| 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 |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$14.28907$ |
$(a^3-a^2-6a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$233.0832829$ |
1.672191079 |
\( -\frac{46448}{27} a^{3} - \frac{6578}{9} a^{2} + \frac{120947}{27} a - \frac{52073}{27} \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{3} + a^{2} + 4 a + 3\) , \( 0\) , \( -2 a^{3} - 2 a^{2} + 20 a + 24\) , \( 2 a^{3} - 15 a^{2} + 14 a + 41\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+4a+3\right){x}^{2}+\left(-2a^{3}-2a^{2}+20a+24\right){x}+2a^{3}-15a^{2}+14a+41$ |
| 3.1-b1 |
3.1-b |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
3.1 |
\( 3 \) |
\( 3^{13} \) |
$14.28907$ |
$(a^3-a^2-6a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$132.9092858$ |
0.953520644 |
\( -\frac{554873342}{1594323} a^{3} + \frac{311661553}{531441} a^{2} + \frac{2398493765}{1594323} a + \frac{609744973}{1594323} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 5 a - 2\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -8 a^{3} + 2 a^{2} + 58 a + 52\) , \( 13 a^{3} - 3 a^{2} - 94 a - 86\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-2\right){x}^{2}+\left(-8a^{3}+2a^{2}+58a+52\right){x}+13a^{3}-3a^{2}-94a-86$ |
| 7.1-a1 |
7.1-a |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( 7^{18} \) |
$15.88551$ |
$(-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$36$ |
\( 2 \) |
$1$ |
$19.78378823$ |
2.554798989 |
\( \frac{100165921885866154644443}{1628413597910449} a^{3} - \frac{218751517839776350343023}{1628413597910449} a^{2} - \frac{442356410169189336781644}{1628413597910449} a + \frac{423653670518817494908720}{1628413597910449} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( 0\) , \( 4 a^{3} + 2 a^{2} - 46 a - 63\) , \( -42 a^{3} - 3 a^{2} + 207 a + 150\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(4a^{3}+2a^{2}-46a-63\right){x}-42a^{3}-3a^{2}+207a+150$ |
| 7.1-a2 |
7.1-a |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{9} \) |
$15.88551$ |
$(-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 1 \) |
$1$ |
$158.2703058$ |
2.554798989 |
\( \frac{6289411669802935}{40353607} a^{3} + \frac{13573529834312165}{40353607} a^{2} - \frac{1161695785537324}{40353607} a - \frac{9957925782743205}{40353607} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( 0\) , \( -a^{3} - 3 a^{2} - a + 2\) , \( -12 a^{3} - 22 a^{2} + 9 a + 21\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(-a^{3}-3a^{2}-a+2\right){x}-12a^{3}-22a^{2}+9a+21$ |
| 7.1-a3 |
7.1-a |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( 7^{6} \) |
$15.88551$ |
$(-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$178.0540941$ |
2.554798989 |
\( -\frac{115032152675294839569}{117649} a^{3} + \frac{26705737851356832029}{117649} a^{2} + \frac{825730831896751210084}{117649} a + \frac{749062224223013997305}{117649} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( a^{3} - 3 a^{2} - 3 a + 6\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 2 a^{3} + 2 a^{2} - 2 a + 1\) , \( a^{3} - 28 a^{2} - 10 a + 24\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{3}-2a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+6\right){x}^{2}+\left(2a^{3}+2a^{2}-2a+1\right){x}+a^{3}-28a^{2}-10a+24$ |
| 7.1-a4 |
7.1-a |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{3} \) |
$15.88551$ |
$(-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$1424.432752$ |
2.554798989 |
\( \frac{3144883617}{343} a^{3} - \frac{748332541}{343} a^{2} - \frac{22553629876}{343} a - \frac{20362895270}{343} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( a^{3} - 3 a^{2} - 3 a + 6\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 2 a^{3} - 8 a^{2} - 7 a + 11\) , \( a^{3} - 5 a^{2} - 4 a + 6\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{3}-2a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+6\right){x}^{2}+\left(2a^{3}-8a^{2}-7a+11\right){x}+a^{3}-5a^{2}-4a+6$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{19} \) |
$16.39246$ |
$(a^3-a^2-6a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.395646413$ |
$86.81605376$ |
3.942775354 |
\( -\frac{554873342}{1594323} a^{3} + \frac{311661553}{531441} a^{2} + \frac{2398493765}{1594323} a + \frac{609744973}{1594323} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a\) , \( a\) , \( -4 a^{3} + 5 a^{2} + 20 a + 8\) , \( -59 a^{3} + 126 a^{2} + 261 a - 228\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-4a^{3}+5a^{2}+20a+8\right){x}-59a^{3}+126a^{2}+261a-228$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{9} \) |
$16.39246$ |
$(a^3-a^2-6a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.075320738$ |
$350.0394464$ |
3.026399907 |
\( -\frac{46448}{27} a^{3} - \frac{6578}{9} a^{2} + \frac{120947}{27} a - \frac{52073}{27} \) |
\( \bigl[a + 1\) , \( a^{2} - 5\) , \( a^{2} - 3\) , \( a^{3} - 3 a^{2} + 10\) , \( -a^{3} + 5 a^{2} - 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(a^{3}-3a^{2}+10\right){x}-a^{3}+5a^{2}-11$ |
| 13.2-a1 |
13.2-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
13.2 |
\( 13 \) |
\( 13^{9} \) |
$17.16353$ |
$(a^3-2a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3^{2} \) |
$0.129965314$ |
$134.7871138$ |
4.524317908 |
\( \frac{10675275784047}{10604499373} a^{3} - \frac{24693223163008}{10604499373} a^{2} - \frac{2658013593557}{815730721} a + \frac{28117159034480}{10604499373} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} - a - 4\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 8 a^{3} - 15 a^{2} - 27 a + 18\) , \( -3 a^{3} + 17 a^{2} + 4 a - 18\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(8a^{3}-15a^{2}-27a+18\right){x}-3a^{3}+17a^{2}+4a-18$ |
| 13.2-a2 |
13.2-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
13.2 |
\( 13 \) |
\( 13^{3} \) |
$17.16353$ |
$(a^3-2a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.043321771$ |
$1213.084024$ |
4.524317908 |
\( -\frac{17957475}{2197} a^{3} + \frac{16316600}{2197} a^{2} + \frac{4325517}{169} a - \frac{43390305}{2197} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( a\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 3 a\) , \( a^{3} - 2 a^{2} - 2 a + 5\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+a{x}^{2}+3a{x}+a^{3}-2a^{2}-2a+5$ |
| 15.1-a1 |
15.1-a |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( 3^{5} \cdot 5^{8} \) |
$17.47331$ |
$(a^3-a^2-6a-3), (a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.029011974$ |
$557.7177430$ |
3.714638079 |
\( \frac{781724277705181}{94921875} a^{3} - \frac{555508336894817}{31640625} a^{2} - \frac{2678638810564222}{94921875} a + \frac{458545814178934}{94921875} \) |
\( \bigl[a^{2} - 4\) , \( -a^{3} + a^{2} + 6 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( 6 a^{3} - 20 a^{2} - 9 a + 46\) , \( -12 a^{3} + 27 a^{2} + 43 a - 13\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+2\right){x}^{2}+\left(6a^{3}-20a^{2}-9a+46\right){x}-12a^{3}+27a^{2}+43a-13$ |
| 15.1-b1 |
15.1-b |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{2} \cdot 5^{18} \) |
$17.47331$ |
$(a^3-a^2-6a-3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$3.381673946$ |
$4.360205132$ |
3.808166515 |
\( -\frac{1300291556615138823016269401}{34332275390625} a^{3} + \frac{100624798074642126012745357}{11444091796875} a^{2} + \frac{9333832160711289804443643362}{34332275390625} a + \frac{8467190911724378545014419611}{34332275390625} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{2} - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 309 a^{3} - 72 a^{2} - 2185 a - 1987\) , \( 7113 a^{3} - 1652 a^{2} - 50990 a - 46248\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(309a^{3}-72a^{2}-2185a-1987\right){x}+7113a^{3}-1652a^{2}-50990a-46248$ |
| 15.1-b2 |
15.1-b |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3 \cdot 5^{9} \) |
$17.47331$ |
$(a^3-a^2-6a-3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$6.763347893$ |
$8.720410265$ |
3.808166515 |
\( \frac{119137389126578221309}{5859375} a^{3} - \frac{86698515630905988863}{1953125} a^{2} - \frac{526227987388143191908}{5859375} a + \frac{503472535833511060426}{5859375} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{2} - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 19 a^{3} + 8 a^{2} - 125 a - 142\) , \( 122 a^{3} - 8 a^{2} - 824 a - 774\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(19a^{3}+8a^{2}-125a-142\right){x}+122a^{3}-8a^{2}-824a-774$ |
| 15.1-b3 |
15.1-b |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{6} \cdot 5^{6} \) |
$17.47331$ |
$(a^3-a^2-6a-3), (a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.127224648$ |
$353.1766157$ |
3.808166515 |
\( \frac{18147641422639}{11390625} a^{3} + \frac{19952862625327}{3796875} a^{2} + \frac{5692827679382}{11390625} a - \frac{47001016250204}{11390625} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{2} - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 9 a^{3} - 2 a^{2} - 35 a - 22\) , \( 16 a^{3} + 14 a^{2} - 76 a - 84\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(9a^{3}-2a^{2}-35a-22\right){x}+16a^{3}+14a^{2}-76a-84$ |
| 15.1-b4 |
15.1-b |
$4$ |
$6$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
15.1 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5^{3} \) |
$17.47331$ |
$(a^3-a^2-6a-3), (a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$2.254449297$ |
$706.3532314$ |
3.808166515 |
\( \frac{33036079}{3375} a^{3} - \frac{25019753}{1125} a^{2} - \frac{147490648}{3375} a + \frac{142543831}{3375} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{2} - 4\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( 4 a^{3} + 3 a^{2} - 5 a - 2\) , \( 8 a^{3} + 14 a^{2} - 5 a - 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(4a^{3}+3a^{2}-5a-2\right){x}+8a^{3}+14a^{2}-5a-13$ |
| 16.1-a1 |
16.1-a |
$2$ |
$5$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{12} \) |
$17.61484$ |
$(2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 3 \) |
$0.191527155$ |
$1950.891851$ |
1.286706712 |
\( \frac{723080241}{8} a^{3} - \frac{167974687}{8} a^{2} - \frac{2595127967}{4} a - \frac{4708126031}{8} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 3 a^{2} + 22 a - 3\) , \( 3 a^{3} + 4 a^{2} - 4 a + 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-1\right){x}^{2}+\left(-a^{3}+3a^{2}+22a-3\right){x}+3a^{3}+4a^{2}-4a+13$ |
| 16.1-a2 |
16.1-a |
$2$ |
$5$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{60} \) |
$17.61484$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 3 \cdot 5 \) |
$0.957635777$ |
$3.121426962$ |
1.286706712 |
\( \frac{48472489846715269}{32768} a^{3} - \frac{105823032092927769}{32768} a^{2} - \frac{107051109445795807}{16384} a + \frac{51210986423466231}{8192} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( a^{2} - a - 4\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( -83 a^{3} - 136 a^{2} + 41 a + 89\) , \( -4281 a^{3} - 9573 a^{2} + 657 a + 7075\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+\left(a^{3}-2a^{2}-4a+2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-83a^{3}-136a^{2}+41a+89\right){x}-4281a^{3}-9573a^{2}+657a+7075$ |
| 17.1-a1 |
17.1-a |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{7} \) |
$17.74884$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$216.1624678$ |
1.550797405 |
\( -\frac{7537733716222}{410338673} a^{3} - \frac{17713052806839}{410338673} a^{2} + \frac{342617095246}{410338673} a + \frac{13421317293292}{410338673} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 4\) , \( 34 a^{3} - 90 a^{2} - 77 a + 93\) , \( 309 a^{3} - 844 a^{2} - 685 a + 882\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(34a^{3}-90a^{2}-77a+93\right){x}+309a^{3}-844a^{2}-685a+882$ |
| 19.1-a1 |
19.1-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{3} \) |
$17.99733$ |
$(a^3-2a^2-3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$5.213218045$ |
$9.445456551$ |
4.239209370 |
\( \frac{7129476305597641}{6859} a^{3} + \frac{15385219754928597}{6859} a^{2} - \frac{1318640904638952}{6859} a - \frac{11289087776040390}{6859} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( a^{3} - a^{2} - 5 a - 1\) , \( a^{2} - 3\) , \( 5 a^{3} - 19 a^{2} - 50 a - 3\) , \( -12 a^{3} - 83 a^{2} - 130 a - 38\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-1\right){x}^{2}+\left(5a^{3}-19a^{2}-50a-3\right){x}-12a^{3}-83a^{2}-130a-38$ |
| 19.1-a2 |
19.1-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( -19 \) |
$17.99733$ |
$(a^3-2a^2-3a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.737739348$ |
$765.0819806$ |
4.239209370 |
\( -\frac{68215}{19} a^{3} + \frac{212859}{19} a^{2} + \frac{356594}{19} a - \frac{373245}{19} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( a^{3} - a^{2} - 5 a - 1\) , \( a^{2} - 3\) , \( a^{2} - 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a-1\right){x}^{2}+\left(a^{2}-3\right){x}$ |
| 21.1-a1 |
21.1-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3 \cdot 7 \) |
$18.22389$ |
$(a^3-a^2-6a-3), (-a^3+2a^2+4a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$236.1106487$ |
1.693910072 |
\( -\frac{1114978}{21} a^{3} - \frac{4826211}{7} a^{2} + \frac{37229539}{21} a + \frac{61945592}{21} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 2\) , \( -a^{2} + a + 5\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -3 a^{2} + 10 a + 18\) , \( a^{3} - 3 a^{2} + 6 a + 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-2\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(-3a^{2}+10a+18\right){x}+a^{3}-3a^{2}+6a+13$ |
| 21.1-a2 |
21.1-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{3} \cdot 7^{3} \) |
$18.22389$ |
$(a^3-a^2-6a-3), (-a^3+2a^2+4a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$26.23451652$ |
1.693910072 |
\( -\frac{33150705155085730}{9261} a^{3} + \frac{24124392662472635}{3087} a^{2} + \frac{146426138916738739}{9261} a - \frac{140094335282833273}{9261} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{2} - a - 5\) , \( 1\) , \( -2 a^{3} + 13 a^{2} + 15 a - 16\) , \( 4 a^{3} - 6 a^{2} - 2 a + 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-2a^{3}+13a^{2}+15a-16\right){x}+4a^{3}-6a^{2}-2a+3$ |
| 21.1-b1 |
21.1-b |
$2$ |
$2$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7 \) |
$18.22389$ |
$(a^3-a^2-6a-3), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$583.9857413$ |
4.189643010 |
\( -\frac{1153393}{567} a^{3} + \frac{1706339}{189} a^{2} + \frac{1831483}{567} a - \frac{3901039}{567} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - 4\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( a^{3} + 3 a^{2} + 9 a + 12\) , \( 95 a^{3} - 11 a^{2} - 654 a - 604\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-2a^{2}-4a+3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(a^{3}+3a^{2}+9a+12\right){x}+95a^{3}-11a^{2}-654a-604$ |
| 21.1-b2 |
21.1-b |
$2$ |
$2$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{8} \cdot 7^{2} \) |
$18.22389$ |
$(a^3-a^2-6a-3), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$145.9964353$ |
4.189643010 |
\( -\frac{27627823091618}{321489} a^{3} + \frac{25516086347917}{107163} a^{2} + \frac{58300054839860}{321489} a - \frac{77377999088708}{321489} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( a^{3} - 3 a^{2} - 4 a + 6\) , \( a^{3} - 2 a^{2} - 3 a + 2\) , \( -3 a^{3} + 9 a^{2} + 16 a - 18\) , \( -15 a^{3} - 44 a^{2} - 8 a + 36\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{3}-2a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-3a^{2}-4a+6\right){x}^{2}+\left(-3a^{3}+9a^{2}+16a-18\right){x}-15a^{3}-44a^{2}-8a+36$ |
| 21.1-c1 |
21.1-c |
$2$ |
$2$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{7} \) |
$18.22389$ |
$(a^3-a^2-6a-3), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$349.4370096$ |
1.253469238 |
\( \frac{5281441846206265133}{7411887} a^{3} - \frac{3843404696649657658}{2470629} a^{2} - \frac{23328045236613662015}{7411887} a + \frac{22319286690602679812}{7411887} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 18 a^{3} - 51 a^{2} - 37 a + 52\) , \( -64 a^{3} + 177 a^{2} + 139 a - 192\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{3}-2a^{2}-4a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-1\right){x}^{2}+\left(18a^{3}-51a^{2}-37a+52\right){x}-64a^{3}+177a^{2}+139a-192$ |
| 21.1-c2 |
21.1-c |
$2$ |
$2$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{14} \) |
$18.22389$ |
$(a^3-a^2-6a-3), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$174.7185048$ |
1.253469238 |
\( -\frac{188378777270571433586336}{54936068900769} a^{3} + \frac{172221277899016496186863}{18312022966923} a^{2} + \frac{418263087591758146642865}{54936068900769} a - \frac{540489773325086265685493}{54936068900769} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 323 a^{3} - 876 a^{2} - 747 a + 872\) , \( -5328 a^{3} + 14692 a^{2} + 11700 a - 15551\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{3}-2a^{2}-4a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-1\right){x}^{2}+\left(323a^{3}-876a^{2}-747a+872\right){x}-5328a^{3}+14692a^{2}+11700a-15551$ |
| 27.2-a1 |
27.2-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{3} \) |
$18.80547$ |
$(a^3-a^2-6a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$0.636124946$ |
$177.7063204$ |
3.243994198 |
\( -536 a^{3} - 18557 a^{2} - 7100 a + 18417 \) |
\( \bigl[a\) , \( a^{3} - 3 a^{2} - 2 a + 6\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 56 a^{3} - 123 a^{2} - 248 a + 243\) , \( 56 a^{3} - 122 a^{2} - 248 a + 232\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{3}-3a^{2}-2a+6\right){x}^{2}+\left(56a^{3}-123a^{2}-248a+243\right){x}+56a^{3}-122a^{2}-248a+232$ |
| 27.2-a2 |
27.2-a |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{5} \) |
$18.80547$ |
$(a^3-a^2-6a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$9$ |
\( 1 \) |
$1.908374838$ |
$6.581715571$ |
3.243994198 |
\( -173677418750950 a^{3} - 374807260039148 a^{2} + 32075839859846 a + 274974489285247 \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 2\) , \( -a + 1\) , \( a\) , \( 377 a^{3} - 83 a^{2} - 2714 a - 2476\) , \( 9849 a^{3} - 2274 a^{2} - 70718 a - 64201\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+2\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(377a^{3}-83a^{2}-2714a-2476\right){x}+9849a^{3}-2274a^{2}-70718a-64201$ |
| 27.2-b1 |
27.2-b |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{11} \) |
$18.80547$ |
$(a^3-a^2-6a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$4.981949922$ |
$4.057454085$ |
5.220718159 |
\( -173677418750950 a^{3} - 374807260039148 a^{2} + 32075839859846 a + 274974489285247 \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - a^{2} - 5 a - 1\) , \( -31 a^{3} + 17 a^{2} + 39 a - 37\) , \( -291 a^{3} - 422 a^{2} + 142 a + 265\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-31a^{3}+17a^{2}+39a-37\right){x}-291a^{3}-422a^{2}+142a+265$ |
| 27.2-b2 |
27.2-b |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$18.80547$ |
$(a^3-a^2-6a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.660649974$ |
$328.6537809$ |
5.220718159 |
\( -536 a^{3} - 18557 a^{2} - 7100 a + 18417 \) |
\( \bigl[1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( 4 a^{3} - 5 a^{2} - 24 a - 3\) , \( -a^{3} + 4 a^{2} + a - 18\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-2a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(4a^{3}-5a^{2}-24a-3\right){x}-a^{3}+4a^{2}+a-18$ |
| 31.1-a1 |
31.1-a |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( 31 \) |
$19.13304$ |
$(-a^3+3a^2+3a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$200.9572608$ |
1.441711883 |
\( \frac{11868633481}{31} a^{3} - \frac{2755411463}{31} a^{2} - \frac{85196157894}{31} a - \frac{77285681812}{31} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 1\) , \( 1\) , \( 15 a^{3} - 42 a^{2} - 32 a + 47\) , \( -31 a^{3} + 85 a^{2} + 69 a - 89\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a^{2}-5a+1\right){x}^{2}+\left(15a^{3}-42a^{2}-32a+47\right){x}-31a^{3}+85a^{2}+69a-89$ |
| 31.2-a1 |
31.2-a |
$2$ |
$5$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$1.266605636$ |
$1405.057457$ |
2.042820744 |
\( \frac{451835}{31} a^{3} - \frac{1226337}{31} a^{2} - \frac{1036600}{31} a + \frac{1325868}{31} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( a^{3} - 3 a^{2} - 4 a + 5\) , \( 1\) , \( -4 a^{3} - a^{2} + 31 a + 38\) , \( -25 a^{3} + 4 a^{2} + 182 a + 172\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-3a^{2}-4a+5\right){x}^{2}+\left(-4a^{3}-a^{2}+31a+38\right){x}-25a^{3}+4a^{2}+182a+172$ |
| 31.2-a2 |
31.2-a |
$2$ |
$5$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{5} \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 5 \) |
$6.333028183$ |
$2.248091932$ |
2.042820744 |
\( \frac{177186713545573092587453}{28629151} a^{3} - \frac{386826328376080881029057}{28629151} a^{2} - \frac{782630903629631679379010}{28629151} a + \frac{748788140254434775831953}{28629151} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( -a^{2} + 2 a + 3\) , \( a^{2} - a - 3\) , \( -128 a^{3} + 123 a^{2} + 190 a - 158\) , \( 658 a^{3} - 6075 a^{2} - 3236 a + 5765\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-128a^{3}+123a^{2}+190a-158\right){x}+658a^{3}-6075a^{2}-3236a+5765$ |
| 31.2-b1 |
31.2-b |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{3} \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.569728935$ |
$212.5216238$ |
3.191106525 |
\( -\frac{1167585234944}{29791} a^{3} + \frac{2535737331712}{29791} a^{2} + \frac{5143661780992}{29791} a - \frac{4915965276160}{29791} \) |
\( \bigl[0\) , \( a^{2} - 3\) , \( a + 1\) , \( 6 a^{3} - a^{2} - 41 a - 37\) , \( -174 a^{3} + 40 a^{2} + 1249 a + 1134\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(6a^{3}-a^{2}-41a-37\right){x}-174a^{3}+40a^{2}+1249a+1134$ |
| 31.2-b2 |
31.2-b |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$4.709186805$ |
$2.623723751$ |
3.191106525 |
\( -\frac{89309854486593630208}{31} a^{3} - \frac{192733933833511915520}{31} a^{2} + \frac{16495283168405676032}{31} a + \frac{141397211766893629440}{31} \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 5 a + 2\) , \( a^{2} - a - 4\) , \( -68 a^{3} + 129 a^{2} + 347 a - 237\) , \( 2433 a^{3} - 5075 a^{2} - 11004 a + 8732\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a+2\right){x}^{2}+\left(-68a^{3}+129a^{2}+347a-237\right){x}+2433a^{3}-5075a^{2}-11004a+8732$ |
| 31.2-c1 |
31.2-c |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{3} \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$688.6642494$ |
4.940629804 |
\( -\frac{106790912}{29791} a^{3} + \frac{291057664}{29791} a^{2} + \frac{240222208}{29791} a - \frac{298799104}{29791} \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 4 a + 1\) , \( a^{2} - a - 3\) , \( -2 a^{3} + 2 a^{2} + 16 a + 13\) , \( -9 a^{3} + a^{2} + 62 a + 57\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a+1\right){x}^{2}+\left(-2a^{3}+2a^{2}+16a+13\right){x}-9a^{3}+a^{2}+62a+57$ |
| 31.2-d1 |
31.2-d |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{3} \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$4$ |
\( 1 \) |
$1$ |
$46.80217730$ |
1.343076730 |
\( \frac{344812070453}{29791} a^{3} - \frac{945876018415}{29791} a^{2} - \frac{765523593556}{29791} a + \frac{989488284262}{29791} \) |
\( \bigl[a^{3} - a^{2} - 5 a - 2\) , \( -a^{3} + 3 a^{2} + 3 a - 7\) , \( a^{3} - a^{2} - 5 a - 1\) , \( 6 a^{3} - 13 a^{2} - 26 a + 20\) , \( -8 a^{3} + 16 a^{2} + 40 a - 37\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-2\right){x}{y}+\left(a^{3}-a^{2}-5a-1\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+3a-7\right){x}^{2}+\left(6a^{3}-13a^{2}-26a+20\right){x}-8a^{3}+16a^{2}+40a-37$ |
| 31.2-d2 |
31.2-d |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$36$ |
\( 1 \) |
$1$ |
$5.200241922$ |
1.343076730 |
\( -\frac{48808733959823}{31} a^{3} - \frac{105147071605249}{31} a^{2} + \frac{9556697342758}{31} a + \frac{77658579474387}{31} \) |
\( \bigl[a^{3} - 2 a^{2} - 4 a + 3\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{3} - 2 a^{2} - 4 a + 2\) , \( 36 a^{3} - 99 a^{2} - 80 a + 99\) , \( 1600 a^{3} - 4388 a^{2} - 3553 a + 4587\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-4a+3\right){x}{y}+\left(a^{3}-2a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(36a^{3}-99a^{2}-80a+99\right){x}+1600a^{3}-4388a^{2}-3553a+4587$ |
| 31.2-e1 |
31.2-e |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{3} \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.488505840$ |
$98.29472079$ |
5.511816010 |
\( -\frac{170816818639}{29791} a^{3} + \frac{300002375680}{29791} a^{2} + \frac{857697912619}{29791} a - \frac{320551417008}{29791} \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 0\) , \( -a^{3} + 5 a^{2} + 4 a - 13\) , \( -5 a^{3} + 15 a^{2} + 23 a - 32\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+3\right){x}^{2}+\left(-a^{3}+5a^{2}+4a-13\right){x}-5a^{3}+15a^{2}+23a-32$ |
| 31.2-e2 |
31.2-e |
$2$ |
$3$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.054278426$ |
$884.6524871$ |
5.511816010 |
\( -\frac{515037}{31} a^{3} - \frac{1095674}{31} a^{2} + \frac{136915}{31} a + \frac{842717}{31} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( -a^{3} + 2 a^{2} + 5 a - 2\) , \( a^{2} - 4\) , \( a^{2} - a + 2\) , \( 2 a^{3} - 4 a^{2} - 4 a + 2\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-2\right){x}^{2}+\left(a^{2}-a+2\right){x}+2a^{3}-4a^{2}-4a+2$ |
| 31.2-f1 |
31.2-f |
$1$ |
$1$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$19.13304$ |
$(a^3-a^2-6a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$193.8019908$ |
1.390378392 |
\( \frac{1242972976}{31} a^{3} - \frac{302663049}{31} a^{2} - \frac{8902555702}{31} a - \frac{8016984347}{31} \) |
\( \bigl[a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( 0\) , \( 3 a^{3} + a^{2} - 25 a - 26\) , \( 14 a^{3} - 3 a^{2} - 101 a - 92\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-1\right){x}^{2}+\left(3a^{3}+a^{2}-25a-26\right){x}+14a^{3}-3a^{2}-101a-92$ |
| 35.1-a1 |
35.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{8} \cdot 7 \) |
$19.42550$ |
$(a), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$29.17384925$ |
1.674397231 |
\( -\frac{100255502540005874709187}{2734375} a^{3} + \frac{23275207034409755199227}{2734375} a^{2} + \frac{719660178385656737802469}{2734375} a + \frac{652840166649681351347332}{2734375} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 6 a + 2\) , \( a^{2} - a - 3\) , \( -2556 a^{3} + 5586 a^{2} + 11283 a - 10832\) , \( -107481 a^{3} + 234637 a^{2} + 474739 a - 454186\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+2\right){x}^{2}+\left(-2556a^{3}+5586a^{2}+11283a-10832\right){x}-107481a^{3}+234637a^{2}+474739a-454186$ |
| 35.1-a2 |
35.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{2} \cdot 7^{4} \) |
$19.42550$ |
$(a), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$14.58692462$ |
1.674397231 |
\( \frac{645279763380607380450959197}{60025} a^{3} - \frac{1408746720517512725683353637}{60025} a^{2} - \frac{2850190480907465175140486139}{60025} a + \frac{2726941677934766675435994708}{60025} \) |
\( \bigl[a\) , \( a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( 1599 a^{3} - 369 a^{2} - 11472 a - 10405\) , \( -81252 a^{3} + 18867 a^{2} + 583260 a + 529100\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-2a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+3\right){x}^{2}+\left(1599a^{3}-369a^{2}-11472a-10405\right){x}-81252a^{3}+18867a^{2}+583260a+529100$ |
| 35.1-a3 |
35.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( 5^{4} \cdot 7^{2} \) |
$19.42550$ |
$(a), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$233.3907940$ |
1.674397231 |
\( \frac{1456808499743089}{30625} a^{3} - \frac{3183784624344094}{30625} a^{2} - \frac{6429967712771018}{30625} a + \frac{6174851382822971}{30625} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{3} - 3 a^{2} - 3 a + 7\) , \( a^{2} - 3\) , \( 157 a^{3} - 28 a^{2} - 1140 a - 1073\) , \( -2566 a^{3} + 656 a^{2} + 18333 a + 16375\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+7\right){x}^{2}+\left(157a^{3}-28a^{2}-1140a-1073\right){x}-2566a^{3}+656a^{2}+18333a+16375$ |
| 35.1-a4 |
35.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7 \) |
$19.42550$ |
$(a), (-a^3+2a^2+4a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1867.126352$ |
1.674397231 |
\( \frac{14451746}{175} a^{3} - \frac{26597716}{175} a^{2} - \frac{71580727}{175} a + \frac{59751219}{175} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 3\) , \( a^{3} - 3 a^{2} - 3 a + 7\) , \( a^{2} - 3\) , \( 12 a^{3} - 8 a^{2} - 80 a - 53\) , \( -32 a^{3} + 7 a^{2} + 229 a + 209\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-3a^{2}-3a+7\right){x}^{2}+\left(12a^{3}-8a^{2}-80a-53\right){x}-32a^{3}+7a^{2}+229a+209$ |
| 35.1-b1 |
35.1-b |
$2$ |
$2$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{4} \cdot 7^{2} \) |
$19.42550$ |
$(a), (-a^3+2a^2+4a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.073245134$ |
$878.2762479$ |
3.692110423 |
\( -\frac{1197912953664681}{30625} a^{3} + \frac{3285531224258951}{30625} a^{2} + \frac{2659539643865722}{30625} a - \frac{3436676436163809}{30625} \) |
\( \bigl[a\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a^{2} - 3\) , \( 62 a^{3} - 24 a^{2} - 434 a - 354\) , \( -562 a^{3} + 138 a^{2} + 4021 a + 3614\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(62a^{3}-24a^{2}-434a-354\right){x}-562a^{3}+138a^{2}+4021a+3614$ |
| 35.1-b2 |
35.1-b |
$2$ |
$2$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
35.1 |
\( 5 \cdot 7 \) |
\( - 5^{2} \cdot 7 \) |
$19.42550$ |
$(a), (-a^3+2a^2+4a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.146490268$ |
$1756.552495$ |
3.692110423 |
\( -\frac{12903931}{175} a^{3} + \frac{36539451}{175} a^{2} + \frac{28792972}{175} a - \frac{38151534}{175} \) |
\( \bigl[a\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a^{2} - 3\) , \( 2 a^{3} + a^{2} - 19 a - 24\) , \( -9 a^{3} + 65 a + 66\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(2a^{3}+a^{2}-19a-24\right){x}-9a^{3}+65a+66$ |
| 39.1-a1 |
39.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{4} \cdot 13 \) |
$19.69005$ |
$(a^3-a^2-6a-3), (a^2-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$717.0311951$ |
2.572070277 |
\( -\frac{27313302212065478}{1053} a^{3} + \frac{1701198117074023}{351} a^{2} + \frac{197813372299103087}{1053} a + \frac{184668256548956923}{1053} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( 1\) , \( -329 a^{3} + 715 a^{2} + 1441 a - 1382\) , \( 5108 a^{3} - 11151 a^{2} - 22566 a + 21591\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(-329a^{3}+715a^{2}+1441a-1382\right){x}+5108a^{3}-11151a^{2}-22566a+21591$ |
| 39.1-a2 |
39.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{2} \) |
$19.69005$ |
$(a^3-a^2-6a-3), (a^2-a-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$1434.062390$ |
2.572070277 |
\( -\frac{2565473777}{1521} a^{3} + \frac{550765558}{507} a^{2} + \frac{15884685779}{1521} a + \frac{14803057468}{1521} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( 1\) , \( -24 a^{3} + 50 a^{2} + 96 a - 92\) , \( 85 a^{3} - 193 a^{2} - 388 a + 373\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(-24a^{3}+50a^{2}+96a-92\right){x}+85a^{3}-193a^{2}-388a+373$ |
| 39.1-a3 |
39.1-a |
$4$ |
$4$ |
4.4.19429.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13 \) |
$19.69005$ |
$(a^3-a^2-6a-3), (a^2-a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$1434.062390$ |
2.572070277 |
\( -\frac{9986}{39} a^{3} + \frac{17619}{13} a^{2} - \frac{23329}{39} a - \frac{145238}{39} \) |
\( \bigl[a^{3} - 2 a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( 1\) , \( -4 a^{3} + 10 a^{2} + 11 a - 12\) , \( a^{2} - 5 a + 3\bigr] \) |
${y}^2+\left(a^{3}-2a^{2}-3a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(-4a^{3}+10a^{2}+11a-12\right){x}+a^{2}-5a+3$ |
*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.
