| 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 |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1255.113391$ |
2.484364996 |
\( -\frac{4151552}{27} a^{3} + \frac{8435392}{27} a^{2} + \frac{6494080}{27} a - \frac{372736}{27} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - a^{2} - 6 a + 1\) , \( a^{2} - 2\) , \( 3 a^{3} - 10 a^{2} - 41 a - 12\) , \( -3 a^{3} - 35 a^{2} - 74 a - 27\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+1\right){x}^{2}+\left(3a^{3}-10a^{2}-41a-12\right){x}-3a^{3}-35a^{2}-74a-27$ |
| 3.1-a2 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1255.113391$ |
2.484364996 |
\( \frac{1460648042528}{27} a^{3} - \frac{931716587896}{27} a^{2} - \frac{8169571713064}{27} a + \frac{2289903595912}{27} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( a^{3} - 7 a - 1\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( -33 a^{3} + 77 a^{2} + 25 a - 11\) , \( -292 a^{3} + 645 a^{2} + 324 a - 132\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(-33a^{3}+77a^{2}+25a-11\right){x}-292a^{3}+645a^{2}+324a-132$ |
| 3.1-a3 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$2510.226783$ |
2.484364996 |
\( \frac{78693248}{729} a^{3} - \frac{50155840}{729} a^{2} - \frac{440181376}{729} a + \frac{124901248}{729} \) |
\( \bigl[a^{3} - 5 a\) , \( -2 a^{3} + a^{2} + 10 a - 2\) , \( 1\) , \( -13 a^{3} + 8 a^{2} + 72 a - 19\) , \( 69 a^{3} - 44 a^{2} - 386 a + 108\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+a^{2}+10a-2\right){x}^{2}+\left(-13a^{3}+8a^{2}+72a-19\right){x}+69a^{3}-44a^{2}-386a+108$ |
| 3.1-a4 |
3.1-a |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{12} \) |
$12.94751$ |
$(a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$627.5566957$ |
2.484364996 |
\( \frac{29828142752}{531441} a^{3} + \frac{82452366632}{531441} a^{2} + \frac{20522751224}{531441} a - \frac{11772688328}{531441} \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( a^{3} - 8 a^{2} + 9 a + 15\) , \( a^{3} - 12 a^{2} + 19 a + 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(a^{3}-8a^{2}+9a+15\right){x}+a^{3}-12a^{2}+19a+15$ |
| 3.1-b1 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.226328221$ |
$255.3488237$ |
1.372733573 |
\( -\frac{4151552}{27} a^{3} + \frac{8435392}{27} a^{2} + \frac{6494080}{27} a - \frac{372736}{27} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 6 a + 1\) , \( 1\) , \( -7 a^{3} + 3 a^{2} + 40 a - 3\) , \( -35 a^{3} + 22 a^{2} + 196 a - 53\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+6a+1\right){x}^{2}+\left(-7a^{3}+3a^{2}+40a-3\right){x}-35a^{3}+22a^{2}+196a-53$ |
| 3.1-b2 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.113164110$ |
$1021.395294$ |
1.372733573 |
\( \frac{78693248}{729} a^{3} - \frac{50155840}{729} a^{2} - \frac{440181376}{729} a + \frac{124901248}{729} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 2\) , \( a^{3} - 6 a\) , \( 26 a^{3} + 7 a^{2} - 158 a - 91\) , \( -10 a^{3} - 4 a^{2} + 61 a + 35\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(26a^{3}+7a^{2}-158a-91\right){x}-10a^{3}-4a^{2}+61a+35$ |
| 3.1-b3 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.226328221$ |
$255.3488237$ |
1.372733573 |
\( \frac{1460648042528}{27} a^{3} - \frac{931716587896}{27} a^{2} - \frac{8169571713064}{27} a + \frac{2289903595912}{27} \) |
\( \bigl[-a^{3} + a^{2} + 6 a - 2\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a\) , \( 7 a^{3} - 3 a^{2} - 39 a + 7\) , \( -a^{3} + a^{2} + 8 a\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+6a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(7a^{3}-3a^{2}-39a+7\right){x}-a^{3}+a^{2}+8a$ |
| 3.1-b4 |
3.1-b |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{12} \) |
$12.94751$ |
$(a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.056582055$ |
$1021.395294$ |
1.372733573 |
\( \frac{29828142752}{531441} a^{3} + \frac{82452366632}{531441} a^{2} + \frac{20522751224}{531441} a - \frac{11772688328}{531441} \) |
\( \bigl[a + 1\) , \( -2 a^{3} + a^{2} + 12 a - 2\) , \( a\) , \( 4 a^{3} - 6 a^{2} - 28 a + 10\) , \( -4 a^{3} + 5 a^{2} + 28 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-2a^{3}+a^{2}+12a-2\right){x}^{2}+\left(4a^{3}-6a^{2}-28a+10\right){x}-4a^{3}+5a^{2}+28a-7$ |
| 9.1-a1 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$0.161818153$ |
$1804.345305$ |
2.311741792 |
\( -\frac{438400}{9} a^{3} - \frac{104512}{9} a^{2} + \frac{2660480}{9} a + \frac{1670272}{9} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - 2 a - 3\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( -4 a^{3} - 3 a^{2} + 5 a - 2\) , \( 3 a^{3} + 15 a^{2} + 12 a - 2\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-4a^{3}-3a^{2}+5a-2\right){x}+3a^{3}+15a^{2}+12a-2$ |
| 9.1-a2 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$0.485454460$ |
$601.4484350$ |
2.311741792 |
\( -\frac{36287841664}{729} a^{3} + \frac{80628764864}{729} a^{2} + \frac{39188491136}{729} a - \frac{16137971840}{729} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 7 a - 1\) , \( a^{2} - a - 3\) , \( 49 a^{3} - 63 a^{2} - 144 a - 49\) , \( -181 a^{3} + 533 a^{2} - 57 a - 298\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(49a^{3}-63a^{2}-144a-49\right){x}-181a^{3}+533a^{2}-57a-298$ |
| 9.1-a3 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{7} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.323636306$ |
$451.0863262$ |
2.311741792 |
\( \frac{11187968}{3} a^{3} - \frac{7136320}{3} a^{2} - \frac{62574976}{3} a + \frac{17544448}{3} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 7 a - 1\) , \( 1\) , \( -38 a^{3} + 25 a^{2} + 209 a - 57\) , \( -264 a^{3} + 170 a^{2} + 1472 a - 413\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(-38a^{3}+25a^{2}+209a-57\right){x}-264a^{3}+170a^{2}+1472a-413$ |
| 9.1-a4 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{10} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.080909076$ |
$902.1726525$ |
2.311741792 |
\( \frac{3309323120}{81} a^{3} + \frac{8519772056}{81} a^{2} + \frac{2070998504}{81} a - \frac{1285496552}{81} \) |
\( \bigl[a^{2} - a - 2\) , \( -2 a^{3} + a^{2} + 11 a - 2\) , \( a^{3} - 6 a\) , \( 3 a^{3} - 2 a^{2} - 20 a + 5\) , \( a^{2}\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+11a-2\right){x}^{2}+\left(3a^{3}-2a^{2}-20a+5\right){x}+a^{2}$ |
| 9.1-a5 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{18} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.242727230$ |
$300.7242175$ |
2.311741792 |
\( \frac{256837198071872960}{531441} a^{3} - \frac{163829910240580936}{531441} a^{2} - \frac{1436520261264916744}{531441} a + \frac{402645315341124664}{531441} \) |
\( \bigl[a^{2} - a - 2\) , \( -a\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( -17 a^{3} - 16 a^{2} + 53 a + 33\) , \( 117 a^{3} + 77 a^{2} - 583 a - 377\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}-a{x}^{2}+\left(-17a^{3}-16a^{2}+53a+33\right){x}+117a^{3}+77a^{2}-583a-377$ |
| 9.1-a6 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{9} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.970908920$ |
$300.7242175$ |
2.311741792 |
\( -\frac{1269423627687280448}{27} a^{3} + \frac{2807474833639714648}{27} a^{2} + \frac{1407491635332950488}{27} a - \frac{573980691553088632}{27} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -2 a^{3} + a^{2} + 11 a - 1\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( 32 a^{3} + 4 a^{2} - 191 a - 100\) , \( -118 a^{3} - 114 a^{2} + 460 a + 299\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+11a-1\right){x}^{2}+\left(32a^{3}+4a^{2}-191a-100\right){x}-118a^{3}-114a^{2}+460a+299$ |
| 9.1-a7 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{7} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.323636306$ |
$902.1726525$ |
2.311741792 |
\( -\frac{40153079888}{3} a^{3} - \frac{11056865288}{3} a^{2} + \frac{237872824648}{3} a + \frac{145810516136}{3} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( 2 a^{3} - a^{2} - 10 a\) , \( a^{2} - 2\) , \( 8 a^{3} - 2 a^{2} - 46 a - 5\) , \( -2 a^{3} - 4 a^{2} + 12 a + 25\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{3}-a^{2}-10a\right){x}^{2}+\left(8a^{3}-2a^{2}-46a-5\right){x}-2a^{3}-4a^{2}+12a+25$ |
| 9.1-a8 |
9.1-a |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.970908920$ |
$150.3621087$ |
2.311741792 |
\( \frac{3451351040}{27} a^{3} + \frac{10947869888}{27} a^{2} + \frac{8041774976}{27} a + \frac{1570955776}{27} \) |
\( \bigl[a^{3} - 5 a\) , \( 2 a^{3} - a^{2} - 11 a\) , \( a^{3} - 6 a\) , \( 58 a^{3} + 13 a^{2} - 345 a - 198\) , \( 320 a^{3} + 85 a^{2} - 1900 a - 1158\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(2a^{3}-a^{2}-11a\right){x}^{2}+\left(58a^{3}+13a^{2}-345a-198\right){x}+320a^{3}+85a^{2}-1900a-1158$ |
| 9.1-b1 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.984606476$ |
$40.42071502$ |
2.540567960 |
\( \frac{3451351040}{27} a^{3} + \frac{10947869888}{27} a^{2} + \frac{8041774976}{27} a + \frac{1570955776}{27} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 5 a - 1\) , \( -1001 a^{3} + 638 a^{2} + 5600 a - 1573\) , \( -10275 a^{3} + 6552 a^{2} + 57475 a - 16111\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(-1001a^{3}+638a^{2}+5600a-1573\right){x}-10275a^{3}+6552a^{2}+57475a-16111$ |
| 9.1-b2 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$3.969212953$ |
$80.84143005$ |
2.540567960 |
\( -\frac{36287841664}{729} a^{3} + \frac{80628764864}{729} a^{2} + \frac{39188491136}{729} a - \frac{16137971840}{729} \) |
\( \bigl[a^{2} - 3\) , \( -2 a^{3} + a^{2} + 10 a - 1\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( -2 a^{3} - a^{2} + 6 a - 1\) , \( -2 a^{2} - 7 a - 8\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+10a-1\right){x}^{2}+\left(-2a^{3}-a^{2}+6a-1\right){x}-2a^{2}-7a-8$ |
| 9.1-b3 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.323070984$ |
$2182.718611$ |
2.540567960 |
\( -\frac{438400}{9} a^{3} - \frac{104512}{9} a^{2} + \frac{2660480}{9} a + \frac{1670272}{9} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a\) , \( -24 a^{3} + 15 a^{2} + 133 a - 36\) , \( 42 a^{3} - 27 a^{2} - 235 a + 66\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+1\right){x}^{2}+\left(-24a^{3}+15a^{2}+133a-36\right){x}+42a^{3}-27a^{2}-235a+66$ |
| 9.1-b4 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{10} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.661535492$ |
$1091.359305$ |
2.540567960 |
\( \frac{3309323120}{81} a^{3} + \frac{8519772056}{81} a^{2} + \frac{2070998504}{81} a - \frac{1285496552}{81} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( 25 a^{3} - 49 a^{2} - 42 a + 6\) , \( -129 a^{3} + 261 a^{2} + 193 a - 26\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(25a^{3}-49a^{2}-42a+6\right){x}-129a^{3}+261a^{2}+193a-26$ |
| 9.1-b5 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{9} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$7.938425906$ |
$20.21035751$ |
2.540567960 |
\( -\frac{1269423627687280448}{27} a^{3} + \frac{2807474833639714648}{27} a^{2} + \frac{1407491635332950488}{27} a - \frac{573980691553088632}{27} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 7 a - 1\) , \( 1\) , \( -69 a^{3} + 46 a^{2} + 382 a - 112\) , \( -1020 a^{3} + 645 a^{2} + 5717 a - 1593\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(-69a^{3}+46a^{2}+382a-112\right){x}-1020a^{3}+645a^{2}+5717a-1593$ |
| 9.1-b6 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{7} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.646141968$ |
$545.6796528$ |
2.540567960 |
\( -\frac{40153079888}{3} a^{3} - \frac{11056865288}{3} a^{2} + \frac{237872824648}{3} a + \frac{145810516136}{3} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( 0\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( 7 a^{3} - a^{2} - 32 a + 8\) , \( 125 a^{3} + 281 a^{2} - 24\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(-a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(7a^{3}-a^{2}-32a+8\right){x}+125a^{3}+281a^{2}-24$ |
| 9.1-b7 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{7} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.661535492$ |
$1091.359305$ |
2.540567960 |
\( \frac{11187968}{3} a^{3} - \frac{7136320}{3} a^{2} - \frac{62574976}{3} a + \frac{17544448}{3} \) |
\( \bigl[a^{3} - 5 a\) , \( a + 1\) , \( a^{2} - a - 3\) , \( 4 a^{3} + a^{2} - 21 a - 8\) , \( 2 a^{3} - 11 a - 7\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a^{3}+a^{2}-21a-8\right){x}+2a^{3}-11a-7$ |
| 9.1-b8 |
9.1-b |
$8$ |
$12$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{18} \) |
$14.85342$ |
$(a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.984606476$ |
$40.42071502$ |
2.540567960 |
\( \frac{256837198071872960}{531441} a^{3} - \frac{163829910240580936}{531441} a^{2} - \frac{1436520261264916744}{531441} a + \frac{402645315341124664}{531441} \) |
\( \bigl[a + 1\) , \( 2 a^{3} - a^{2} - 10 a + 2\) , \( 1\) , \( -225 a^{3} + 142 a^{2} + 1265 a - 355\) , \( -3465 a^{3} + 2207 a^{2} + 19387 a - 5434\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(2a^{3}-a^{2}-10a+2\right){x}^{2}+\left(-225a^{3}+142a^{2}+1265a-355\right){x}-3465a^{3}+2207a^{2}+19387a-5434$ |
| 11.2-a1 |
11.2-a |
$1$ |
$1$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
11.2 |
\( 11 \) |
\( 11^{3} \) |
$15.23071$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.060049997$ |
$456.5189372$ |
2.604626564 |
\( \frac{486469}{1331} a^{3} - \frac{356137}{1331} a^{2} - \frac{2631892}{1331} a + \frac{838049}{1331} \) |
\( \bigl[a^{3} - 6 a\) , \( a^{2} - a - 3\) , \( a^{2} - a - 2\) , \( -a + 2\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-a+2\right){x}$ |
| 11.2-b1 |
11.2-b |
$1$ |
$1$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
11.2 |
\( 11 \) |
\( 11^{3} \) |
$15.23071$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.027248650$ |
$1009.135851$ |
2.612572269 |
\( \frac{486469}{1331} a^{3} - \frac{356137}{1331} a^{2} - \frac{2631892}{1331} a + \frac{838049}{1331} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 2\) , \( -2 a^{3} + a^{2} + 10 a - 1\) , \( a^{2} - a - 2\) , \( 2 a^{3} - 5 a^{2} + 1\) , \( -14 a^{3} + 31 a^{2} + 15 a - 7\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+10a-1\right){x}^{2}+\left(2a^{3}-5a^{2}+1\right){x}-14a^{3}+31a^{2}+15a-7$ |
| 12.1-a1 |
12.1-a |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$661.8983126$ |
2.620316235 |
\( -\frac{182272}{9} a^{3} + \frac{413696}{9} a^{2} + \frac{167648}{9} a - \frac{63296}{9} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 2\) , \( -5 a^{3} + 2 a^{2} + 30 a\) , \( -5 a^{3} + 6 a^{2} + 39 a - 3\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-5a^{3}+2a^{2}+30a\right){x}-5a^{3}+6a^{2}+39a-3$ |
| 12.1-a2 |
12.1-a |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1323.796625$ |
2.620316235 |
\( -\frac{128}{3} a^{3} + \frac{3328}{3} a^{2} + \frac{7936}{3} a + \frac{5120}{3} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 1\) , \( a^{3} - 7 a\) , \( a^{2} - 3\) , \( 2 a^{3} - 16 a + 4\) , \( 2 a^{3} - 16 a + 4\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-7a\right){x}^{2}+\left(2a^{3}-16a+4\right){x}+2a^{3}-16a+4$ |
| 12.1-b1 |
12.1-b |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
|
\( 3 \) |
$1$ |
$142.1819311$ |
2.789943808 |
\( -\frac{5419648}{3} a^{3} - \frac{1911040}{3} a^{2} + \frac{32401664}{3} a + \frac{21703936}{3} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 1\) , \( -a^{2} + 2\) , \( a^{2} - 3\) , \( -8 a^{3} + 60 a - 15\) , \( -41 a^{3} + 11 a^{2} + 271 a - 77\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-8a^{3}+60a-15\right){x}-41a^{3}+11a^{2}+271a-77$ |
| 12.1-b2 |
12.1-b |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
|
\( 2 \cdot 3 \) |
$1$ |
$142.1819311$ |
2.789943808 |
\( -\frac{24573103360}{9} a^{3} + \frac{54346232384}{9} a^{2} + \frac{27245744480}{9} a - \frac{11110951616}{9} \) |
\( \bigl[a^{3} - 5 a\) , \( -2 a^{3} + a^{2} + 10 a - 2\) , \( -a^{3} + a^{2} + 6 a - 2\) , \( -68 a^{3} + 43 a^{2} + 379 a - 106\) , \( 28 a^{3} - 18 a^{2} - 157 a + 44\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(-a^{3}+a^{2}+6a-2\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+10a-2\right){x}^{2}+\left(-68a^{3}+43a^{2}+379a-106\right){x}+28a^{3}-18a^{2}-157a+44$ |
| 12.1-c1 |
12.1-c |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1174.475780$ |
2.324751322 |
\( -\frac{5419648}{3} a^{3} - \frac{1911040}{3} a^{2} + \frac{32401664}{3} a + \frac{21703936}{3} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 1\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( 4 a^{3} - 4 a^{2} - 22 a - 1\) , \( 9 a^{3} + 3 a^{2} - 51 a - 22\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(4a^{3}-4a^{2}-22a-1\right){x}+9a^{3}+3a^{2}-51a-22$ |
| 12.1-c2 |
12.1-c |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$587.2378902$ |
2.324751322 |
\( -\frac{24573103360}{9} a^{3} + \frac{54346232384}{9} a^{2} + \frac{27245744480}{9} a - \frac{11110951616}{9} \) |
\( \bigl[a^{3} - 5 a\) , \( -a^{3} + a^{2} + 6 a - 3\) , \( a^{2} - a - 2\) , \( 27 a^{3} + 63 a^{2} + 3 a - 10\) , \( 277 a^{3} + 712 a^{2} + 173 a - 110\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-3\right){x}^{2}+\left(27a^{3}+63a^{2}+3a-10\right){x}+277a^{3}+712a^{2}+173a-110$ |
| 12.1-d1 |
12.1-d |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$0.061212405$ |
$1805.219984$ |
2.624721727 |
\( -\frac{128}{3} a^{3} + \frac{3328}{3} a^{2} + \frac{7936}{3} a + \frac{5120}{3} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 1\) , \( -a^{3} + 7 a + 1\) , \( a^{3} - 5 a\) , \( 34 a^{3} - 75 a^{2} - 40 a + 19\) , \( -35 a^{3} + 76 a^{2} + 41 a - 16\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{3}+7a+1\right){x}^{2}+\left(34a^{3}-75a^{2}-40a+19\right){x}-35a^{3}+76a^{2}+41a-16$ |
| 12.1-d2 |
12.1-d |
$2$ |
$2$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.030606202$ |
$1805.219984$ |
2.624721727 |
\( -\frac{182272}{9} a^{3} + \frac{413696}{9} a^{2} + \frac{167648}{9} a - \frac{63296}{9} \) |
\( \bigl[a^{3} - 5 a\) , \( a^{3} - 5 a - 1\) , \( a + 1\) , \( a^{3} - 3 a + 3\) , \( a^{3} + a^{2} - 3 a\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(a^{3}-3a+3\right){x}+a^{3}+a^{2}-3a$ |
| 12.1-e1 |
12.1-e |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3^{8} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.427708668$ |
$121.2698972$ |
2.464022915 |
\( \frac{2184122113854284}{6561} a^{3} + \frac{5622172866174788}{6561} a^{2} + \frac{1367364650535164}{6561} a - \frac{848495668384328}{6561} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - a - 2\) , \( -3 a^{3} - 4 a^{2} + 23 a + 18\) , \( -17 a^{3} - 7 a^{2} + 105 a + 65\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-3a^{3}-4a^{2}+23a+18\right){x}-17a^{3}-7a^{2}+105a+65$ |
| 12.1-e2 |
12.1-e |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.106927167$ |
$485.0795891$ |
2.464022915 |
\( -\frac{4662190228012}{9} a^{3} + \frac{10311772519196}{9} a^{2} + \frac{5166967210148}{9} a - \frac{2107475944232}{9} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - 6 a - 1\) , \( 31 a^{3} - 38 a^{2} - 92 a - 39\) , \( -70 a^{3} + 251 a^{2} - 109 a - 189\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(31a^{3}-38a^{2}-92a-39\right){x}-70a^{3}+251a^{2}-109a-189$ |
| 12.1-e3 |
12.1-e |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{4} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.213854334$ |
$970.1591782$ |
2.464022915 |
\( -\frac{5392192}{81} a^{3} + \frac{49830176}{81} a^{2} + \frac{18667328}{81} a - \frac{8391248}{81} \) |
\( \bigl[a^{3} - 5 a\) , \( -2 a^{3} + a^{2} + 11 a\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( -20 a^{3} + 12 a^{2} + 112 a - 27\) , \( 104 a^{3} - 68 a^{2} - 583 a + 165\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(-a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+11a\right){x}^{2}+\left(-20a^{3}+12a^{2}+112a-27\right){x}+104a^{3}-68a^{2}-583a+165$ |
| 12.1-e4 |
12.1-e |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.106927167$ |
$1940.318356$ |
2.464022915 |
\( -\frac{32448512}{9} a^{3} - \frac{8935424}{9} a^{2} + \frac{192237568}{9} a + \frac{117837824}{9} \) |
\( \bigl[0\) , \( 2 a^{3} - a^{2} - 11 a\) , \( a^{2} - a - 2\) , \( -7 a^{2} - 13 a + 7\) , \( -a^{3} - 11 a^{2} - 15 a + 3\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){y}={x}^{3}+\left(2a^{3}-a^{2}-11a\right){x}^{2}+\left(-7a^{2}-13a+7\right){x}-a^{3}-11a^{2}-15a+3$ |
| 12.1-f1 |
12.1-f |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$33.02840503$ |
2.092040165 |
\( -\frac{4662190228012}{9} a^{3} + \frac{10311772519196}{9} a^{2} + \frac{5166967210148}{9} a - \frac{2107475944232}{9} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{2} + a + 4\) , \( a^{2} - 3\) , \( -3 a^{3} - 2 a^{2} + 11 a + 1\) , \( -13 a^{3} - 2 a^{2} + 51 a - 19\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-3a^{3}-2a^{2}+11a+1\right){x}-13a^{3}-2a^{2}+51a-19$ |
| 12.1-f2 |
12.1-f |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3^{8} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$132.1136201$ |
2.092040165 |
\( \frac{2184122113854284}{6561} a^{3} + \frac{5622172866174788}{6561} a^{2} + \frac{1367364650535164}{6561} a - \frac{848495668384328}{6561} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 5 a\) , \( -20 a^{3} - 46 a^{2} + 2 a + 15\) , \( -173 a^{3} - 439 a^{2} - 91 a + 74\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-20a^{3}-46a^{2}+2a+15\right){x}-173a^{3}-439a^{2}-91a+74$ |
| 12.1-f3 |
12.1-f |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{4} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$264.2272403$ |
2.092040165 |
\( -\frac{5392192}{81} a^{3} + \frac{49830176}{81} a^{2} + \frac{18667328}{81} a - \frac{8391248}{81} \) |
\( \bigl[a^{3} - 5 a\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( a^{3} - 5 a\) , \( 8 a^{3} - 22 a^{2} + 4 a - 2\) , \( 69 a^{3} - 156 a^{2} - 67 a + 29\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-2\right){x}^{2}+\left(8a^{3}-22a^{2}+4a-2\right){x}+69a^{3}-156a^{2}-67a+29$ |
| 12.1-f4 |
12.1-f |
$4$ |
$4$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$15.39727$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$528.4544806$ |
2.092040165 |
\( -\frac{32448512}{9} a^{3} - \frac{8935424}{9} a^{2} + \frac{192237568}{9} a + \frac{117837824}{9} \) |
\( \bigl[0\) , \( -a^{2} + 2\) , \( a^{2} - a - 2\) , \( 2 a^{3} - 2 a^{2} - 5 a - 1\) , \( 3 a^{3} - 7 a^{2} - 4 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(2a^{3}-2a^{2}-5a-1\right){x}+3a^{3}-7a^{2}-4a+1$ |
| 18.1-a1 |
18.1-a |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{7} \cdot 3^{3} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 7 \) |
$0.010343952$ |
$1164.329885$ |
5.340021755 |
\( -20770 a^{3} - \frac{254045}{4} a^{2} - \frac{110489}{4} a + 2324 \) |
\( \bigl[-a^{3} + a^{2} + 6 a - 1\) , \( a + 1\) , \( a^{3} - 6 a - 1\) , \( 35 a^{3} + 6 a^{2} - 209 a - 114\) , \( -163 a^{3} - 45 a^{2} + 967 a + 595\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+6a-1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(35a^{3}+6a^{2}-209a-114\right){x}-163a^{3}-45a^{2}+967a+595$ |
| 18.1-a2 |
18.1-a |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{21} \cdot 3^{9} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \cdot 7 \) |
$0.031031858$ |
$129.3699872$ |
5.340021755 |
\( \frac{43167119510275}{64} a^{3} - \frac{95468997957743}{64} a^{2} - \frac{47862162173503}{64} a + \frac{19518380378309}{64} \) |
\( \bigl[a^{3} - 6 a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a + 1\) , \( 115 a^{3} - 71 a^{2} - 643 a + 184\) , \( 1804 a^{3} - 1140 a^{2} - 10073 a + 2822\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(115a^{3}-71a^{2}-643a+184\right){x}+1804a^{3}-1140a^{2}-10073a+2822$ |
| 18.1-b1 |
18.1-b |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{9} \cdot 3^{11} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$162.7769454$ |
2.577598793 |
\( \frac{10385}{1944} a^{3} + \frac{206141}{1944} a^{2} - \frac{655615}{1944} a + \frac{162457}{1944} \) |
\( \bigl[-a^{3} + a^{2} + 6 a - 1\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( a^{2} - 2\) , \( -8 a^{3} + 18 a^{2} + 13 a - 4\) , \( -25 a^{3} + 57 a^{2} + 29 a - 12\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+6a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-1\right){x}^{2}+\left(-8a^{3}+18a^{2}+13a-4\right){x}-25a^{3}+57a^{2}+29a-12$ |
| 18.1-b2 |
18.1-b |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{3} \cdot 3^{21} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \cdot 3 \) |
$1$ |
$6.028775757$ |
2.577598793 |
\( -\frac{6009749595601}{28697814} a^{3} - \frac{7184763433423}{28697814} a^{2} + \frac{6156345216232}{14348907} a - \frac{1360131836029}{14348907} \) |
\( \bigl[a\) , \( -2 a^{3} + a^{2} + 10 a - 1\) , \( a^{2} - 3\) , \( a^{3} - 6 a^{2} - 2 a + 14\) , \( 115 a^{3} + 16 a^{2} - 651 a - 403\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+10a-1\right){x}^{2}+\left(a^{3}-6a^{2}-2a+14\right){x}+115a^{3}+16a^{2}-651a-403$ |
| 18.1-c1 |
18.1-c |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{3} \cdot 3^{21} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.283713926$ |
$64.68439691$ |
2.324837487 |
\( -\frac{6009749595601}{28697814} a^{3} - \frac{7184763433423}{28697814} a^{2} + \frac{6156345216232}{14348907} a - \frac{1360131836029}{14348907} \) |
\( \bigl[a^{3} - 6 a\) , \( -a\) , \( -a^{3} + a^{2} + 6 a - 2\) , \( -42 a^{3} + 16 a^{2} + 265 a - 73\) , \( 212 a^{3} - 87 a^{2} - 1322 a + 361\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(-a^{3}+a^{2}+6a-2\right){y}={x}^{3}-a{x}^{2}+\left(-42a^{3}+16a^{2}+265a-73\right){x}+212a^{3}-87a^{2}-1322a+361$ |
| 18.1-c2 |
18.1-c |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{9} \cdot 3^{11} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.094571308$ |
$194.0531907$ |
2.324837487 |
\( \frac{10385}{1944} a^{3} + \frac{206141}{1944} a^{2} - \frac{655615}{1944} a + \frac{162457}{1944} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 9 a + 3\) , \( -a^{3} + a^{2} + 2 a - 2\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(a^{3}-9a+3\right){x}-a^{3}+a^{2}+2a-2$ |
| 18.1-d1 |
18.1-d |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{21} \cdot 3^{9} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$9.995064472$ |
1.424460905 |
\( \frac{43167119510275}{64} a^{3} - \frac{95468997957743}{64} a^{2} - \frac{47862162173503}{64} a + \frac{19518380378309}{64} \) |
\( \bigl[a^{2} - a - 3\) , \( 0\) , \( -a^{3} + a^{2} + 6 a - 1\) , \( 8 a^{3} - 18 a^{2} - 30 a + 58\) , \( -80 a^{3} + 26 a^{2} + 502 a - 104\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(-a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(8a^{3}-18a^{2}-30a+58\right){x}-80a^{3}+26a^{2}+502a-104$ |
| 18.1-d2 |
18.1-d |
$2$ |
$3$ |
4.4.15952.1 |
$4$ |
$[4, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{7} \cdot 3^{3} \) |
$16.19777$ |
$(a+1), (a^2+2a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$809.6002222$ |
1.424460905 |
\( -20770 a^{3} - \frac{254045}{4} a^{2} - \frac{110489}{4} a + 2324 \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 2\) , \( a^{3} - 5 a - 2\) , \( a\) , \( 2 a^{3} - 3 a^{2} - 8 a + 6\) , \( 3 a^{3} - 4 a^{2} - 11 a + 2\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(2a^{3}-3a^{2}-8a+6\right){x}+3a^{3}-4a^{2}-11a+2$ |
*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.
