| 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 |
| 3.1-a1 |
3.1-a |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$14.03192$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$569.7552813$ |
2.081229269 |
\( \frac{71680}{9} a^{3} - \frac{204352}{9} a^{2} - \frac{63104}{3} a + \frac{463744}{9} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - 3 a^{2} - 2 a + 6\) , \( a^{3} - a^{2} - 4 a\) , \( 7 a^{3} + a^{2} - 17 a - 2\) , \( 17 a^{3} + 13 a^{2} - 35 a - 23\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+6\right){x}^{2}+\left(7a^{3}+a^{2}-17a-2\right){x}+17a^{3}+13a^{2}-35a-23$ |
| 3.1-a2 |
3.1-a |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( -3 \) |
$14.03192$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1139.510562$ |
2.081229269 |
\( -\frac{24704}{3} a^{3} + \frac{52544}{3} a^{2} + 22656 a + \frac{7744}{3} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( a + 1\) , \( a^{2} - 2\) , \( 2 a^{3} - 4 a + 2\) , \( -11 a^{3} + 43 a^{2} + 33 a - 93\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a^{3}-4a+2\right){x}-11a^{3}+43a^{2}+33a-93$ |
| 3.1-b1 |
3.1-b |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( -3 \) |
$14.03192$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.281538453$ |
$1152.267055$ |
2.370022274 |
\( -\frac{24704}{3} a^{3} + \frac{52544}{3} a^{2} + 22656 a + \frac{7744}{3} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a\) , \( a + 1\) , \( 7 a^{3} + 7 a^{2} - 13 a - 11\) , \( 30 a^{3} + 5 a^{2} - 70 a - 9\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(7a^{3}+7a^{2}-13a-11\right){x}+30a^{3}+5a^{2}-70a-9$ |
| 3.1-b2 |
3.1-b |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$14.03192$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.140769226$ |
$1152.267055$ |
2.370022274 |
\( \frac{71680}{9} a^{3} - \frac{204352}{9} a^{2} - \frac{63104}{3} a + \frac{463744}{9} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{3} - 3 a^{2} - 3 a + 6\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( 3 a^{3} - 7 a^{2} - 8 a + 15\) , \( 3 a^{3} - 6 a^{2} - 8 a + 13\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+6\right){x}^{2}+\left(3a^{3}-7a^{2}-8a+15\right){x}+3a^{3}-6a^{2}-8a+13$ |
| 5.1-a1 |
5.1-a |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$14.95713$ |
$(a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.025169196$ |
$1710.699691$ |
2.516486936 |
\( -\frac{2932}{25} a^{3} + \frac{1054}{25} a^{2} + \frac{3481}{5} a + \frac{11822}{25} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a + 2\) , \( 0\) , \( -2 a^{3} + 2 a^{2} + 10 a + 6\) , \( -a^{3} + 7 a + 8\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+4a+2\right){x}^{2}+\left(-2a^{3}+2a^{2}+10a+6\right){x}-a^{3}+7a+8$ |
| 5.1-b1 |
5.1-b |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$14.95713$ |
$(a)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.014773454$ |
$868.0760329$ |
2.998136238 |
\( -\frac{2932}{25} a^{3} + \frac{1054}{25} a^{2} + \frac{3481}{5} a + \frac{11822}{25} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 3 a - 1\) , \( -a^{3} + 7 a + 5\) , \( -2 a^{3} + 9 a + 5\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-a^{3}+7a+5\right){x}-2a^{3}+9a+5$ |
| 9.1-a1 |
9.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.829086642$ |
$1495.134004$ |
4.024936935 |
\( -\frac{45516800}{729} a^{3} + \frac{29152448}{729} a^{2} + \frac{90284416}{243} a + \frac{180319360}{729} \) |
\( \bigl[a^{2} - a - 3\) , \( 0\) , \( a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{3} + 3 a^{2} - 3 a - 4\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}+a^{3}+3a^{2}-3a-4$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{15} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.974519854$ |
$55.37533348$ |
4.024936935 |
\( \frac{2317333946807936}{19683} a^{3} - \frac{6501316612789952}{19683} a^{2} - \frac{2116585852959616}{6561} a + \frac{14384164831730240}{19683} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{2} - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 38 a^{3} - 126 a^{2} + 17 a + 110\) , \( 248 a^{3} - 868 a^{2} + 117 a + 785\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(38a^{3}-126a^{2}+17a+110\right){x}+248a^{3}-868a^{2}+117a+785$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{24} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$2.487259927$ |
$55.37533348$ |
4.024936935 |
\( \frac{53721649457152}{387420489} a^{3} - \frac{151271036310592}{387420489} a^{2} - \frac{48770427817856}{129140163} a + \frac{336211429843840}{387420489} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{2} - 3\) , \( -8 a^{3} + 26 a^{2} + 20 a - 56\) , \( -17 a^{3} + 56 a^{2} + 46 a - 140\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(-8a^{3}+26a^{2}+20a-56\right){x}-17a^{3}+56a^{2}+46a-140$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1.658173284$ |
$1495.134004$ |
4.024936935 |
\( \frac{3320192}{27} a^{3} + \frac{7213888}{27} a^{2} - \frac{2750848}{9} a - \frac{9330880}{27} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 4 a - 1\) , \( 4 a^{3} + 4 a^{2} - 8 a - 9\) , \( 8 a^{3} + 13 a^{2} - 12 a - 20\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(4a^{3}+4a^{2}-8a-9\right){x}+8a^{3}+13a^{2}-12a-20$ |
| 9.1-b1 |
9.1-b |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.083768328$ |
$1014.813305$ |
2.484205714 |
\( \frac{3320192}{27} a^{3} + \frac{7213888}{27} a^{2} - \frac{2750848}{9} a - \frac{9330880}{27} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} - a^{2} - 1\) , \( 2 a^{3} + a^{2} - 7 a - 6\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+2\right){x}^{2}+\left(-a^{3}-a^{2}-1\right){x}+2a^{3}+a^{2}-7a-6$ |
| 9.1-b2 |
9.1-b |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{24} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.502609973$ |
$169.1355508$ |
2.484205714 |
\( \frac{53721649457152}{387420489} a^{3} - \frac{151271036310592}{387420489} a^{2} - \frac{48770427817856}{129140163} a + \frac{336211429843840}{387420489} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - a^{2} - 4 a\) , \( -17 a^{3} + 47 a^{2} + 45 a - 102\) , \( 62 a^{3} - 175 a^{2} - 170 a + 384\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-17a^{3}+47a^{2}+45a-102\right){x}+62a^{3}-175a^{2}-170a+384$ |
| 9.1-b3 |
9.1-b |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{15} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.251304986$ |
$338.2711016$ |
2.484205714 |
\( \frac{2317333946807936}{19683} a^{3} - \frac{6501316612789952}{19683} a^{2} - \frac{2116585852959616}{6561} a + \frac{14384164831730240}{19683} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + 4 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 233 a^{3} - 804 a^{2} + 111 a + 733\) , \( 3935 a^{3} - 13896 a^{2} + 1935 a + 12631\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-2\right){x}^{2}+\left(233a^{3}-804a^{2}+111a+733\right){x}+3935a^{3}-13896a^{2}+1935a+12631$ |
| 9.1-b4 |
9.1-b |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$16.09746$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.167536657$ |
$507.4066525$ |
2.484205714 |
\( -\frac{45516800}{729} a^{3} + \frac{29152448}{729} a^{2} + \frac{90284416}{243} a + \frac{180319360}{729} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( a - 1\) , \( a^{2} - 3\) , \( 4 a^{3} - 6 a^{2} - 9 a + 2\) , \( -5 a^{3} + 31 a^{2} - 16 a - 37\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4a^{3}-6a^{2}-9a+2\right){x}-5a^{3}+31a^{2}-16a-37$ |
| 12.1-a1 |
12.1-a |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{7} \) |
$16.68686$ |
$(-a+1), (-a^2+a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.102977648$ |
$163.2704564$ |
3.439300876 |
\( -\frac{33074735}{8748} a^{3} + \frac{19908608}{2187} a^{2} + \frac{14311181}{1458} a - \frac{176076143}{8748} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - 3 a\) , \( a + 1\) , \( -3 a^{3} + 18 a^{2} + 12 a - 36\) , \( -11 a^{3} + 52 a^{2} + 34 a - 113\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(-3a^{3}+18a^{2}+12a-36\right){x}-11a^{3}+52a^{2}+34a-113$ |
| 12.1-b1 |
12.1-b |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{7} \) |
$16.68686$ |
$(-a+1), (-a^2+a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.030183649$ |
$895.0430197$ |
2.368418585 |
\( -\frac{33074735}{8748} a^{3} + \frac{19908608}{2187} a^{2} + \frac{14311181}{1458} a - \frac{176076143}{8748} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 2\) , \( a^{2} - 2\) , \( -7 a^{3} + 11 a^{2} + 27 a - 6\) , \( -2 a^{3} - 13 a^{2} + 30 a + 72\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-2\right){x}^{2}+\left(-7a^{3}+11a^{2}+27a-6\right){x}-2a^{3}-13a^{2}+30a+72$ |
| 15.1-a1 |
15.1-a |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{4} \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.396475710$ |
$180.6735082$ |
4.186611012 |
\( -\frac{38162367695555072}{2989355625} a^{3} + \frac{25930133465825984}{2989355625} a^{2} + \frac{15004444299070592}{199290375} a + \frac{144533247929706112}{2989355625} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 5 a - 2\) , \( a\) , \( -20 a^{3} - 22 a^{2} + 45 a + 40\) , \( 388 a^{3} + 406 a^{2} - 729 a - 648\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a-2\right){x}^{2}+\left(-20a^{3}-22a^{2}+45a+40\right){x}+388a^{3}+406a^{2}-729a-648$ |
| 15.1-a2 |
15.1-a |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{7} \cdot 5^{8} \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.198237855$ |
$361.3470164$ |
4.186611012 |
\( -\frac{865425473391488}{854296875} a^{3} + \frac{3190457881430336}{854296875} a^{2} - \frac{32546676610432}{56953125} a - \frac{2914969854940352}{854296875} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{3} - 3 a^{2} - 2 a + 6\) , \( a^{2} - 2\) , \( 5 a^{3} - 14 a^{2} - 14 a + 29\) , \( 4 a^{3} - 11 a^{2} - 12 a + 24\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+6\right){x}^{2}+\left(5a^{3}-14a^{2}-14a+29\right){x}+4a^{3}-11a^{2}-12a+24$ |
| 15.1-b1 |
15.1-b |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5 \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.334965738$ |
$325.1826247$ |
3.978865753 |
\( \frac{1296459818}{1215} a^{3} + \frac{8210945029}{1215} a^{2} - \frac{56048924}{81} a - \frac{16189439938}{1215} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -3 a^{3} + 10 a^{2} - a - 12\) , \( -16 a^{3} + 57 a^{2} - 13 a - 54\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-3a^{3}+10a^{2}-a-12\right){x}-16a^{3}+57a^{2}-13a-54$ |
| 15.1-b2 |
15.1-b |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{10} \cdot 5^{2} \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.167482869$ |
$162.5913123$ |
3.978865753 |
\( \frac{19436318048457602401}{1476225} a^{3} - \frac{54528894369140239672}{1476225} a^{2} - \frac{3550514187097184041}{98415} a + \frac{120645193213920105829}{1476225} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 4 a - 1\) , \( 22 a^{3} - 85 a^{2} + 39 a + 43\) , \( -130 a^{3} + 480 a^{2} - 150 a - 348\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(22a^{3}-85a^{2}+39a+43\right){x}-130a^{3}+480a^{2}-150a-348$ |
| 15.1-c1 |
15.1-c |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{7} \cdot 5^{8} \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 7 \) |
$0.046191611$ |
$664.5297892$ |
3.139548291 |
\( -\frac{865425473391488}{854296875} a^{3} + \frac{3190457881430336}{854296875} a^{2} - \frac{32546676610432}{56953125} a - \frac{2914969854940352}{854296875} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 7 a^{3} - 6 a^{2} - 19 a - 3\) , \( 4 a^{3} + 11 a^{2} - 11 a - 14\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(7a^{3}-6a^{2}-19a-3\right){x}+4a^{3}+11a^{2}-11a-14$ |
| 15.1-c2 |
15.1-c |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{14} \cdot 5^{4} \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.023095805$ |
$664.5297892$ |
3.139548291 |
\( -\frac{38162367695555072}{2989355625} a^{3} + \frac{25930133465825984}{2989355625} a^{2} + \frac{15004444299070592}{199290375} a + \frac{144533247929706112}{2989355625} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 3\) , \( -a^{2} + a + 4\) , \( a\) , \( 7 a^{3} - 24 a^{2} + 27\) , \( -17 a^{3} + 64 a^{2} - 12 a - 57\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(7a^{3}-24a^{2}+27\right){x}-17a^{3}+64a^{2}-12a-57$ |
| 15.1-d1 |
15.1-d |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5 \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.543864797$ |
$119.7424561$ |
2.225377787 |
\( \frac{1296459818}{1215} a^{3} + \frac{8210945029}{1215} a^{2} - \frac{56048924}{81} a - \frac{16189439938}{1215} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( -10 a^{3} + 36 a^{2} - 8 a - 27\) , \( -223 a^{3} + 791 a^{2} - 111 a - 721\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-10a^{3}+36a^{2}-8a-27\right){x}-223a^{3}+791a^{2}-111a-721$ |
| 15.1-d2 |
15.1-d |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{10} \cdot 5^{2} \) |
$17.15886$ |
$(-a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.271932398$ |
$59.87122808$ |
2.225377787 |
\( \frac{19436318048457602401}{1476225} a^{3} - \frac{54528894369140239672}{1476225} a^{2} - \frac{3550514187097184041}{98415} a + \frac{120645193213920105829}{1476225} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( 195 a^{3} - 694 a^{2} + 107 a + 618\) , \( -3007 a^{3} + 10657 a^{2} - 1438 a - 9753\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(195a^{3}-694a^{2}+107a+618\right){x}-3007a^{3}+10657a^{2}-1438a-9753$ |
| 20.1-a1 |
20.1-a |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{3} \) |
$17.78712$ |
$(a), (-a^2+a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$347.0334078$ |
2.535320372 |
\( \frac{19146052}{125} a^{3} - \frac{270022301}{500} a^{2} + \frac{6599311}{100} a + \frac{242669857}{500} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + a^{2} + 3 a\) , \( 0\) , \( -3 a^{3} - a^{2} + 17 a + 13\) , \( a^{2} + 12 a + 9\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-3a^{3}-a^{2}+17a+13\right){x}+a^{2}+12a+9$ |
| 20.1-a2 |
20.1-a |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{18} \cdot 5 \) |
$17.78712$ |
$(a), (-a^2+a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3^{2} \) |
$1$ |
$4.284363059$ |
2.535320372 |
\( \frac{53389253593770657461}{160} a^{3} - \frac{189492305508745213317}{160} a^{2} + \frac{5325290200110207693}{32} a + \frac{86152868600922316927}{80} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + a^{2} + 3 a\) , \( 0\) , \( 47 a^{3} - 146 a^{2} + 27 a + 138\) , \( 366 a^{3} - 1321 a^{2} + 208 a + 1214\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(47a^{3}-146a^{2}+27a+138\right){x}+366a^{3}-1321a^{2}+208a+1214$ |
| 20.1-b1 |
20.1-b |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{3} \) |
$17.78712$ |
$(a), (-a^2+a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$114.4035608$ |
0.835797568 |
\( \frac{19146052}{125} a^{3} - \frac{270022301}{500} a^{2} + \frac{6599311}{100} a + \frac{242669857}{500} \) |
\( \bigl[1\) , \( -a^{2} + a + 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 3 a^{3} - 12 a^{2} + 4 a + 14\) , \( 8 a^{3} - 28 a^{2} + 3 a + 24\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(3a^{3}-12a^{2}+4a+14\right){x}+8a^{3}-28a^{2}+3a+24$ |
| 20.1-b2 |
20.1-b |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{18} \cdot 5 \) |
$17.78712$ |
$(a), (-a^2+a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$12.71150675$ |
0.835797568 |
\( \frac{53389253593770657461}{160} a^{3} - \frac{189492305508745213317}{160} a^{2} + \frac{5325290200110207693}{32} a + \frac{86152868600922316927}{80} \) |
\( \bigl[1\) , \( -a^{2} + a + 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 273 a^{3} - 972 a^{2} + 144 a + 889\) , \( 5495 a^{3} - 19506 a^{2} + 2746 a + 17739\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(273a^{3}-972a^{2}+144a+889\right){x}+5495a^{3}-19506a^{2}+2746a+17739$ |
| 21.1-a1 |
21.1-a |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{3} \) |
$17.89594$ |
$(-a+1), (a^2-a-2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \cdot 3 \) |
$1$ |
$56.29670740$ |
3.419270455 |
\( -\frac{539105035486}{3087} a^{3} - \frac{295938980621}{441} a^{2} + \frac{2897336874164}{1029} a + \frac{8063938576502}{3087} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 10 a^{3} - 14 a^{2} - 42 a - 20\) , \( 23 a^{3} - 28 a^{2} - 106 a - 60\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(10a^{3}-14a^{2}-42a-20\right){x}+23a^{3}-28a^{2}-106a-60$ |
| 21.1-a2 |
21.1-a |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{6} \) |
$17.89594$ |
$(-a+1), (a^2-a-2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{2} \cdot 3 \) |
$1$ |
$14.07417685$ |
3.419270455 |
\( -\frac{7160824368248177247534251}{9529569} a^{3} + \frac{3630803224952966312784116}{1361367} a^{2} - \frac{1190422705762490880799601}{3176523} a - \frac{23110477084613048835734003}{9529569} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 35 a^{3} - 109 a^{2} - 2 a + 35\) , \( 252 a^{3} - 868 a^{2} + 126 a + 573\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(35a^{3}-109a^{2}-2a+35\right){x}+252a^{3}-868a^{2}+126a+573$ |
| 21.1-b1 |
21.1-b |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{6} \) |
$17.89594$ |
$(-a+1), (a^2-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$57.52380789$ |
2.521508513 |
\( -\frac{7160824368248177247534251}{9529569} a^{3} + \frac{3630803224952966312784116}{1361367} a^{2} - \frac{1190422705762490880799601}{3176523} a - \frac{23110477084613048835734003}{9529569} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{3} - 3 a^{2} - a + 6\) , \( a^{2} - a - 3\) , \( -17 a^{3} + 8 a^{2} + 54 a - 51\) , \( 87 a^{3} - 57 a^{2} - 160 a + 109\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+6\right){x}^{2}+\left(-17a^{3}+8a^{2}+54a-51\right){x}+87a^{3}-57a^{2}-160a+109$ |
| 21.1-b2 |
21.1-b |
$2$ |
$2$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{3} \) |
$17.89594$ |
$(-a+1), (a^2-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$230.0952315$ |
2.521508513 |
\( -\frac{539105035486}{3087} a^{3} - \frac{295938980621}{441} a^{2} + \frac{2897336874164}{1029} a + \frac{8063938576502}{3087} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{3} - 3 a^{2} - a + 6\) , \( a^{2} - a - 3\) , \( -7 a^{3} - 17 a^{2} + 9 a + 29\) , \( 47 a^{3} + 48 a^{2} - 88 a - 77\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+6\right){x}^{2}+\left(-7a^{3}-17a^{2}+9a+29\right){x}+47a^{3}+48a^{2}-88a-77$ |
| 27.2-a1 |
27.2-a |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$18.46705$ |
$(-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$299.0752032$ |
2.184952337 |
\( -64801 a^{3} + 46326 a^{2} + 378735 a + 234712 \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - a - 4\) , \( a^{3} - a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 2 a^{3} + a^{2} - 7 a - 6\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(a^{3}+a^{2}-2a-2\right){x}+2a^{3}+a^{2}-7a-6$ |
| 27.2-a2 |
27.2-a |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{11} \) |
$18.46705$ |
$(-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$33.23057814$ |
2.184952337 |
\( -28205392301321970 a^{3} + 19168817394981516 a^{2} + 166337173865044673 a + 106807482865238932 \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - a - 4\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -9 a^{3} - 24 a^{2} + 3 a + 28\) , \( -195 a^{3} - 187 a^{2} + 397 a + 338\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-9a^{3}-24a^{2}+3a+28\right){x}-195a^{3}-187a^{2}+397a+338$ |
| 27.2-b1 |
27.2-b |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{11} \) |
$18.46705$ |
$(-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1.307802499$ |
$4.888027039$ |
1.681279139 |
\( -28205392301321970 a^{3} + 19168817394981516 a^{2} + 166337173865044673 a + 106807482865238932 \) |
\( \bigl[a^{3} - a^{2} - 3 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 1\) , \( a\) , \( 44 a^{3} - 28 a^{2} - 254 a - 156\) , \( 478 a^{3} - 323 a^{2} - 2809 a - 1807\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a-1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-1\right){x}^{2}+\left(44a^{3}-28a^{2}-254a-156\right){x}+478a^{3}-323a^{2}-2809a-1807$ |
| 27.2-b2 |
27.2-b |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$18.46705$ |
$(-a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.435934166$ |
$395.9301901$ |
1.681279139 |
\( -64801 a^{3} + 46326 a^{2} + 378735 a + 234712 \) |
\( \bigl[a^{3} - a^{2} - 3 a - 1\) , \( -a^{3} + 2 a^{2} + 2 a - 1\) , \( a\) , \( -a^{3} + 7 a^{2} + 6 a - 1\) , \( 3 a^{3} + 6 a^{2} - 9 a - 9\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a-1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-1\right){x}^{2}+\left(-a^{3}+7a^{2}+6a-1\right){x}+3a^{3}+6a^{2}-9a-9$ |
| 28.1-a1 |
28.1-a |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{10} \cdot 7^{8} \) |
$18.55119$ |
$(a^2-a-2), (-a^2+a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$118.5765875$ |
1.732568861 |
\( \frac{738092002225}{23059204} a^{3} - \frac{607944512351}{6588344} a^{2} - \frac{3532776509883}{46118408} a + \frac{8686252915607}{46118408} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( -a^{3} + a^{2} + 4 a + 2\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( -8 a^{3} + 19 a^{2} + 28 a - 32\) , \( -24 a^{3} + 59 a^{2} + 70 a - 125\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{3}-2a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a+2\right){x}^{2}+\left(-8a^{3}+19a^{2}+28a-32\right){x}-24a^{3}+59a^{2}+70a-125$ |
| 28.1-b1 |
28.1-b |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{10} \cdot 7^{8} \) |
$18.55119$ |
$(a^2-a-2), (-a^2+a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$0.030411820$ |
$367.4328612$ |
3.265441906 |
\( \frac{738092002225}{23059204} a^{3} - \frac{607944512351}{6588344} a^{2} - \frac{3532776509883}{46118408} a + \frac{8686252915607}{46118408} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( -a^{3} + 3 a^{2} + 2 a - 6\) , \( a^{2} - 3\) , \( -4 a^{3} + 9 a^{2} + 13 a - 10\) , \( -17 a^{3} - 9 a^{2} + 131 a + 165\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-6\right){x}^{2}+\left(-4a^{3}+9a^{2}+13a-10\right){x}-17a^{3}-9a^{2}+131a+165$ |
| 33.1-a1 |
33.1-a |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11 \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$321.5090041$ |
2.348846853 |
\( \frac{68377}{33} a^{3} - \frac{184027}{33} a^{2} - \frac{71647}{11} a + \frac{384361}{33} \) |
\( \bigl[a^{2} - 2\) , \( -a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{2} - a - 1\) , \( a^{3} + a^{2} - 3 a - 3\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a^{2}-a-1\right){x}+a^{3}+a^{2}-3a-3$ |
| 33.1-b1 |
33.1-b |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{9} \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$23.29724996$ |
1.531823507 |
\( \frac{29304895037561053705}{7073843073} a^{3} - \frac{104007861442395688351}{7073843073} a^{2} + \frac{4867559735628413724}{2357947691} a + \frac{94588503033013182916}{7073843073} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( -21 a^{3} + 43 a^{2} + 131 a - 193\) , \( 342 a^{3} - 1041 a^{2} - 571 a + 1785\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(-21a^{3}+43a^{2}+131a-193\right){x}+342a^{3}-1041a^{2}-571a+1785$ |
| 33.1-b2 |
33.1-b |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{3} \cdot 11^{3} \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$209.6752496$ |
1.531823507 |
\( \frac{28822066}{35937} a^{3} - \frac{89478406}{35937} a^{2} - \frac{10842131}{11979} a + \frac{172857379}{35937} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( a^{2} - 2\) , \( -a^{3} + 3 a^{2} + 6 a - 3\) , \( -2 a^{3} + 8 a^{2} + 8 a - 15\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+2\right){x}^{2}+\left(-a^{3}+3a^{2}+6a-3\right){x}-2a^{3}+8a^{2}+8a-15$ |
| 33.1-c1 |
33.1-c |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{5} \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$0.022197859$ |
$644.3450868$ |
2.089879183 |
\( \frac{33407652673139647}{483153} a^{3} - \frac{93725901150560797}{483153} a^{2} - \frac{30513280892570535}{161051} a + \frac{207367282753699441}{483153} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + a^{2} + 3 a + 1\) , \( a^{2} - 2\) , \( 178 a^{3} - 649 a^{2} + 121 a + 613\) , \( -2853 a^{3} + 10100 a^{2} - 1368 a - 9147\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a+1\right){x}^{2}+\left(178a^{3}-649a^{2}+121a+613\right){x}-2853a^{3}+10100a^{2}-1368a-9147$ |
| 33.1-d1 |
33.1-d |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{5} \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 5 \) |
$1$ |
$165.1907169$ |
6.034165307 |
\( \frac{33407652673139647}{483153} a^{3} - \frac{93725901150560797}{483153} a^{2} - \frac{30513280892570535}{161051} a + \frac{207367282753699441}{483153} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - a - 3\) , \( a + 1\) , \( 31 a^{3} - 102 a^{2} + 13 a + 93\) , \( -178 a^{3} + 651 a^{2} - 100 a - 596\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(31a^{3}-102a^{2}+13a+93\right){x}-178a^{3}+651a^{2}-100a-596$ |
| 33.1-e1 |
33.1-e |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3^{3} \cdot 11^{3} \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.015953558$ |
$312.6710951$ |
6.140002606 |
\( \frac{28822066}{35937} a^{3} - \frac{89478406}{35937} a^{2} - \frac{10842131}{11979} a + \frac{172857379}{35937} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( -a^{3} - 4 a^{2} + 15 a + 11\) , \( 2 a^{3} - 13 a^{2} + 11 a + 23\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(-a^{3}-4a^{2}+15a+11\right){x}+2a^{3}-13a^{2}+11a+23$ |
| 33.1-e2 |
33.1-e |
$2$ |
$3$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11^{9} \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$6.047860674$ |
$3.860136977$ |
6.140002606 |
\( \frac{29304895037561053705}{7073843073} a^{3} - \frac{104007861442395688351}{7073843073} a^{2} + \frac{4867559735628413724}{2357947691} a + \frac{94588503033013182916}{7073843073} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( 114 a^{3} - 424 a^{2} + 135 a + 301\) , \( 1854 a^{3} - 6635 a^{2} + 1197 a + 5653\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(114a^{3}-424a^{2}+135a+301\right){x}+1854a^{3}-6635a^{2}+1197a+5653$ |
| 33.1-f1 |
33.1-f |
$1$ |
$1$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
33.1 |
\( 3 \cdot 11 \) |
\( - 3 \cdot 11 \) |
$18.93613$ |
$(-a+1), (a^2-2a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.190739686$ |
$540.4790662$ |
3.012600164 |
\( \frac{68377}{33} a^{3} - \frac{184027}{33} a^{2} - \frac{71647}{11} a + \frac{384361}{33} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 3 a^{2} + 2 a - 6\) , \( a + 1\) , \( a^{2} + 3 a + 6\) , \( a^{3} + 2 a^{2} - a - 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-6\right){x}^{2}+\left(a^{2}+3a+6\right){x}+a^{3}+2a^{2}-a-3$ |
| 35.1-a1 |
35.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7^{9} \) |
$19.07592$ |
$(a), (a^2-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$6.979359937$ |
2.065057651 |
\( \frac{18650538221112678912}{1008840175} a^{3} - \frac{10198668275108799552}{144120025} a^{2} - \frac{4557729469645476736}{201768035} a + \frac{205399518982498488448}{1008840175} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( a^{2} - 3\) , \( 5 a^{3} + 57 a^{2} - 117 a - 307\) , \( 93 a^{3} + 249 a^{2} - 1010 a - 1834\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(5a^{3}+57a^{2}-117a-307\right){x}+93a^{3}+249a^{2}-1010a-1834$ |
| 35.1-a2 |
35.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{6} \cdot 7^{3} \) |
$19.07592$ |
$(a), (a^2-a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$565.3281549$ |
2.065057651 |
\( -\frac{20946263552}{5359375} a^{3} + \frac{10137369792}{765625} a^{2} - \frac{2225184384}{1071875} a - \frac{32303431808}{5359375} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( a^{2} - 3\) , \( 2 a^{2} - 2 a - 7\) , \( a^{2} - a - 4\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2a^{2}-2a-7\right){x}+a^{2}-a-4$ |
| 35.1-a3 |
35.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5 \cdot 7^{18} \) |
$19.07592$ |
$(a), (a^2-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$6.979359937$ |
2.065057651 |
\( -\frac{491228665354000707712}{8142067989552245} a^{3} + \frac{285058561817380729152}{1163152569936035} a^{2} + \frac{470895259440932646528}{1628413597910449} a - \frac{2968352427121729803328}{8142067989552245} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 2 a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -78 a^{3} - 82 a^{2} + 142 a + 129\) , \( -1639 a^{3} - 1837 a^{2} + 3123 a + 2849\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+1\right){x}^{2}+\left(-78a^{3}-82a^{2}+142a+129\right){x}-1639a^{3}-1837a^{2}+3123a+2849$ |
| 35.1-a4 |
35.1-a |
$4$ |
$6$ |
4.4.18736.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{3} \cdot 7^{6} \) |
$19.07592$ |
$(a), (a^2-a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$565.3281549$ |
2.065057651 |
\( -\frac{14042412928}{14706125} a^{3} + \frac{1802132288}{2100875} a^{2} + \frac{19381610624}{2941225} a + \frac{67643741888}{14706125} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( a^{2} - 2\) , \( 7 a^{3} - 17 a + 6\) , \( -27 a^{3} + 167 a^{2} - 50 a - 163\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+3\right){x}^{2}+\left(7a^{3}-17a+6\right){x}-27a^{3}+167a^{2}-50a-163$ |
*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.
