| 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 |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{9} \) |
$1.92270$ |
$(a+1)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$145.2317184$ |
0.900671562 |
\( -\frac{693022720}{19683} a^{2} - \frac{855752704}{19683} a + \frac{246431744}{19683} \) |
\( \bigl[0\) , \( 1\) , \( a^{2} - a - 3\) , \( -767 a^{2} + 741 a - 133\) , \( 17362 a^{2} - 19422 a + 3649\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){y}={x}^{3}+{x}^{2}+\left(-767a^{2}+741a-133\right){x}+17362a^{2}-19422a+3649$ |
| 3.1-a2 |
3.1-a |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
3.1 |
\( 3 \) |
\( -3 \) |
$1.92270$ |
$(a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$1.792984178$ |
0.900671562 |
\( \frac{1074386927992107008}{3} a^{2} - \frac{817476003654639616}{3} a - \frac{4493025253858410496}{3} \) |
\( \bigl[0\) , \( 1\) , \( a^{2} - a - 2\) , \( -9 a^{2} - 9 a\) , \( -62 a^{2} - 82 a + 19\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){y}={x}^{3}+{x}^{2}+\left(-9a^{2}-9a\right){x}-62a^{2}-82a+19$ |
| 3.1-a3 |
3.1-a |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{3} \) |
$1.92270$ |
$(a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$48.41057281$ |
0.900671562 |
\( \frac{7245824}{27} a^{2} - \frac{5521408}{27} a - \frac{30269440}{27} \) |
\( \bigl[0\) , \( -a^{2} + 2\) , \( a + 1\) , \( 1197 a^{2} + 1718 a - 478\) , \( 80232 a^{2} + 117179 a - 32608\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(1197a^{2}+1718a-478\right){x}+80232a^{2}+117179a-32608$ |
| 7.1-a1 |
7.1-a |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$2.21432$ |
$(-a^2+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$148.4403187$ |
0.460285036 |
\( -\frac{442543462262862405}{49} a^{2} + \frac{336721018795482936}{49} a + \frac{1850691682923692027}{49} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( a\) , \( -30 a^{2} + 108 a - 83\) , \( 38 a^{2} - 228 a + 331\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-30a^{2}+108a-83\right){x}+38a^{2}-228a+331$ |
| 7.1-a2 |
7.1-a |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$2.21432$ |
$(-a^2+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$296.8806374$ |
0.460285036 |
\( -\frac{298287543}{7} a^{2} + \frac{276426141}{7} a + \frac{1331506496}{7} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( a\) , \( 10 a^{2} - 27 a + 7\) , \( 9 a^{2} - 27 a + 12\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(10a^{2}-27a+7\right){x}+9a^{2}-27a+12$ |
| 7.1-a3 |
7.1-a |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$2.21432$ |
$(-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$1.832596527$ |
0.460285036 |
\( \frac{233914256282065367506622}{49} a^{2} - \frac{631481510541950289587895}{49} a + \frac{137626725304254175085673}{49} \) |
\( \bigl[a\) , \( -a^{2} + 4\) , \( 0\) , \( -36666 a^{2} + 85524 a - 18349\) , \( -6922636 a^{2} + 17769810 a - 3853337\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-36666a^{2}+85524a-18349\right){x}-6922636a^{2}+17769810a-3853337$ |
| 7.1-a4 |
7.1-a |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{6} \) |
$2.21432$ |
$(-a^2+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$49.48010624$ |
0.460285036 |
\( \frac{483310163943}{117649} a^{2} - \frac{1322660735001}{117649} a + \frac{328592697002}{117649} \) |
\( \bigl[a\) , \( -a^{2} + 4\) , \( 0\) , \( -226 a^{2} + 1389 a - 314\) , \( -10536 a^{2} + 23790 a - 5084\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-226a^{2}+1389a-314\right){x}-10536a^{2}+23790a-5084$ |
| 7.1-a5 |
7.1-a |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{3} \) |
$2.21432$ |
$(-a^2+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$98.96021248$ |
0.460285036 |
\( -\frac{282564}{343} a^{2} + \frac{796113}{343} a + \frac{419975}{343} \) |
\( \bigl[a\) , \( -a^{2} + 4\) , \( 0\) , \( -76 a^{2} - 6 a + 11\) , \( -302 a^{2} + 270 a - 45\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-76a^{2}-6a+11\right){x}-302a^{2}+270a-45$ |
| 7.1-a6 |
7.1-a |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$2.21432$ |
$(-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$3.665193055$ |
0.460285036 |
\( -\frac{169782428405}{7} a^{2} + \frac{460934072994}{7} a - \frac{99672285689}{7} \) |
\( \bigl[a\) , \( -a^{2} + 4\) , \( 0\) , \( -5351 a^{2} + 879 a + 101\) , \( -322934 a^{2} - 32031 a + 26127\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-5351a^{2}+879a+101\right){x}-322934a^{2}-32031a+26127$ |
| 9.2-a1 |
9.2-a |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{21} \) |
$2.30904$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$36.61664665$ |
1.021870960 |
\( -\frac{56237456370572}{19683} a^{2} + \frac{128369230367570}{59049} a + \frac{705545800652999}{59049} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 660 a^{2} - 490 a - 2793\) , \( -14053 a^{2} + 10729 a + 58676\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(660a^{2}-490a-2793\right){x}-14053a^{2}+10729a+58676$ |
| 9.2-a2 |
9.2-a |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$2.30904$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$18.30832332$ |
1.021870960 |
\( \frac{9746567996631364}{81} a^{2} - \frac{26312108564672162}{81} a + \frac{5734529463243313}{81} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -30 a^{2} + 220 a - 363\) , \( -1063 a^{2} + 3089 a - 1168\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-30a^{2}+220a-363\right){x}-1063a^{2}+3089a-1168$ |
| 9.2-a3 |
9.2-a |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{12} \) |
$2.30904$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$73.23329331$ |
1.021870960 |
\( \frac{932449124}{243} a^{2} - \frac{2528900252}{243} a + \frac{578538179}{243} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 35 a^{2} - 15 a - 178\) , \( -252 a^{2} + 231 a + 954\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(35a^{2}-15a-178\right){x}-252a^{2}+231a+954$ |
| 9.2-a4 |
9.2-a |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.30904$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$73.23329331$ |
1.021870960 |
\( \frac{20552}{27} a^{2} + \frac{32584}{9} a - \frac{24773}{27} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -3\) , \( -9 a^{2} + 7 a + 34\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}-3{x}-9a^{2}+7a+34$ |
| 9.3-a1 |
9.3-a |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{15} \) |
$2.30904$ |
$(a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$10.01686977$ |
1.118174302 |
\( -\frac{693022720}{19683} a^{2} - \frac{855752704}{19683} a + \frac{246431744}{19683} \) |
\( \bigl[0\) , \( a^{2} - a - 2\) , \( a^{2} - 3\) , \( -708 a^{2} - 719 a + 214\) , \( -19813 a^{2} - 25500 a + 7228\bigr] \) |
${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-708a^{2}-719a+214\right){x}-19813a^{2}-25500a+7228$ |
| 9.3-a2 |
9.3-a |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{7} \) |
$2.30904$ |
$(a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$90.15182800$ |
1.118174302 |
\( \frac{1074386927992107008}{3} a^{2} - \frac{817476003654639616}{3} a - \frac{4493025253858410496}{3} \) |
\( \bigl[0\) , \( a^{2} - a - 2\) , \( a + 1\) , \( 2 a^{2} - 38 a - 70\) , \( 73 a^{2} + 280 a + 265\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2a^{2}-38a-70\right){x}+73a^{2}+280a+265$ |
| 9.3-a3 |
9.3-a |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$2.30904$ |
$(a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$1$ |
$90.15182800$ |
1.118174302 |
\( \frac{7245824}{27} a^{2} - \frac{5521408}{27} a - \frac{30269440}{27} \) |
\( \bigl[0\) , \( -a - 1\) , \( a^{2} - a - 3\) , \( 2354 a^{2} + 3433 a - 955\) , \( -230385 a^{2} - 336480 a + 93631\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2354a^{2}+3433a-955\right){x}-230385a^{2}-336480a+93631$ |
| 9.3-b1 |
9.3-b |
$2$ |
$3$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{3} \) |
$2.30904$ |
$(a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.015440669$ |
$148.8699425$ |
0.769789308 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} - 4\) , \( 1\) , \( -a^{2} + a + 5\) , \( -a - 2\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-a^{2}+a+5\right){x}-a-2$ |
| 9.3-b2 |
9.3-b |
$2$ |
$3$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{9} \) |
$2.30904$ |
$(a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.046322007$ |
$49.62331419$ |
0.769789308 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} - 4\) , \( a + 1\) , \( -a^{2} + a + 5\) , \( 67166 a^{2} + 98095 a - 27300\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-a^{2}+a+5\right){x}+67166a^{2}+98095a-27300$ |
| 21.1-a1 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{3} \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$3.290731816$ |
$4.711122302$ |
1.297942974 |
\( -\frac{8240555964142487370725000}{3087} a^{2} + \frac{6270047207340133912728349}{3087} a + \frac{34461538099612476582028741}{3087} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 2\) , \( 332 a^{2} - 279 a - 1437\) , \( 5695 a^{2} - 4506 a - 24104\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(332a^{2}-279a-1437\right){x}+5695a^{2}-4506a-24104$ |
| 21.1-a2 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{12} \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$3.290731816$ |
$2.355561151$ |
1.297942974 |
\( \frac{3503012292854062725880}{90812685325761} a^{2} - \frac{9443468677358551072595}{90812685325761} a + \frac{2028182297320680201637}{90812685325761} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 2\) , \( 22 a^{2} + a - 87\) , \( 161 a^{2} - 10 a - 516\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(22a^{2}+a-87\right){x}+161a^{2}-10a-516$ |
| 21.1-a3 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{6} \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1.645365908$ |
$18.84448920$ |
1.297942974 |
\( -\frac{237252334798556183}{9529569} a^{2} + \frac{180519760374696370}{9529569} a + \frac{992175964399818553}{9529569} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 2\) , \( 17 a^{2} - 19 a - 82\) , \( 118 a^{2} - 90 a - 494\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(17a^{2}-19a-82\right){x}+118a^{2}-90a-494$ |
| 21.1-a4 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{3} \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.822682954$ |
$75.37795683$ |
1.297942974 |
\( \frac{30548955041}{3087} a^{2} + \frac{45139407305}{3087} a - \frac{11527542412}{3087} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 2\) , \( -3 a^{2} - 4 a + 3\) , \( 3 a^{2} - a - 11\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-3a^{2}-4a+3\right){x}+3a^{2}-a-11$ |
| 21.1-a5 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{6} \cdot 7 \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.274227651$ |
$226.1338705$ |
1.297942974 |
\( -\frac{13266793}{5103} a^{2} + \frac{12282170}{5103} a + \frac{59011895}{5103} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( -13 a^{2} - 14 a + 11\) , \( -19 a^{2} - 37 a + 11\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-13a^{2}-14a+11\right){x}-19a^{2}-37a+11$ |
| 21.1-a6 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{24} \cdot 7^{4} \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1.096910605$ |
$7.066683453$ |
1.297942974 |
\( \frac{70673606857848385}{678113317090881} a^{2} - \frac{365854117750060865}{678113317090881} a + \frac{368634701270491612}{678113317090881} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( -108 a^{2} - 54 a + 26\) , \( -3207 a^{2} - 4661 a + 1299\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-108a^{2}-54a+26\right){x}-3207a^{2}-4661a+1299$ |
| 21.1-a7 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{12} \cdot 7^{2} \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$0.548455302$ |
$56.53346762$ |
1.297942974 |
\( \frac{1706346154348}{26040609} a^{2} + \frac{106849738099}{26040609} a + \frac{21071435377}{26040609} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( -148 a^{2} - 149 a + 51\) , \( -1877 a^{2} - 3047 a + 837\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-148a^{2}-149a+51\right){x}-1877a^{2}-3047a+837$ |
| 21.1-a8 |
21.1-a |
$8$ |
$12$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{6} \cdot 7 \) |
$2.65926$ |
$(a+1), (-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1.096910605$ |
$14.13336690$ |
1.297942974 |
\( \frac{112427409603601}{5103} a^{2} + \frac{76711266681583}{5103} a - \frac{24656866452020}{5103} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 3\) , \( -2348 a^{2} - 2404 a + 716\) , \( -111419 a^{2} - 180773 a + 49599\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-2348a^{2}-2404a+716\right){x}-111419a^{2}-180773a+49599$ |
| 21.2-a1 |
21.2-a |
$2$ |
$2$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$50.92285610$ |
1.421118332 |
\( -\frac{986185}{7} a^{2} + \frac{3522404}{21} a + \frac{9282536}{21} \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a + 1\) , \( 4 a^{2} - 3 a - 19\) , \( 11 a^{2} - 9 a - 48\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(4a^{2}-3a-19\right){x}+11a^{2}-9a-48$ |
| 21.2-a2 |
21.2-a |
$2$ |
$2$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{2} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$25.46142805$ |
1.421118332 |
\( \frac{3438672879094}{441} a^{2} - \frac{9283017103189}{441} a + \frac{2023162282732}{441} \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a + 1\) , \( -a^{2} + 7 a - 14\) , \( 9 a^{2} + 6 a - 72\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{2}+7a-14\right){x}+9a^{2}+6a-72$ |
| 21.2-b1 |
21.2-b |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{8} \cdot 7 \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$12.98194311$ |
1.449162813 |
\( -\frac{21518964113006}{567} a^{2} + \frac{16237574878610}{567} a + \frac{90324984331975}{567} \) |
\( \bigl[a + 1\) , \( a^{2} - 4\) , \( a^{2} - a - 3\) , \( -230 a^{2} + 701 a - 325\) , \( 5115 a^{2} - 13492 a + 2235\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-230a^{2}+701a-325\right){x}+5115a^{2}-13492a+2235$ |
| 21.2-b2 |
21.2-b |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{4} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.490971556$ |
1.449162813 |
\( \frac{4635466624220592710}{7203} a^{2} + \frac{6770121579472320070}{7203} a - \frac{1883949379942211203}{7203} \) |
\( \bigl[a + 1\) , \( a^{2} - 4\) , \( a^{2} - a - 3\) , \( -40 a^{2} + 111 a - 25\) , \( -163 a^{2} + 450 a - 119\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-40a^{2}+111a-25\right){x}-163a^{2}+450a-119$ |
| 21.2-b3 |
21.2-b |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$51.92777245$ |
1.449162813 |
\( \frac{404487996}{49} a^{2} + \frac{5401742792}{441} a - \frac{1320930403}{441} \) |
\( \bigl[a + 1\) , \( a^{2} - 4\) , \( a^{2} - a - 3\) , \( -15 a^{2} + 46 a - 15\) , \( 82 a^{2} - 217 a + 42\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-15a^{2}+46a-15\right){x}+82a^{2}-217a+42$ |
| 21.2-b4 |
21.2-b |
$4$ |
$4$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$207.7110898$ |
1.449162813 |
\( \frac{13192}{21} a^{2} + \frac{38144}{21} a + \frac{26797}{21} \) |
\( \bigl[a + 1\) , \( a^{2} - 4\) , \( a^{2} - a - 3\) , \( a + 5\) , \( 5 a^{2} - 10 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(a+5\right){x}+5a^{2}-10a-1$ |
| 21.2-c1 |
21.2-c |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{6} \cdot 7^{3} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.148172137$ |
$107.6953075$ |
1.335986495 |
\( -\frac{8009167}{9261} a^{2} + \frac{14398346}{9261} a + \frac{2140743}{343} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( 1\) , \( -76353 a^{2} - 111512 a + 31036\) , \( -19874923 a^{2} - 29027423 a + 8077577\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-76353a^{2}-111512a+31036\right){x}-19874923a^{2}-29027423a+8077577$ |
| 21.2-c2 |
21.2-c |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{12} \cdot 7^{6} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.074086068$ |
$53.84765375$ |
1.335986495 |
\( \frac{153773153798}{85766121} a^{2} + \frac{122235572063}{85766121} a - \frac{1138766603}{9529569} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( 1\) , \( -384778 a^{2} - 561967 a + 156386\) , \( 261324115 a^{2} + 381665141 a - 106207520\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-384778a^{2}-561967a+156386\right){x}+261324115a^{2}+381665141a-106207520$ |
| 21.2-c3 |
21.2-c |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{9} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.444516411$ |
$3.988715092$ |
1.335986495 |
\( \frac{287246227651866040}{40353607} a^{2} + \frac{1258627780088614091}{121060821} a - \frac{350136466697422879}{121060821} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( 1\) , \( -5836753 a^{2} - 8524602 a + 2372181\) , \( -16506416515 a^{2} - 24107701706 a + 6708548603\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-5836753a^{2}-8524602a+2372181\right){x}-16506416515a^{2}-24107701706a+6708548603$ |
| 21.2-c4 |
21.2-c |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{18} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.222258205$ |
$1.994357546$ |
1.335986495 |
\( \frac{14027125943428555215709}{14655722381194041} a^{2} - \frac{36168481134011244336109}{14655722381194041} a + \frac{7841170043827907289922}{14655722381194041} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( 1\) , \( -5842223 a^{2} - 8532557 a + 2374396\) , \( -16473996726 a^{2} - 24060352337 a + 6695372505\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-5842223a^{2}-8532557a+2374396\right){x}-16473996726a^{2}-24060352337a+6695372505$ |
| 21.2-c5 |
21.2-c |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$0.444516411$ |
$323.0859225$ |
1.335986495 |
\( -\frac{943960905697}{21} a^{2} + \frac{718246160195}{21} a + \frac{1315867845900}{7} \) |
\( \bigl[a\) , \( 0\) , \( a^{2} - a - 2\) , \( -5 a^{2} - 19 a - 16\) , \( 23 a^{2} + 53 a + 22\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-5a^{2}-19a-16\right){x}+23a^{2}+53a+22$ |
| 21.2-c6 |
21.2-c |
$6$ |
$18$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{2} \) |
$2.65926$ |
$(-a^2+a+3), (-a^2+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.222258205$ |
$161.5429612$ |
1.335986495 |
\( \frac{117277617455633}{441} a^{2} + \frac{170011024942118}{441} a - \frac{15786571572883}{147} \) |
\( \bigl[a\) , \( 0\) , \( a^{2} - a - 2\) , \( -155 a^{2} - 234 a + 44\) , \( 2257 a^{2} + 3312 a - 885\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-155a^{2}-234a+44\right){x}+2257a^{2}+3312a-885$ |
| 24.1-a1 |
24.1-a |
$1$ |
$1$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{3} \cdot 3^{11} \) |
$2.71911$ |
$(a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$28.99484901$ |
1.618334659 |
\( -\frac{22667068225}{354294} a^{2} + \frac{28943473325}{354294} a - \frac{5325723943}{354294} \) |
\( \bigl[a^{2} - 3\) , \( 1\) , \( a^{2} - 2\) , \( -6 a^{2} + 8 a + 14\) , \( 4 a^{2} - 9 a - 5\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}+\left(-6a^{2}+8a+14\right){x}+4a^{2}-9a-5$ |
| 24.1-b1 |
24.1-b |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{9} \cdot 3^{3} \) |
$2.71911$ |
$(a+1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$49.51356978$ |
0.921192657 |
\( -\frac{1507}{216} a^{2} - \frac{19555}{216} a + \frac{604}{27} \) |
\( \bigl[a^{2} - a - 3\) , \( -a - 1\) , \( a\) , \( -319 a^{2} - 476 a + 133\) , \( -18887 a^{2} - 27538 a + 7665\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-319a^{2}-476a+133\right){x}-18887a^{2}-27538a+7665$ |
| 24.1-b2 |
24.1-b |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{27} \cdot 3 \) |
$2.71911$ |
$(a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$1.833835918$ |
0.921192657 |
\( -\frac{74235981265}{1536} a^{2} - \frac{107726342491}{1536} a + \frac{30004684793}{1536} \) |
\( \bigl[a^{2} - a - 3\) , \( -a - 1\) , \( a\) , \( -37934 a^{2} - 54796 a + 15273\) , \( -8554625 a^{2} - 12485773 a + 3474792\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-37934a^{2}-54796a+15273\right){x}-8554625a^{2}-12485773a+3474792$ |
| 24.1-b3 |
24.1-b |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( - 2^{3} \cdot 3^{9} \) |
$2.71911$ |
$(a+1), (2)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$148.5407093$ |
0.921192657 |
\( \frac{78543279647}{39366} a^{2} - \frac{29879044112}{19683} a - \frac{164231943596}{19683} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 3\) , \( a\) , \( -3 a^{2} + 4 a + 9\) , \( 5 a^{2} - 16 a + 8\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-3a^{2}+4a+9\right){x}+5a^{2}-16a+8$ |
| 27.1-a1 |
27.1-a |
$4$ |
$6$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{15} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$91.01420459$ |
1.693305843 |
\( -\frac{118553573237}{729} a^{2} + \frac{90107483119}{729} a + \frac{496039265614}{729} \) |
\( \bigl[a^{2} - 2\) , \( a + 1\) , \( a^{2} - a - 3\) , \( -1100 a^{2} + 2991 a - 682\) , \( 40475 a^{2} - 109276 a + 23851\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1100a^{2}+2991a-682\right){x}+40475a^{2}-109276a+23851$ |
| 27.1-a2 |
27.1-a |
$4$ |
$6$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{12} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$182.0284091$ |
1.693305843 |
\( \frac{171076}{27} a^{2} - \frac{105203}{27} a - \frac{679445}{27} \) |
\( \bigl[a^{2} - 2\) , \( a + 1\) , \( a^{2} - a - 3\) , \( -45 a^{2} + 131 a - 32\) , \( 1041 a^{2} - 2802 a + 608\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-45a^{2}+131a-32\right){x}+1041a^{2}-2802a+608$ |
| 27.1-a3 |
27.1-a |
$4$ |
$6$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{5} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$30.33806819$ |
1.693305843 |
\( \frac{9005094731866}{9} a^{2} + \frac{13151984680969}{9} a - \frac{3659856471863}{9} \) |
\( \bigl[1\) , \( a^{2} - 2 a - 3\) , \( 1\) , \( -14 a^{2} + 36 a - 6\) , \( -66 a^{2} + 178 a - 39\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-14a^{2}+36a-6\right){x}-66a^{2}+178a-39$ |
| 27.1-a4 |
27.1-a |
$4$ |
$6$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{4} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$60.67613639$ |
1.693305843 |
\( \frac{986359}{3} a^{2} + \frac{1442698}{3} a - \frac{396629}{3} \) |
\( \bigl[1\) , \( a^{2} - 2 a - 3\) , \( 1\) , \( a^{2} - 4 a + 4\) , \( -2 a^{2} + 6 a - 3\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(a^{2}-4a+4\right){x}-2a^{2}+6a-3$ |
| 27.1-b1 |
27.1-b |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{9} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 1 \) |
$1$ |
$20.84190847$ |
1.163281892 |
\( \frac{7245824}{27} a^{2} - \frac{5521408}{27} a - \frac{30269440}{27} \) |
\( \bigl[0\) , \( -a + 1\) , \( a^{2} - a - 2\) , \( 76 a^{2} + 83 a - 24\) , \( -1175 a^{2} - 1748 a + 484\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(76a^{2}+83a-24\right){x}-1175a^{2}-1748a+484$ |
| 27.1-b2 |
27.1-b |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{7} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$2.315767607$ |
1.163281892 |
\( \frac{1074386927992107008}{3} a^{2} - \frac{817476003654639616}{3} a - \frac{4493025253858410496}{3} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 122 a^{2} - 92 a - 516\) , \( 1124 a^{2} - 848 a - 4709\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(122a^{2}-92a-516\right){x}+1124a^{2}-848a-4709$ |
| 27.1-b3 |
27.1-b |
$3$ |
$9$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{15} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$20.84190847$ |
1.163281892 |
\( -\frac{693022720}{19683} a^{2} - \frac{855752704}{19683} a + \frac{246431744}{19683} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -409 a^{2} + 1027 a - 222\) , \( 8470 a^{2} - 23221 a + 5068\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-409a^{2}+1027a-222\right){x}+8470a^{2}-23221a+5068$ |
| 27.1-c1 |
27.1-c |
$4$ |
$6$ |
3.3.321.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{9} \) |
$2.77301$ |
$(a+1), (-a^2+a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.033576913$ |
$249.6833541$ |
1.403780161 |
\( -\frac{118553573237}{729} a^{2} + \frac{90107483119}{729} a + \frac{496039265614}{729} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - a - 4\) , \( a^{2} - 2\) , \( -141 a^{2} + 401 a - 138\) , \( 1980 a^{2} - 5420 a + 1345\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-141a^{2}+401a-138\right){x}+1980a^{2}-5420a+1345$ |
*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.
