| 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 |
| 4.1-a1 |
4.1-a |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$12.86482$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 3 \) |
$0.783461947$ |
$145.9365131$ |
2.833318517 |
\( 3072 a^{3} - 11264 a^{2} + 6016 a + 3632 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{2} + 2\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{3} - 5 a^{2} + 4 a + 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(a^{3}-4a^{2}+2a+2\right){x}+a^{3}-5a^{2}+4a+2$ |
| 4.1-a2 |
4.1-a |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$12.86482$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 3 \) |
$0.391730973$ |
$291.8730263$ |
2.833318517 |
\( -155041788 a^{3} + 557301756 a^{2} - 268140568 a - 193838348 \) |
\( \bigl[a\) , \( a^{3} - 3 a^{2} - 3 a + 5\) , \( 0\) , \( -13 a^{3} - 21 a^{2} + 10 a + 17\) , \( -121 a^{3} - 96 a^{2} + 215 a + 97\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a^{3}-3a^{2}-3a+5\right){x}^{2}+\left(-13a^{3}-21a^{2}+10a+17\right){x}-121a^{3}-96a^{2}+215a+97$ |
| 4.1-b1 |
4.1-b |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$12.86482$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 3 \) |
$0.047636491$ |
$1312.761052$ |
1.549669110 |
\( 3072 a^{3} - 11264 a^{2} + 6016 a + 3632 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 3 a^{2} - 3 a + 4\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( 2 a^{3} - 6 a^{2} - 2 a + 7\) , \( -2 a^{3} + 7 a^{2} - 5 a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-2a^{2}-2a+2\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+4\right){x}^{2}+\left(2a^{3}-6a^{2}-2a+7\right){x}-2a^{3}+7a^{2}-5a$ |
| 4.1-b2 |
4.1-b |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$12.86482$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 3 \) |
$0.023818245$ |
$2625.522104$ |
1.549669110 |
\( -155041788 a^{3} + 557301756 a^{2} - 268140568 a - 193838348 \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 3 a^{2} + 3 a - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -15 a^{3} - 15 a^{2} + 16 a + 6\) , \( 107 a^{3} + 78 a^{2} - 202 a - 86\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+3a-4\right){x}^{2}+\left(-15a^{3}-15a^{2}+16a+6\right){x}+107a^{3}+78a^{2}-202a-86$ |
| 6.1-a1 |
6.1-a |
$2$ |
$7$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{7} \cdot 3^{7} \) |
$13.53366$ |
$(a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 7 \) |
$0.032880609$ |
$392.8001750$ |
2.987183022 |
\( -\frac{3135665}{8748} a^{3} + \frac{663160}{729} a^{2} - \frac{449851}{8748} a + \frac{1243190}{2187} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -3 a^{3} - a^{2} + 23 a + 11\) , \( -5 a^{3} - 2 a^{2} + 41 a + 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(-3a^{3}-a^{2}+23a+11\right){x}-5a^{3}-2a^{2}+41a+13$ |
| 6.1-a2 |
6.1-a |
$2$ |
$7$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3 \) |
$13.53366$ |
$(a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$0.230164264$ |
$392.8001750$ |
2.987183022 |
\( -\frac{194556648524736783475}{6} a^{3} - 26782370872497663107 a^{2} + \frac{162054508393092817655}{3} a + \frac{68846155117469089886}{3} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 1182 a^{3} - 3496 a^{2} + 188 a + 641\) , \( -43453 a^{3} + 145326 a^{2} - 48476 a - 42727\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(1182a^{3}-3496a^{2}+188a+641\right){x}-43453a^{3}+145326a^{2}-48476a-42727$ |
| 6.1-b1 |
6.1-b |
$1$ |
$1$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{9} \) |
$13.53366$ |
$(a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$0.007590559$ |
$1088.928200$ |
2.457920479 |
\( \frac{1863988241}{39366} a^{3} - \frac{213084893}{6561} a^{2} - \frac{4829844136}{19683} a - \frac{2270471680}{19683} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - a^{2} - 3 a - 1\) , \( -9 a^{3} + 22 a^{2} + 28 a - 51\) , \( 17 a^{3} - 40 a^{2} - 55 a + 86\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+2\right){x}^{2}+\left(-9a^{3}+22a^{2}+28a-51\right){x}+17a^{3}-40a^{2}-55a+86$ |
| 6.1-c1 |
6.1-c |
$1$ |
$1$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{9} \) |
$13.53366$ |
$(a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$106.6022619$ |
0.880559469 |
\( \frac{1863988241}{39366} a^{3} - \frac{213084893}{6561} a^{2} - \frac{4829844136}{19683} a - \frac{2270471680}{19683} \) |
\( \bigl[a^{3} - a^{2} - 3 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -11 a^{3} + 26 a^{2} + 41 a - 53\) , \( -24 a^{3} + 57 a^{2} + 80 a - 119\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(-11a^{3}+26a^{2}+41a-53\right){x}-24a^{3}+57a^{2}+80a-119$ |
| 6.1-d1 |
6.1-d |
$2$ |
$7$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{7} \cdot 3^{7} \) |
$13.53366$ |
$(a), (a+1)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7^{2} \) |
$1$ |
$348.6746347$ |
2.880133551 |
\( -\frac{3135665}{8748} a^{3} + \frac{663160}{729} a^{2} - \frac{449851}{8748} a + \frac{1243190}{2187} \) |
\( \bigl[1\) , \( -a^{3} + 3 a^{2} + a - 6\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - 5 a^{2} + 2 a + 11\) , \( -a^{3} + 2 a^{2} + 4 a - 7\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+a-6\right){x}^{2}+\left(a^{3}-5a^{2}+2a+11\right){x}-a^{3}+2a^{2}+4a-7$ |
| 6.1-d2 |
6.1-d |
$2$ |
$7$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3 \) |
$13.53366$ |
$(a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$2401$ |
\( 1 \) |
$1$ |
$0.145220589$ |
2.880133551 |
\( -\frac{194556648524736783475}{6} a^{3} - 26782370872497663107 a^{2} + \frac{162054508393092817655}{3} a + \frac{68846155117469089886}{3} \) |
\( \bigl[1\) , \( -a^{3} + 3 a^{2} + a - 6\) , \( a^{3} - 2 a^{2} - 2 a + 2\) , \( 1186 a^{3} - 3500 a^{2} + 167 a + 641\) , \( 46312 a^{3} - 153591 a^{2} + 48491 a + 44173\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+a-6\right){x}^{2}+\left(1186a^{3}-3500a^{2}+167a+641\right){x}+46312a^{3}-153591a^{2}+48491a+44173$ |
| 9.1-a1 |
9.1-a |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1.261073430$ |
$1024.663326$ |
2.371927042 |
\( \frac{3631744}{9} a^{3} + \frac{1013632}{3} a^{2} - \frac{6085888}{9} a - \frac{2584768}{9} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( 1\) , \( a^{3} + a^{2} - a + 1\) , \( a^{3} + a^{2} + a\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-1\right){x}^{2}+\left(a^{3}+a^{2}-a+1\right){x}+a^{3}+a^{2}+a$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.630536715$ |
$1024.663326$ |
2.371927042 |
\( -\frac{3682432}{81} a^{3} + \frac{1006208}{27} a^{2} + \frac{18407680}{81} a + \frac{6783808}{81} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 3\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( -a^{3} - 4 a^{2} + 7\) , \( 3 a^{3} + 3 a^{2} - 4 a - 4\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-2a^{2}-2a+3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{3}-4a^{2}+7\right){x}+3a^{3}+3a^{2}-4a-4$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$3.783220292$ |
$37.95049356$ |
2.371927042 |
\( -\frac{188835460226432}{729} a^{3} + \frac{226153272995200}{243} a^{2} - \frac{325351272405760}{729} a - \frac{237102042732736}{729} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a^{2} - a + 5\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( -7 a^{3} + 12 a^{2} + 18 a - 21\) , \( -48 a^{3} + 87 a^{2} + 134 a - 208\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-2a^{2}-2a+3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-a+5\right){x}^{2}+\left(-7a^{3}+12a^{2}+18a-21\right){x}-48a^{3}+87a^{2}+134a-208$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{18} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.891610146$ |
$37.95049356$ |
2.371927042 |
\( \frac{250955794304}{531441} a^{3} - \frac{228423336832}{177147} a^{2} - \frac{431914909952}{531441} a + \frac{1030181115712}{531441} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{2} + 3\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( -41 a^{3} + 95 a^{2} + 132 a - 205\) , \( -286 a^{3} + 679 a^{2} + 887 a - 1477\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-41a^{3}+95a^{2}+132a-205\right){x}-286a^{3}+679a^{2}+887a-1477$ |
| 9.1-b1 |
9.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.173880723$ |
$436.6447434$ |
1.254301500 |
\( 287496 \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + 4\) , \( a^{3} - a^{2} - 4 a - 1\) , \( 5 a^{3} - 13 a^{2} - 40 a - 7\) , \( -31 a^{3} - a^{2} + 102 a + 40\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(5a^{3}-13a^{2}-40a-7\right){x}-31a^{3}-a^{2}+102a+40$ |
| 9.1-b2 |
9.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.173880723$ |
$436.6447434$ |
1.254301500 |
\( 1728 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a^{2} - 2 a + 4\) , \( a^{2} - a - 3\) , \( 6 a^{3} - 12 a^{2} - 20 a + 29\) , \( 10 a^{3} - 21 a^{2} - 29 a + 44\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+4\right){x}^{2}+\left(6a^{3}-12a^{2}-20a+29\right){x}+10a^{3}-21a^{2}-29a+44$ |
| 9.1-b3 |
9.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.086940361$ |
$1746.578973$ |
1.254301500 |
\( 1728 \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( -5 a^{3} + 7 a^{2} + 21 a - 5\) , \( -3 a^{3} + 2 a^{2} + 15 a + 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-2a^{2}-2a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(-5a^{3}+7a^{2}+21a-5\right){x}-3a^{3}+2a^{2}+15a+4$ |
| 9.1-b4 |
9.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.043470180$ |
$1746.578973$ |
1.254301500 |
\( 287496 \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a + 1\) , \( 1\) , \( 4 a^{3} - 9 a^{2} - 35 a - 12\) , \( 41 a^{3} - 22 a^{2} - 180 a - 64\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4a^{3}-9a^{2}-35a-12\right){x}+41a^{3}-22a^{2}-180a-64$ |
| 9.1-c1 |
9.1-c |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.622495638$ |
$234.7381985$ |
2.414028246 |
\( \frac{3631744}{9} a^{3} + \frac{1013632}{3} a^{2} - \frac{6085888}{9} a - \frac{2584768}{9} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( 2 a^{3} - a^{2} - 4 a + 3\) , \( 2 a^{3} + 2 a^{2} - 4 a - 5\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-2a^{2}-2a+3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(2a^{3}-a^{2}-4a+3\right){x}+2a^{3}+2a^{2}-4a-5$ |
| 9.1-c2 |
9.1-c |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.311247819$ |
$234.7381985$ |
2.414028246 |
\( -\frac{3682432}{81} a^{3} + \frac{1006208}{27} a^{2} + \frac{18407680}{81} a + \frac{6783808}{81} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a + 2\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -3 a^{3} + 2 a^{2} + 13 a + 4\) , \( -11 a^{3} - 7 a^{2} + 22 a + 8\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+2\right){x}^{2}+\left(-3a^{3}+2a^{2}+13a+4\right){x}-11a^{3}-7a^{2}+22a+8$ |
| 9.1-c3 |
9.1-c |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.207498546$ |
$704.2145956$ |
2.414028246 |
\( -\frac{188835460226432}{729} a^{3} + \frac{226153272995200}{243} a^{2} - \frac{325351272405760}{729} a - \frac{237102042732736}{729} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a^{2} - 3 a + 6\) , \( a^{2} - 3\) , \( -7 a^{3} + 8 a^{2} + 19 a - 18\) , \( 12 a^{3} - 33 a^{2} - 34 a + 78\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+6\right){x}^{2}+\left(-7a^{3}+8a^{2}+19a-18\right){x}+12a^{3}-33a^{2}-34a+78$ |
| 9.1-c4 |
9.1-c |
$4$ |
$6$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{18} \) |
$14.23727$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.103749273$ |
$704.2145956$ |
2.414028246 |
\( \frac{250955794304}{531441} a^{3} - \frac{228423336832}{177147} a^{2} - \frac{431914909952}{531441} a + \frac{1030181115712}{531441} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 3 a^{2} - 3 a + 6\) , \( a^{2} - a - 3\) , \( -37 a^{3} + 87 a^{2} + 117 a - 192\) , \( 146 a^{3} - 348 a^{2} - 453 a + 763\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+6\right){x}^{2}+\left(-37a^{3}+87a^{2}+117a-192\right){x}+146a^{3}-348a^{2}-453a+763$ |
| 12.1-a1 |
12.1-a |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3 \) |
$14.75856$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.863642939$ |
$410.6406837$ |
2.929466076 |
\( \frac{180850208}{3} a^{3} - 143794624 a^{2} - \frac{557219456}{3} a + \frac{938280544}{3} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 3 a^{2} + 3 a - 6\) , \( a^{3} - a^{2} - 3 a\) , \( 4 a^{3} - 3 a^{2} - 12 a + 4\) , \( 4 a^{3} + 2 a^{2} - 11 a - 7\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+3a-6\right){x}^{2}+\left(4a^{3}-3a^{2}-12a+4\right){x}+4a^{3}+2a^{2}-11a-7$ |
| 12.1-a2 |
12.1-a |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$14.75856$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.431821469$ |
$410.6406837$ |
2.929466076 |
\( -\frac{17192320}{9} a^{3} + \frac{4841984}{3} a^{2} + \frac{85619200}{9} a + \frac{30042496}{9} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( a^{2} - 2\) , \( a^{3} - 2 a + 2\) , \( 3 a^{3} + 5 a^{2} - 4 a - 7\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+1\right){x}^{2}+\left(a^{3}-2a+2\right){x}+3a^{3}+5a^{2}-4a-7$ |
| 12.1-b1 |
12.1-b |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3^{3} \) |
$14.75856$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.049019664$ |
$1792.315705$ |
2.177200227 |
\( -\frac{788542432}{27} a^{3} + \frac{1054227008}{9} a^{2} - \frac{1611714560}{27} a - \frac{1136314016}{27} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 3 a^{2} + a - 5\) , \( a^{3} - a^{2} - 3 a\) , \( -a^{2} - 4 a + 1\) , \( -a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+a-5\right){x}^{2}+\left(-a^{2}-4a+1\right){x}-a-1$ |
| 12.1-b2 |
12.1-b |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{6} \) |
$14.75856$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.024509832$ |
$1792.315705$ |
2.177200227 |
\( -\frac{7939179904}{729} a^{3} + \frac{2241559040}{243} a^{2} + \frac{39518829568}{729} a + \frac{13785278848}{729} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + 5 a + 1\) , \( a^{2} - 2\) , \( 127 a^{3} - 464 a^{2} + 236 a + 166\) , \( -2126 a^{3} + 7632 a^{2} - 3650 a - 2663\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+1\right){x}^{2}+\left(127a^{3}-464a^{2}+236a+166\right){x}-2126a^{3}+7632a^{2}-3650a-2663$ |
| 12.1-c1 |
12.1-c |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3 \) |
$14.75856$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1337.050246$ |
2.761086476 |
\( \frac{180850208}{3} a^{3} - 143794624 a^{2} - \frac{557219456}{3} a + \frac{938280544}{3} \) |
\( \bigl[a^{2} - 2\) , \( -a - 1\) , \( a^{2} - a - 2\) , \( a^{2} - a - 6\) , \( -a^{3} + 7 a + 5\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a^{2}-a-6\right){x}-a^{3}+7a+5$ |
| 12.1-c2 |
12.1-c |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$14.75856$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$668.5251232$ |
2.761086476 |
\( -\frac{17192320}{9} a^{3} + \frac{4841984}{3} a^{2} + \frac{85619200}{9} a + \frac{30042496}{9} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( -a^{3} + a^{2} + 5 a\) , \( a^{3} - a^{2} - 4 a\) , \( -a^{3} + 4 a^{2} + 6 a - 3\) , \( -3 a^{3} - 3 a^{2} + 5 a + 4\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a\right){x}^{2}+\left(-a^{3}+4a^{2}+6a-3\right){x}-3a^{3}-3a^{2}+5a+4$ |
| 12.1-d1 |
12.1-d |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3^{3} \) |
$14.75856$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3^{2} \) |
$1$ |
$101.9701111$ |
1.895167859 |
\( -\frac{788542432}{27} a^{3} + \frac{1054227008}{9} a^{2} - \frac{1611714560}{27} a - \frac{1136314016}{27} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( -a^{3} + 2 a^{2} + 2 a - 1\) , \( a\) , \( -4 a^{3} + 4 a^{2} + 15 a + 2\) , \( -a^{3} - 2 a^{2} + 6 a + 6\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-1\right){x}^{2}+\left(-4a^{3}+4a^{2}+15a+2\right){x}-a^{3}-2a^{2}+6a+6$ |
| 12.1-d2 |
12.1-d |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{6} \) |
$14.75856$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$50.98505555$ |
1.895167859 |
\( -\frac{7939179904}{729} a^{3} + \frac{2241559040}{243} a^{2} + \frac{39518829568}{729} a + \frac{13785278848}{729} \) |
\( \bigl[a^{3} - 2 a^{2} - 2 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{2} - 2\) , \( 131 a^{3} - 470 a^{2} + 220 a + 170\) , \( 2383 a^{3} - 8564 a^{2} + 4110 a + 2990\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(131a^{3}-470a^{2}+220a+170\right){x}+2383a^{3}-8564a^{2}+4110a+2990$ |
| 15.1-a1 |
15.1-a |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.241375737$ |
$565.9719863$ |
2.256892210 |
\( -\frac{5575391984}{405} a^{3} + \frac{7791489232}{135} a^{2} - \frac{22905268816}{405} a + \frac{3886464536}{405} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( -a^{3} + a^{2} + 4 a + 1\) , \( a^{2} - 3\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+1\right){x}^{2}+\left(-a^{3}+a^{2}+4a+1\right){x}+a^{2}-3$ |
| 15.1-a2 |
15.1-a |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.241375737$ |
$1131.943972$ |
2.256892210 |
\( -\frac{255712768}{15} a^{3} + \frac{72232704}{5} a^{2} + \frac{1272667648}{15} a + \frac{443931712}{15} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 3 a^{2} + 2 a - 4\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( -a^{3} + 13 a^{2} - 14 a\) , \( 62 a^{3} - 214 a^{2} + 103 a + 70\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-4\right){x}^{2}+\left(-a^{3}+13a^{2}-14a\right){x}+62a^{3}-214a^{2}+103a+70$ |
| 15.1-a3 |
15.1-a |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.120687868$ |
$2263.887945$ |
2.256892210 |
\( \frac{743936}{225} a^{3} + \frac{534272}{75} a^{2} - \frac{1327616}{225} a + \frac{903616}{225} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a^{2} - 2 a + 6\) , \( a^{2} - a - 3\) , \( -32 a^{3} + 70 a^{2} + 95 a - 154\) , \( 78 a^{3} - 180 a^{2} - 238 a + 392\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+6\right){x}^{2}+\left(-32a^{3}+70a^{2}+95a-154\right){x}+78a^{3}-180a^{2}-238a+392$ |
| 15.1-a4 |
15.1-a |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.060343934$ |
$1131.943972$ |
2.256892210 |
\( \frac{1101258894992}{1875} a^{3} + \frac{303198274384}{625} a^{2} - \frac{1834571225552}{1875} a - \frac{779387798648}{1875} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 5 a - 2\) , \( a^{2} - a - 3\) , \( -a^{3} - 2 a^{2} + 2\) , \( a^{3} + a^{2} - 2 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-2\right){x}^{2}+\left(-a^{3}-2a^{2}+2\right){x}+a^{3}+a^{2}-2a-2$ |
| 15.1-b1 |
15.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$0.979076986$ |
$26.48376256$ |
1.713479003 |
\( \frac{253302772771602296925242251}{9265100944259205} a^{3} - \frac{201421065927893686032509663}{3088366981419735} a^{2} - \frac{780245045289219324108485791}{9265100944259205} a + \frac{1314024696214074721104250151}{9265100944259205} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( 1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( -142 a^{3} + 336 a^{2} + 476 a - 828\) , \( -1906 a^{3} + 4610 a^{2} + 5808 a - 10220\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+{x}^{2}+\left(-142a^{3}+336a^{2}+476a-828\right){x}-1906a^{3}+4610a^{2}+5808a-10220$ |
| 15.1-b2 |
15.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{2} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.489538493$ |
$423.7402010$ |
1.713479003 |
\( \frac{29394768971684594}{1076168025} a^{3} + \frac{8046876524598263}{358722675} a^{2} - \frac{49139816543224214}{1076168025} a - \frac{20413651456077461}{1076168025} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( 1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( -2 a^{3} - 4 a^{2} + 41 a - 43\) , \( -59 a^{3} + 173 a^{2} + 53 a - 206\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+{x}^{2}+\left(-2a^{3}-4a^{2}+41a-43\right){x}-59a^{3}+173a^{2}+53a-206$ |
| 15.1-b3 |
15.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{4} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.244769246$ |
$423.7402010$ |
1.713479003 |
\( -\frac{4476041726}{4100625} a^{3} - \frac{1085592902}{1366875} a^{2} + \frac{8122914656}{4100625} a + \frac{2738887319}{4100625} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( 1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( -2 a^{3} + a^{2} + 11 a + 2\) , \( -5 a^{3} + 7 a^{2} + 18 a\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+{x}^{2}+\left(-2a^{3}+a^{2}+11a+2\right){x}-5a^{3}+7a^{2}+18a$ |
| 15.1-b4 |
15.1-b |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.979076986$ |
$423.7402010$ |
1.713479003 |
\( \frac{579253472299654271509}{32805} a^{3} + \frac{159479380095303625663}{10935} a^{2} - \frac{964972208071904658049}{32805} a - \frac{409952734319362969591}{32805} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( 1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 138 a^{3} - 424 a^{2} + 86 a + 22\) , \( -1988 a^{3} + 6820 a^{2} - 2622 a - 2196\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a-1\right){y}={x}^{3}+{x}^{2}+\left(138a^{3}-424a^{2}+86a+22\right){x}-1988a^{3}+6820a^{2}-2622a-2196$ |
| 15.1-c1 |
15.1-c |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{4} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$229.1229489$ |
1.892608830 |
\( \frac{1101258894992}{1875} a^{3} + \frac{303198274384}{625} a^{2} - \frac{1834571225552}{1875} a - \frac{779387798648}{1875} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + 3 a^{2} + a - 5\) , \( a^{2} - a - 3\) , \( -a^{3} - 2 a^{2} - 6 a + 3\) , \( -9 a^{3} - a^{2} + 16 a\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+a-5\right){x}^{2}+\left(-a^{3}-2a^{2}-6a+3\right){x}-9a^{3}-a^{2}+16a$ |
| 15.1-c2 |
15.1-c |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$916.4917956$ |
1.892608830 |
\( -\frac{255712768}{15} a^{3} + \frac{72232704}{5} a^{2} + \frac{1272667648}{15} a + \frac{443931712}{15} \) |
\( \bigl[a^{2} - 2\) , \( 0\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( -2 a^{3} + 12 a^{2} - 10 a - 2\) , \( -66 a^{3} + 240 a^{2} - 114 a - 87\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(-2a^{3}+12a^{2}-10a-2\right){x}-66a^{3}+240a^{2}-114a-87$ |
| 15.1-c3 |
15.1-c |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$916.4917956$ |
1.892608830 |
\( \frac{743936}{225} a^{3} + \frac{534272}{75} a^{2} - \frac{1327616}{225} a + \frac{903616}{225} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( a + 1\) , \( -33 a^{3} + 71 a^{2} + 97 a - 155\) , \( -229 a^{3} + 502 a^{2} + 686 a - 1108\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+5\right){x}^{2}+\left(-33a^{3}+71a^{2}+97a-155\right){x}-229a^{3}+502a^{2}+686a-1108$ |
| 15.1-c4 |
15.1-c |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{4} \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$229.1229489$ |
1.892608830 |
\( -\frac{5575391984}{405} a^{3} + \frac{7791489232}{135} a^{2} - \frac{22905268816}{405} a + \frac{3886464536}{405} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + a^{2} + 3 a + 1\) , \( 1\) , \( -4 a^{3} + 6 a^{2} + 16 a - 1\) , \( -3 a^{3} + 4 a^{2} + 12 a - 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+3a+1\right){x}^{2}+\left(-4a^{3}+6a^{2}+16a-1\right){x}-3a^{3}+4a^{2}+12a-2$ |
| 15.1-d1 |
15.1-d |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{32} \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$95.02139609$ |
1.569797649 |
\( \frac{253302772771602296925242251}{9265100944259205} a^{3} - \frac{201421065927893686032509663}{3088366981419735} a^{2} - \frac{780245045289219324108485791}{9265100944259205} a + \frac{1314024696214074721104250151}{9265100944259205} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -142 a^{3} + 337 a^{2} + 475 a - 829\) , \( 1764 a^{3} - 4274 a^{2} - 5332 a + 9390\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-142a^{3}+337a^{2}+475a-829\right){x}+1764a^{3}-4274a^{2}-5332a+9390$ |
| 15.1-d2 |
15.1-d |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{16} \cdot 5^{2} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$95.02139609$ |
1.569797649 |
\( \frac{29394768971684594}{1076168025} a^{3} + \frac{8046876524598263}{358722675} a^{2} - \frac{49139816543224214}{1076168025} a - \frac{20413651456077461}{1076168025} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -2 a^{3} - 3 a^{2} + 40 a - 44\) , \( 57 a^{3} - 177 a^{2} - 12 a + 161\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-2a^{3}-3a^{2}+40a-44\right){x}+57a^{3}-177a^{2}-12a+161$ |
| 15.1-d3 |
15.1-d |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5^{4} \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$95.02139609$ |
1.569797649 |
\( -\frac{4476041726}{4100625} a^{3} - \frac{1085592902}{1366875} a^{2} + \frac{8122914656}{4100625} a + \frac{2738887319}{4100625} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -2 a^{3} + 2 a^{2} + 10 a + 1\) , \( 3 a^{3} - 6 a^{2} - 7 a\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-2a^{3}+2a^{2}+10a+1\right){x}+3a^{3}-6a^{2}-7a$ |
| 15.1-d4 |
15.1-d |
$4$ |
$4$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{8} \cdot 5 \) |
$15.17601$ |
$(a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$5.938837255$ |
1.569797649 |
\( \frac{579253472299654271509}{32805} a^{3} + \frac{159479380095303625663}{10935} a^{2} - \frac{964972208071904658049}{32805} a - \frac{409952734319362969591}{32805} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 138 a^{3} - 423 a^{2} + 85 a + 21\) , \( 2126 a^{3} - 7244 a^{2} + 2708 a + 2216\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(138a^{3}-423a^{2}+85a+21\right){x}+2126a^{3}-7244a^{2}+2708a+2216$ |
| 16.1-a1 |
16.1-a |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$15.29894$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$813.1163983$ |
3.358265252 |
\( 3072 a^{3} - 11264 a^{2} + 6016 a + 3632 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 2 a^{2} - 3 a + 1\) , \( a^{3} - a^{2} - 4 a\) , \( -7 a^{3} + 18 a^{2} + 22 a - 36\) , \( -45 a^{3} + 111 a^{2} + 138 a - 244\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+1\right){x}^{2}+\left(-7a^{3}+18a^{2}+22a-36\right){x}-45a^{3}+111a^{2}+138a-244$ |
| 16.1-a2 |
16.1-a |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$15.29894$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1626.232796$ |
3.358265252 |
\( -155041788 a^{3} + 557301756 a^{2} - 268140568 a - 193838348 \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 3 a\) , \( -9 a^{3} - 6 a^{2} + 10 a\) , \( 25 a^{3} + 23 a^{2} - 41 a - 20\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-9a^{3}-6a^{2}+10a\right){x}+25a^{3}+23a^{2}-41a-20$ |
| 16.1-b1 |
16.1-b |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$15.29894$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$471.2234828$ |
0.973103882 |
\( -155041788 a^{3} + 557301756 a^{2} - 268140568 a - 193838348 \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - 2 a - 2\) , \( a^{2} - 2\) , \( -11 a^{3} - 5 a^{2} + 16 a + 1\) , \( -66 a^{3} - 55 a^{2} + 108 a + 49\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-11a^{3}-5a^{2}+16a+1\right){x}-66a^{3}-55a^{2}+108a+49$ |
| 16.1-b2 |
16.1-b |
$2$ |
$2$ |
4.4.14656.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$15.29894$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$235.6117414$ |
0.973103882 |
\( 3072 a^{3} - 11264 a^{2} + 6016 a + 3632 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 4 a - 2\) , \( a^{2} - 2\) , \( -9 a^{3} + 21 a^{2} + 32 a - 38\) , \( 42 a^{3} - 94 a^{2} - 130 a + 201\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a-2\right){x}^{2}+\left(-9a^{3}+21a^{2}+32a-38\right){x}+42a^{3}-94a^{2}-130a+201$ |
*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.
