190\) , \( 111 a^{2} - 111 a - 672\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(29a^{2}-29a-190\right){x}+111a^{2}-111a-672$ |
| 31.2-b5 |
31.2-b |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{4} \) |
$12.67377$ |
$(a^3-4a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$663.2901421$ |
0.897979375 |
\( \frac{9029272560}{961} a^{2} - \frac{9029272560}{961} a - \frac{30517260007}{961} \) |
\( \bigl[1\) , \( a^{2} - a - 6\) , \( a^{2} - a - 5\) , \( -a^{2} + a\) , \( 3 a^{2} - 3 a - 12\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-a^{2}+a\right){x}+3a^{2}-3a-12$ |
| 31.2-b6 |
31.2-b |
$6$ |
$8$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$12.67377$ |
$(a^3-4a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2653.160568$ |
0.897979375 |
\( -\frac{106208}{31} a^{2} + \frac{106208}{31} a + \frac{476287}{31} \) |
\( \bigl[1\) , \( a^{2} - a - 6\) , \( a^{2} - a - 5\) , \( -a^{2} + a + 5\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(-a^{2}+a+5\right){x}$ |
| 41.1-a1 |
41.1-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41 \) |
$13.12453$ |
$(-a^3-a^2+8a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$244.1134195$ |
2.643896564 |
\( \frac{2454982634569862}{41} a^{3} - \frac{8361110477734488}{41} a^{2} - \frac{7886047978061787}{41} a + \frac{33180827083440458}{41} \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 53 a^{3} + 9 a^{2} - 355 a - 365\) , \( -319 a^{3} - 142 a^{2} + 2260 a + 2568\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-1\right){x}^{2}+\left(53a^{3}+9a^{2}-355a-365\right){x}-319a^{3}-142a^{2}+2260a+2568$ |
| 41.1-a2 |
41.1-a |
$2$ |
$2$ |
4.4.8525.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{2} \) |
$13.12453$ |
$(-a^3-a^2+8a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$488.2268390$ |
2.643896564 |
\( \frac{4454855932}{1681} a^{3} - \frac{15175587003}{1681} a^{2} - \frac{14307620592}{1681} a + \frac{60228423488}{1681} \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 3 a^{3} - a^{2} - 20 a - 10\) , \( a^{3} + a^{2} - 7 a - 15\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-1\right){x}^{2}+\left(3a^{3}-a^{2}-20a-10\right){x}+a^{3}+a^{2}-7a-15$ |
*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.
