| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$3.59663$ |
$\textsf{none}$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
✓ |
$3, 5$ |
3B.1.1, 5Ns |
$1$ |
\( 1 \) |
$0.194243419$ |
$685.2000997$ |
1.102262344 |
\( -316368 \) |
\( \bigl[a^{2} - a - 8\) , \( -a^{2} + 3 a + 7\) , \( a^{2} - 2 a - 7\) , \( 3 a^{2} - 5 a - 19\) , \( -2 a^{2} + 6 a + 11\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(3a^{2}-5a-19\right){x}-2a^{2}+6a+11$ |
| 1.1-a2 |
1.1-a |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$3.59663$ |
$\textsf{none}$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
✓ |
$3, 5$ |
3B.1.2, 5Ns |
$1$ |
\( 1 \) |
$0.064747806$ |
$228.4000332$ |
1.102262344 |
\( 432 \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{2} - 2 a - 7\) , \( 4 a^{2} + 4 a - 1\) , \( 14 a^{2} + 55 a + 49\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4a^{2}+4a-1\right){x}+14a^{2}+55a+49$ |
| 2.1-a1 |
2.1-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$4.03708$ |
$(a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$0.107292025$ |
$213.0968611$ |
1.704151671 |
\( \frac{20775}{2} a^{2} - 14425 a - 102972 \) |
\( \bigl[a^{2} - a - 7\) , \( 0\) , \( a + 1\) , \( 2\) , \( a + 2\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}+2{x}+a+2$ |
| 2.1-a2 |
2.1-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$4.03708$ |
$(a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.536460128$ |
$42.61937222$ |
1.704151671 |
\( -\frac{8357155}{2} a^{2} + \frac{40258845}{4} a + \frac{44166501}{2} \) |
\( \bigl[a + 1\) , \( a^{2} - 2 a - 8\) , \( a^{2} - 2 a - 7\) , \( -2875311 a^{2} + 3999587 a + 28939928\) , \( 12258063292 a^{2} - 17049811994 a - 123382082558\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(a^{2}-2a-8\right){x}^{2}+\left(-2875311a^{2}+3999587a+28939928\right){x}+12258063292a^{2}-17049811994a-123382082558$ |
| 2.1-b1 |
2.1-b |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{18} \) |
$4.03708$ |
$(a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$53.04318396$ |
2.635737002 |
\( \frac{109503}{64} \) |
\( \bigl[a^{2} - a - 7\) , \( -a^{2} + 3 a + 9\) , \( a^{2} - 2 a - 7\) , \( -30 a^{2} + 47 a + 317\) , \( -79 a^{2} + 118 a + 807\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a^{2}+3a+9\right){x}^{2}+\left(-30a^{2}+47a+317\right){x}-79a^{2}+118a+807$ |
| 2.1-b2 |
2.1-b |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{6} \) |
$4.03708$ |
$(a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$17.68106132$ |
2.635737002 |
\( -\frac{35937}{4} \) |
\( \bigl[a + 1\) , \( 0\) , \( a^{2} - a - 8\) , \( 705935 a^{2} - 981889 a - 7105502\) , \( 620527706 a^{2} - 863095896 a - 6245846948\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-8\right){y}={x}^{3}+\left(705935a^{2}-981889a-7105502\right){x}+620527706a^{2}-863095896a-6245846948$ |
| 2.1-c1 |
2.1-c |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{3} \) |
$4.03708$ |
$(a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.386670647$ |
$92.92571067$ |
1.067162245 |
\( -\frac{11795720531355}{2} a^{2} + 15054138584820 a + \frac{64698033335715}{2} \) |
\( \bigl[1\) , \( a^{2} - 2 a - 7\) , \( 0\) , \( -15548 a^{2} + 21625 a + 156500\) , \( 12640279 a^{2} - 17581444 a - 127229206\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a^{2}-2a-7\right){x}^{2}+\left(-15548a^{2}+21625a+156500\right){x}+12640279a^{2}-17581444a-127229206$ |
| 2.1-c2 |
2.1-c |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{9} \) |
$4.03708$ |
$(a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.462223549$ |
$30.97523689$ |
1.067162245 |
\( \frac{2964195}{8} a^{2} - \frac{2041605}{4} a - \frac{30011985}{8} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 525 a^{2} - 730 a - 5285\) , \( 11541 a^{2} - 16052 a - 116164\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(525a^{2}-730a-5285\right){x}+11541a^{2}-16052a-116164$ |
| 3.1-a1 |
3.1-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{20} \) |
$4.31933$ |
$(a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$3.718192765$ |
$6.430747379$ |
3.564405409 |
\( -\frac{75586782176}{729} a^{2} + \frac{578800730144}{2187} a + \frac{1243752775952}{2187} \) |
\( \bigl[a^{2} - a - 8\) , \( -a + 1\) , \( 1\) , \( -8 a^{2} + 18 a + 64\) , \( -22 a^{2} + 52 a + 137\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a^{2}+18a+64\right){x}-22a^{2}+52a+137$ |
| 3.1-a2 |
3.1-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{4} \) |
$4.31933$ |
$(a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$0.743638553$ |
$32.15373689$ |
3.564405409 |
\( \frac{196092238880}{9} a^{2} - \frac{272757951200}{9} a - \frac{1973773508944}{9} \) |
\( \bigl[a\) , \( -a^{2} + a + 9\) , \( 1\) , \( 147246626540 a^{2} - 204879225699 a - 1481805471268\) , \( -54100623016095554 a^{2} + 75249756398013750 a + 544539923469576281\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+a+9\right){x}^{2}+\left(147246626540a^{2}-204879225699a-1481805471268\right){x}-54100623016095554a^{2}+75249756398013750a+544539923469576281$ |
| 3.1-b1 |
3.1-b |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{4} \) |
$4.31933$ |
$(a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.037314883$ |
$221.6035131$ |
1.232686037 |
\( -\frac{9568}{9} a^{2} - \frac{28064}{9} a - \frac{14032}{9} \) |
\( \bigl[a^{2} - a - 8\) , \( a^{2} - 3 a - 8\) , \( a^{2} - a - 7\) , \( -67 a^{2} + 90 a + 679\) , \( 570 a^{2} - 795 a - 5734\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-a-7\right){y}={x}^{3}+\left(a^{2}-3a-8\right){x}^{2}+\left(-67a^{2}+90a+679\right){x}+570a^{2}-795a-5734$ |
| 5.2-a1 |
5.2-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( -5 \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$175.0001293$ |
4.347913170 |
\( -\frac{3936}{5} a^{2} + \frac{8864}{5} a + \frac{26416}{5} \) |
\( \bigl[a\) , \( 0\) , \( a^{2} - 2 a - 7\) , \( -2 a^{2} + 3 a + 23\) , \( -3 a^{2} + 5 a + 29\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-2a^{2}+3a+23\right){x}-3a^{2}+5a+29$ |
| 5.2-a2 |
5.2-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{5} \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$1$ |
$35.00002587$ |
4.347913170 |
\( \frac{424693664}{3125} a^{2} - \frac{601608736}{3125} a - \frac{4302545104}{3125} \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 7\) , \( 1\) , \( 638356287 a^{2} - 887891428 a - 6425307656\) , \( 15468801291568 a^{2} - 21515651408828 a - 155699364937229\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(638356287a^{2}-887891428a-6425307656\right){x}+15468801291568a^{2}-21515651408828a-155699364937229$ |
| 5.2-b1 |
5.2-b |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{3} \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$23.68701824$ |
1.765526099 |
\( -\frac{1460249856}{125} a^{2} - \frac{5758304256}{125} a - \frac{5184210384}{125} \) |
\( \bigl[a^{2} - a - 8\) , \( a^{2} - a - 8\) , \( a + 1\) , \( 8 a^{2} - 20 a - 44\) , \( 219 a^{2} - 561 a - 1204\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-8\right){x}^{2}+\left(8a^{2}-20a-44\right){x}+219a^{2}-561a-1204$ |
| 5.2-b2 |
5.2-b |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{9} \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$71.06105474$ |
1.765526099 |
\( -\frac{836630580096}{1953125} a^{2} + \frac{2135466333504}{1953125} a + \frac{4588421198256}{1953125} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 8\) , \( 1\) , \( -203 a^{2} + 518 a + 1118\) , \( 3186 a^{2} - 8132 a - 17473\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a^{2}+2a+8\right){x}^{2}+\left(-203a^{2}+518a+1118\right){x}+3186a^{2}-8132a-17473$ |
| 5.2-c1 |
5.2-c |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( 5 \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$395.5952458$ |
2.457160725 |
\( \frac{259362816}{5} a^{2} + \frac{869564416}{5} a + \frac{1184190464}{5} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 6139 a^{2} - 5534 a - 73640\) , \( -639785 a^{2} + 729305 a + 7072891\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(6139a^{2}-5534a-73640\right){x}-639785a^{2}+729305a+7072891$ |
| 5.2-c2 |
5.2-c |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{2} \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$197.7976229$ |
2.457160725 |
\( -\frac{36639951445958464}{25} a^{2} + \frac{50962739372517536}{25} a + \frac{368795021990013104}{25} \) |
\( \bigl[a\) , \( -a^{2} + a + 8\) , \( a^{2} - a - 7\) , \( 163 a^{2} - 227 a - 1649\) , \( -1751 a^{2} + 2433 a + 17629\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-7\right){y}={x}^{3}+\left(-a^{2}+a+8\right){x}^{2}+\left(163a^{2}-227a-1649\right){x}-1751a^{2}+2433a+17629$ |
| 5.2-d1 |
5.2-d |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{2} \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$104.3807111$ |
1.296679809 |
\( \frac{48447659616}{25} a^{2} + \frac{191047705216}{25} a + \frac{172001679024}{25} \) |
\( \bigl[a^{2} - a - 8\) , \( -a^{2} + 2 a + 7\) , \( a + 1\) , \( 339 a^{2} - 473 a - 3413\) , \( 3058 a^{2} - 4254 a - 30783\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(339a^{2}-473a-3413\right){x}+3058a^{2}-4254a-30783$ |
| 5.2-d2 |
5.2-d |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
5.2 |
\( 5 \) |
\( 5 \) |
$4.70317$ |
$(2a^2-3a-19)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$208.7614223$ |
1.296679809 |
\( \frac{14688256}{5} a^{2} - \frac{37330944}{5} a - \frac{79896576}{5} \) |
\( \bigl[0\) , \( -a^{2} + a + 9\) , \( a^{2} - a - 7\) , \( 24 a^{2} - 33 a - 241\) , \( -76 a^{2} + 105 a + 762\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){y}={x}^{3}+\left(-a^{2}+a+9\right){x}^{2}+\left(24a^{2}-33a-241\right){x}-76a^{2}+105a+762$ |
| 6.1-a1 |
6.1-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{10} \cdot 3^{10} \) |
$4.84828$ |
$(a+2), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.120913505$ |
$46.21228132$ |
1.665926974 |
\( -\frac{11038321}{1296} a^{2} - \frac{22299361}{648} a - \frac{5084711}{162} \) |
\( \bigl[a^{2} - a - 7\) , \( -a\) , \( a + 1\) , \( 1725112 a^{2} - 2399476 a - 17363891\) , \( 1475906218126 a^{2} - 2052847260013 a - 14855556426422\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(1725112a^{2}-2399476a-17363891\right){x}+1475906218126a^{2}-2052847260013a-14855556426422$ |
| 6.1-a2 |
6.1-a |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{2} \cdot 3^{2} \) |
$4.84828$ |
$(a+2), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.024182701$ |
$231.0614066$ |
1.665926974 |
\( -\frac{82960675}{3} a^{2} + \frac{141173165}{2} a + \frac{455012647}{3} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 8\) , \( a^{2} - 2 a - 7\) , \( 4 a^{2} - a - 22\) , \( -2 a^{2} + 6 a + 22\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a^{2}+2a+8\right){x}^{2}+\left(4a^{2}-a-22\right){x}-2a^{2}+6a+22$ |
| 7.1-a1 |
7.1-a |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$4.97446$ |
$(a^2-2a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$81.39431602$ |
4.044516080 |
\( \frac{46060621}{49} a^{2} - \frac{9148820}{7} a - \frac{463712533}{49} \) |
\( \bigl[a^{2} - a - 7\) , \( a^{2} - 2 a - 8\) , \( a^{2} - 2 a - 7\) , \( 35 a^{2} - 48 a - 348\) , \( -187 a^{2} + 261 a + 1883\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(a^{2}-2a-8\right){x}^{2}+\left(35a^{2}-48a-348\right){x}-187a^{2}+261a+1883$ |
| 7.1-b1 |
7.1-b |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$4.97446$ |
$(a^2-2a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$86.33305733$ |
1.072481027 |
\( \frac{27425378783195}{49} a^{2} + \frac{15450071265000}{7} a + \frac{97371586041627}{49} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 9\) , \( a^{2} - a - 8\) , \( 7881 a^{2} - 10958 a - 79308\) , \( 677795 a^{2} - 942742 a - 6822249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-8\right){y}={x}^{3}+\left(-a^{2}+3a+9\right){x}^{2}+\left(7881a^{2}-10958a-79308\right){x}+677795a^{2}-942742a-6822249$ |
| 7.1-b2 |
7.1-b |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$4.97446$ |
$(a^2-2a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$172.6661146$ |
1.072481027 |
\( \frac{133740546}{7} a^{2} - 47940242 a - \frac{725540385}{7} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 9\) , \( a^{2} - a - 8\) , \( 476 a^{2} - 658 a - 4773\) , \( 11743 a^{2} - 16325 a - 118180\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-8\right){y}={x}^{3}+\left(-a^{2}+3a+9\right){x}^{2}+\left(476a^{2}-658a-4773\right){x}+11743a^{2}-16325a-118180$ |
| 8.1-a1 |
8.1-a |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$5.08641$ |
$(a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.253808664$ |
$105.1573286$ |
1.989343300 |
\( -206 a^{2} + 273 a + 744 \) |
\( \bigl[a^{2} - a - 8\) , \( a^{2} - 2 a - 7\) , \( a^{2} - 2 a - 8\) , \( -2 a^{2} - 4 a - 6\) , \( -2 a^{2} + 40 a + 58\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-2a-8\right){y}={x}^{3}+\left(a^{2}-2a-7\right){x}^{2}+\left(-2a^{2}-4a-6\right){x}-2a^{2}+40a+58$ |
| 9.1-a1 |
9.1-a |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{6} \) |
$5.18724$ |
$(a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.1, 5Ns |
$9$ |
\( 2 \) |
$1$ |
$54.31587092$ |
2.698977325 |
\( 432 \) |
\( \bigl[a^{2} - a - 8\) , \( 0\) , \( a^{2} - a - 7\) , \( -a^{2} + a + 8\) , \( -a^{2} + a + 8\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-a-7\right){y}={x}^{3}+\left(-a^{2}+a+8\right){x}-a^{2}+a+8$ |
| 9.1-a2 |
9.1-a |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{6} \) |
$5.18724$ |
$(a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.2, 5Ns |
$9$ |
\( 2 \) |
$1$ |
$6.035096769$ |
2.698977325 |
\( -316368 \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{2} - a - 7\) , \( 895 a^{2} - 1248 a - 9005\) , \( 26527 a^{2} - 36898 a - 267004\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-7\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(895a^{2}-1248a-9005\right){x}+26527a^{2}-36898a-267004$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{10} \) |
$5.18724$ |
$(a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$22.11437313$ |
1.098872035 |
\( -\frac{9568}{9} a^{2} - \frac{28064}{9} a - \frac{14032}{9} \) |
\( \bigl[a^{2} - a - 8\) , \( -a^{2} + 2 a + 9\) , \( a + 1\) , \( -12 a^{2} - 23 a + 22\) , \( -63 a^{2} - 218 a - 147\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+9\right){x}^{2}+\left(-12a^{2}-23a+22\right){x}-63a^{2}-218a-147$ |
| 9.1-c1 |
9.1-c |
$4$ |
$6$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{3} \) |
$5.18724$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1.161502114$ |
$49.62331419$ |
2.148025945 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} - a - 8\) , \( 1\) , \( -a^{2} + 2 a + 12\) , \( 6702949 a^{2} - 9323173 a - 67467729\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{2}-a-8\right){x}^{2}+\left(-a^{2}+2a+12\right){x}+6702949a^{2}-9323173a-67467729$ |
| 9.1-c2 |
9.1-c |
$4$ |
$6$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{9} \) |
$5.18724$ |
$(a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$3.484506344$ |
$148.8699425$ |
2.148025945 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a^{2} - 2 a - 7\) , \( 0\) , \( a^{2} - 6 a + 8\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){y}={x}^{3}+a^{2}-6a+8$ |
| 9.1-c3 |
9.1-c |
$4$ |
$6$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{3} \) |
$5.18724$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.580751057$ |
$99.24662838$ |
2.148025945 |
\( 54000 \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{2} - 2 a - 7\) , \( 2109338 a^{2} - 2933823 a - 21231559\) , \( 2889932612 a^{2} - 4019624783 a - 29088271687\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2109338a^{2}-2933823a-21231559\right){x}+2889932612a^{2}-4019624783a-29088271687$ |
| 9.1-c4 |
9.1-c |
$4$ |
$6$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{9} \) |
$5.18724$ |
$(a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.742253172$ |
$297.7398851$ |
2.148025945 |
\( 54000 \) |
\( \bigl[a\) , \( -a^{2} + 3 a + 9\) , \( a^{2} - a - 7\) , \( -100 a^{2} + 407 a - 31\) , \( 2659 a^{2} - 8197 a - 9001\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-7\right){y}={x}^{3}+\left(-a^{2}+3a+9\right){x}^{2}+\left(-100a^{2}+407a-31\right){x}+2659a^{2}-8197a-9001$ |
| 9.1-d1 |
9.1-d |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{10} \) |
$5.18724$ |
$(a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$15.08621097$ |
0.749639850 |
\( \frac{196092238880}{9} a^{2} - \frac{272757951200}{9} a - \frac{1973773508944}{9} \) |
\( \bigl[a^{2} - a - 8\) , \( -a^{2} + 2 a + 9\) , \( a^{2} - 2 a - 7\) , \( 56729977251067 a^{2} - 78906195919453 a - 571008313289912\) , \( 409380059758053249511 a^{2} - 569409306699890861898 a - 4120565719744519744118\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a^{2}+2a+9\right){x}^{2}+\left(56729977251067a^{2}-78906195919453a-571008313289912\right){x}+409380059758053249511a^{2}-569409306699890861898a-4120565719744519744118$ |
| 9.1-d2 |
9.1-d |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{26} \) |
$5.18724$ |
$(a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$15.08621097$ |
0.749639850 |
\( -\frac{75586782176}{729} a^{2} + \frac{578800730144}{2187} a + \frac{1243752775952}{2187} \) |
\( \bigl[a\) , \( -1\) , \( a^{2} - a - 7\) , \( -44 a^{2} + 69 a + 406\) , \( 3541 a^{2} - 4951 a - 35544\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-7\right){y}={x}^{3}-{x}^{2}+\left(-44a^{2}+69a+406\right){x}+3541a^{2}-4951a-35544$ |
| 10.1-a1 |
10.1-a |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{5} \cdot 5^{8} \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$16.43953573$ |
3.267548393 |
\( -\frac{2043384777}{390625} a^{2} + \frac{20770841817}{1562500} a + \frac{11175951447}{390625} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 7\) , \( 0\) , \( -12 a^{2} + 19 a + 141\) , \( -3110 a^{2} + 4332 a + 31309\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(-12a^{2}+19a+141\right){x}-3110a^{2}+4332a+31309$ |
| 10.1-b1 |
10.1-b |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5^{7} \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$39.11553180$ |
3.887332804 |
\( -\frac{889919581}{312500} a^{2} + \frac{1991314697}{156250} a - \frac{378415571}{78125} \) |
\( \bigl[a^{2} - a - 7\) , \( a + 1\) , \( 1\) , \( -3 a^{2} + 14 a + 30\) , \( -4 a^{2} + 22 a + 39\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a^{2}+14a+30\right){x}-4a^{2}+22a+39$ |
| 10.1-c1 |
10.1-c |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5 \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$0.038069758$ |
$200.8361060$ |
5.128946525 |
\( -\frac{32381}{40} a^{2} + \frac{72339}{40} a + \frac{232761}{40} \) |
\( \bigl[a^{2} - a - 7\) , \( a^{2} - a - 8\) , \( a^{2} - 2 a - 8\) , \( 2 a^{2} + 6 a - 1\) , \( 6 a^{2} + 24 a + 19\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+\left(a^{2}-2a-8\right){y}={x}^{3}+\left(a^{2}-a-8\right){x}^{2}+\left(2a^{2}+6a-1\right){x}+6a^{2}+24a+19$ |
| 10.1-d1 |
10.1-d |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{27} \cdot 5^{3} \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$83.06343838$ |
6.191183156 |
\( -\frac{58374455049}{64000} a^{2} - \frac{216373317399}{64000} a - \frac{187800406911}{64000} \) |
\( \bigl[a^{2} - a - 7\) , \( -a^{2} + 3 a + 9\) , \( a + 1\) , \( -316 a^{2} - 383 a + 91\) , \( 6778 a^{2} + 15090 a + 7885\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3a+9\right){x}^{2}+\left(-316a^{2}-383a+91\right){x}+6778a^{2}+15090a+7885$ |
| 10.1-d2 |
10.1-d |
$2$ |
$3$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{9} \cdot 5^{9} \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$27.68781279$ |
6.191183156 |
\( -\frac{4227123700917}{7812500} a^{2} + \frac{11757790775691}{15625000} a + \frac{85100164969599}{15625000} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 19 a^{2} - 35 a - 145\) , \( 74 a^{2} - 154 a - 535\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(19a^{2}-35a-145\right){x}+74a^{2}-154a-535$ |
| 10.1-e1 |
10.1-e |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5 \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$33.05107062$ |
0.821160451 |
\( -\frac{2886076541}{10} a^{2} + \frac{3876339909}{10} a + \frac{28697417841}{10} \) |
\( \bigl[a^{2} - a - 7\) , \( -a^{2} + 2 a + 9\) , \( a\) , \( -3 a^{2} - 3 a + 17\) , \( -9 a^{2} - 21 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2a+9\right){x}^{2}+\left(-3a^{2}-3a+17\right){x}-9a^{2}-21a+1$ |
| 10.1-e2 |
10.1-e |
$2$ |
$5$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{15} \cdot 5^{5} \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 5 \) |
$1$ |
$6.610214125$ |
0.821160451 |
\( -\frac{18064666860861}{100000} a^{2} + \frac{46109413314239}{100000} a + \frac{99083282415921}{100000} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 9\) , \( a\) , \( 3382463 a^{2} - 4628154 a - 34347587\) , \( 5638384033 a^{2} - 7821678993 a - 56834438703\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+9\right){x}^{2}+\left(3382463a^{2}-4628154a-34347587\right){x}+5638384033a^{2}-7821678993a-56834438703$ |
| 10.1-f1 |
10.1-f |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5 \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$29.36895639$ |
0.729677588 |
\( -\frac{139201}{10} a^{2} - \frac{273493}{5} a - \frac{244792}{5} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 7\) , \( a + 1\) , \( -2 a - 1\) , \( -a^{2} - a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+7\right){x}^{2}+\left(-2a-1\right){x}-a^{2}-a+1$ |
| 10.1-g1 |
10.1-g |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{15} \cdot 5 \) |
$5.27913$ |
$(a+2), (2a^2-3a-19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$23.62561048$ |
0.586983011 |
\( \frac{5821540049}{160} a^{2} - \frac{506086881}{10} a - \frac{58596354789}{160} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 8\) , \( a + 1\) , \( 10 a^{2} - 12 a - 85\) , \( 27 a^{2} - 31 a - 264\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3a+8\right){x}^{2}+\left(10a^{2}-12a-85\right){x}+27a^{2}-31a-264$ |
| 10.2-a1 |
10.2-a |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{5} \cdot 5 \) |
$5.27913$ |
$(a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$110.5562097$ |
2.746791115 |
\( \frac{9003}{20} a^{2} - \frac{44329}{20} a - \frac{42951}{5} \) |
\( \bigl[a^{2} - a - 7\) , \( -a^{2} + a + 9\) , \( a\) , \( -5 a^{2} - a + 33\) , \( -9 a^{2} - 11 a + 29\bigr] \) |
${y}^2+\left(a^{2}-a-7\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+9\right){x}^{2}+\left(-5a^{2}-a+33\right){x}-9a^{2}-11a+29$ |
| 10.2-b1 |
10.2-b |
$1$ |
$1$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{19} \cdot 5^{8} \) |
$5.27913$ |
$(a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$6.620812390$ |
1.315963250 |
\( -\frac{18345049}{80000} a^{2} - \frac{1419129}{5000} a + \frac{7693911}{10000} \) |
\( \bigl[a^{2} - 2 a - 7\) , \( a^{2} - a - 7\) , \( a^{2} - a - 8\) , \( 2 a^{2} - 8 a - 18\) , \( -a^{2} - 14 a - 24\bigr] \) |
${y}^2+\left(a^{2}-2a-7\right){x}{y}+\left(a^{2}-a-8\right){y}={x}^{3}+\left(a^{2}-a-7\right){x}^{2}+\left(2a^{2}-8a-18\right){x}-a^{2}-14a-24$ |
| 12.1-a1 |
12.1-a |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$5.44201$ |
$(a+2), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.207144461$ |
$53.85481000$ |
3.741744122 |
\( -\frac{704}{9} a^{2} - \frac{10496}{9} a - \frac{2048}{9} \) |
\( \bigl[a^{2} - 2 a - 8\) , \( a^{2} - 2 a - 9\) , \( a\) , \( -2 a^{2} - 19 a - 24\) , \( -18 a^{2} - 67 a - 59\bigr] \) |
${y}^2+\left(a^{2}-2a-8\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-2a-9\right){x}^{2}+\left(-2a^{2}-19a-24\right){x}-18a^{2}-67a-59$ |
| 12.1-a2 |
12.1-a |
$2$ |
$2$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3^{3} \) |
$5.44201$ |
$(a+2), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3^{2} \) |
$0.414288922$ |
$53.85481000$ |
3.741744122 |
\( 1317888 a^{2} + \frac{15593504}{3} a + \frac{14047360}{3} \) |
\( \bigl[a^{2} - 2 a - 8\) , \( a^{2} - a - 8\) , \( a^{2} - 2 a - 8\) , \( 24 a^{2} - 60 a - 134\) , \( -32 a^{2} + 82 a + 173\bigr] \) |
${y}^2+\left(a^{2}-2a-8\right){x}{y}+\left(a^{2}-2a-8\right){y}={x}^{3}+\left(a^{2}-a-8\right){x}^{2}+\left(24a^{2}-60a-134\right){x}-32a^{2}+82a+173$ |
| 12.1-b1 |
12.1-b |
$4$ |
$4$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$5.44201$ |
$(a+2), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$187.8153120$ |
1.749865853 |
\( \frac{13120}{3} a^{2} - 4000 a - \frac{36736}{3} \) |
\( \bigl[a^{2} - 2 a - 8\) , \( a^{2} - 2 a - 8\) , \( a^{2} - 2 a - 8\) , \( -11 a^{2} - 56 a - 62\) , \( 108 a^{2} + 416 a + 367\bigr] \) |
${y}^2+\left(a^{2}-2a-8\right){x}{y}+\left(a^{2}-2a-8\right){y}={x}^{3}+\left(a^{2}-2a-8\right){x}^{2}+\left(-11a^{2}-56a-62\right){x}+108a^{2}+416a+367$ |
| 12.1-b2 |
12.1-b |
$4$ |
$4$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{4} \cdot 3 \) |
$5.44201$ |
$(a+2), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$93.90765600$ |
1.749865853 |
\( -\frac{79936}{3} a^{2} + \frac{112384}{3} a + \frac{807680}{3} \) |
\( \bigl[a^{2} - 2 a - 8\) , \( 1\) , \( a^{2} - a - 8\) , \( 11 a^{2} - 16 a - 107\) , \( 41 a^{2} - 57 a - 416\bigr] \) |
${y}^2+\left(a^{2}-2a-8\right){x}{y}+\left(a^{2}-a-8\right){y}={x}^{3}+{x}^{2}+\left(11a^{2}-16a-107\right){x}+41a^{2}-57a-416$ |
| 12.1-b3 |
12.1-b |
$4$ |
$4$ |
3.3.1620.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3 \) |
$5.44201$ |
$(a+2), (a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1$ |
$375.6306240$ |
1.749865853 |
\( \frac{133054516}{3} a^{2} + \frac{525999992}{3} a + \frac{475744240}{3} \) |
\( \bigl[a^{2} - a - 8\) , \( a^{2} - 2 a - 7\) , \( a^{2} - 2 a - 8\) , \( 3 a^{2} - 5 a - 27\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-2a-8\right){y}={x}^{3}+\left(a^{2}-2a-7\right){x}^{2}+\left(3a^{2}-5a-27\right){x}$ |
*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.
