| 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 |
| 9.1-a1 |
9.1-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$11.13013$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.031756620$ |
$417.2391168$ |
2.240051449 |
\( -\frac{165708334003432}{729} a^{3} + \frac{75874242565736}{243} a^{2} + \frac{743493901552784}{729} a - \frac{443503509186329}{729} \) |
\( \bigl[a^{3} - 6 a - 2\) , \( a^{2} - 3\) , \( a^{2} - a - 2\) , \( 6 a^{3} - 17 a^{2} + 4 a + 1\) , \( -20 a^{3} + 60 a^{2} - 15 a - 5\bigr] \) |
${y}^2+\left(a^{3}-6a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(6a^{3}-17a^{2}+4a+1\right){x}-20a^{3}+60a^{2}-15a-5$ |
| 9.1-a2 |
9.1-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$11.13013$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.010585540$ |
$417.2391168$ |
2.240051449 |
\( -\frac{1640215}{729} a^{3} + \frac{2483000}{729} a^{2} + \frac{2388430}{243} a - \frac{4353061}{729} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{2} + a + 2\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 9 a^{3} - 4 a^{2} - 47 a - 17\) , \( 1300 a^{3} - 595 a^{2} - 6823 a - 2400\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(9a^{3}-4a^{2}-47a-17\right){x}+1300a^{3}-595a^{2}-6823a-2400$ |
| 9.1-b1 |
9.1-b |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$11.13013$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.031756620$ |
$417.2391168$ |
2.240051449 |
\( -\frac{23133374770600}{729} a^{3} - \frac{38781018923176}{729} a^{2} + \frac{3957644514448}{243} a + \frac{8643418515383}{729} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 22 a^{3} - 11 a^{2} - 114 a - 34\) , \( -65 a^{3} + 30 a^{2} + 341 a + 119\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(22a^{3}-11a^{2}-114a-34\right){x}-65a^{3}+30a^{2}+341a+119$ |
| 9.1-b2 |
9.1-b |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$11.13013$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.010585540$ |
$417.2391168$ |
2.240051449 |
\( -\frac{193000}{729} a^{3} - \frac{216595}{243} a^{2} + \frac{167570}{729} a + \frac{1308079}{729} \) |
\( \bigl[a\) , \( a - 1\) , \( a^{3} - 5 a - 1\) , \( 2 a^{3} - 3 a^{2} - 10 a + 7\) , \( -a^{3} + a^{2} + 5 a - 6\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a^{3}-3a^{2}-10a+7\right){x}-a^{3}+a^{2}+5a-6$ |
| 16.1-a1 |
16.1-a |
$6$ |
$45$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{60} \) |
$11.96010$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3, 5$ |
3Cs, 5B.1.2 |
$1$ |
\( 3 \cdot 5 \) |
$1.873889626$ |
$1.304759140$ |
1.550045042 |
\( -\frac{1680914269}{32768} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -75 a^{3} + 75 a^{2} + 300 a - 175\) , \( -433 a^{3} + 433 a^{2} + 1732 a - 999\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-75a^{3}+75a^{2}+300a-175\right){x}-433a^{3}+433a^{2}+1732a-999$ |
| 16.1-a2 |
16.1-a |
$6$ |
$45$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$11.96010$ |
$(2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3, 5$ |
3Cs, 5B.1.1 |
$1$ |
\( 3 \) |
$0.374777925$ |
$815.4744629$ |
1.550045042 |
\( \frac{1331}{8} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}$ |
| 16.1-a3 |
16.1-a |
$6$ |
$45$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{20} \) |
$11.96010$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 5 \) |
$5.621668880$ |
$1.304759140$ |
1.550045042 |
\( -\frac{1250637664527933}{32} a^{3} + \frac{1250637664527933}{32} a^{2} + \frac{1250637664527933}{8} a - \frac{719984486365939}{8} \) |
\( \bigl[1\) , \( 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -29 a^{3} + 29 a^{2} + 116 a - 27\) , \( -52 a^{3} + 52 a^{2} + 208 a - 158\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+{x}^{2}+\left(-29a^{3}+29a^{2}+116a-27\right){x}-52a^{3}+52a^{2}+208a-158$ |
| 16.1-a4 |
16.1-a |
$6$ |
$45$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$11.96010$ |
$(2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1.124333776$ |
$815.4744629$ |
1.550045042 |
\( \frac{461373}{2} a^{3} - \frac{461373}{2} a^{2} - 922746 a - \frac{601423}{2} \) |
\( \bigl[1\) , \( 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 4 a - 2\) , \( -a^{3} + a^{2} + 4 a\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+{x}^{2}+\left(a^{3}-a^{2}-4a-2\right){x}-a^{3}+a^{2}+4a$ |
| 16.1-a5 |
16.1-a |
$6$ |
$45$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$11.96010$ |
$(2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1.124333776$ |
$815.4744629$ |
1.550045042 |
\( -\frac{461373}{2} a^{3} + \frac{461373}{2} a^{2} + 922746 a - 531398 \) |
\( \bigl[1\) , \( 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -2 a^{3} + 2 a^{2} + 8 a - 4\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+{x}^{2}+\left(-2a^{3}+2a^{2}+8a-4\right){x}$ |
| 16.1-a6 |
16.1-a |
$6$ |
$45$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{20} \) |
$11.96010$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 5 \) |
$5.621668880$ |
$1.304759140$ |
1.550045042 |
\( \frac{1250637664527933}{32} a^{3} - \frac{1250637664527933}{32} a^{2} - \frac{1250637664527933}{8} a - \frac{1629300280935823}{32} \) |
\( \bigl[1\) , \( 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 28 a^{3} - 28 a^{2} - 112 a + 1\) , \( 51 a^{3} - 51 a^{2} - 204 a - 107\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+{x}^{2}+\left(28a^{3}-28a^{2}-112a+1\right){x}+51a^{3}-51a^{2}-204a-107$ |
| 16.1-b1 |
16.1-b |
$1$ |
$1$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$11.96010$ |
$(2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.001558008$ |
$3327.305256$ |
1.752794975 |
\( -\frac{25236809}{4} a^{3} + \frac{25236809}{4} a^{2} + 25236809 a - 14592162 \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -8 a^{3} - 7 a^{2} + 17 a + 3\) , \( 23 a^{3} + 42 a^{2} - 6 a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-8a^{3}-7a^{2}+17a+3\right){x}+23a^{3}+42a^{2}-6a-8$ |
| 27.1-a1 |
27.1-a |
$1$ |
$1$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{9} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$271.4803164$ |
2.868514567 |
\( \frac{49350811760}{2187} a^{3} - \frac{7534140622}{729} a^{2} - \frac{258998922979}{2187} a - \frac{91041478481}{2187} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 4\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} - 2 a^{2} + 6 a + 11\) , \( -2 a^{2} + a + 10\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+4\right){x}^{2}+\left(-a^{3}-2a^{2}+6a+11\right){x}-2a^{2}+a+10$ |
| 27.1-b1 |
27.1-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{10} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$79.69139537$ |
1.684070001 |
\( -\frac{6169601180250083}{6561} a^{3} + \frac{5851628130318007}{2187} a^{2} - \frac{1531990913063891}{6561} a - \frac{3344928807838108}{6561} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 6 a\) , \( a\) , \( -5641 a^{3} + 7744 a^{2} + 25316 a - 15090\) , \( -333527 a^{3} + 458136 a^{2} + 1496429 a - 892633\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+6a\right){x}^{2}+\left(-5641a^{3}+7744a^{2}+25316a-15090\right){x}-333527a^{3}+458136a^{2}+1496429a-892633$ |
| 27.1-b2 |
27.1-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{10} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$19.92284884$ |
1.684070001 |
\( \frac{1468518624302691046387}{81} a^{3} - \frac{2017208248936831882645}{81} a^{2} - \frac{6588893972860435233565}{81} a + \frac{3930358536234082886572}{81} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 2 a^{2} - 5 a + 3\) , \( a + 1\) , \( 1643 a^{3} - 760 a^{2} - 8613 a - 3011\) , \( 53461 a^{3} - 24486 a^{2} - 280601 a - 98625\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+3\right){x}^{2}+\left(1643a^{3}-760a^{2}-8613a-3011\right){x}+53461a^{3}-24486a^{2}-280601a-98625$ |
| 27.1-b3 |
27.1-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$318.7655815$ |
1.684070001 |
\( \frac{170125558651}{81} a^{3} - \frac{77886980204}{27} a^{2} - \frac{763405603094}{81} a + \frac{455367716093}{81} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 5 a - 1\) , \( a^{2} - 3\) , \( 3412 a^{3} - 1563 a^{2} - 17905 a - 6289\) , \( -128049 a^{3} + 58640 a^{2} + 672031 a + 236227\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(3412a^{3}-1563a^{2}-17905a-6289\right){x}-128049a^{3}+58640a^{2}+672031a+236227$ |
| 27.1-b4 |
27.1-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{4} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1275.062326$ |
1.684070001 |
\( \frac{205268}{9} a^{3} - \frac{94052}{3} a^{2} - \frac{913759}{9} a + \frac{551527}{9} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 2\) , \( -a^{3} + 5 a + 3\) , \( 1\) , \( 5 a^{3} - 18 a^{2} + 9 a + 11\) , \( -35 a^{3} + 97 a^{2} - 4 a - 14\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+5a+3\right){x}^{2}+\left(5a^{3}-18a^{2}+9a+11\right){x}-35a^{3}+97a^{2}-4a-14$ |
| 27.1-c1 |
27.1-c |
$2$ |
$2$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{20} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$43.69579399$ |
0.461698377 |
\( -\frac{282789106954879031}{43046721} a^{3} + \frac{267976036291197814}{14348907} a^{2} - \frac{68909997373628906}{43046721} a - \frac{152727633851934643}{43046721} \) |
\( \bigl[a^{2} - a - 3\) , \( a + 1\) , \( a^{3} - 5 a - 1\) , \( -12 a^{3} + 13 a^{2} + 47 a - 36\) , \( -101 a^{3} - 36 a^{2} + 225 a - 98\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-12a^{3}+13a^{2}+47a-36\right){x}-101a^{3}-36a^{2}+225a-98$ |
| 27.1-c2 |
27.1-c |
$2$ |
$2$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$12.76851$ |
$(a^3-a^2-5a+2), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$87.39158798$ |
0.461698377 |
\( \frac{377132050820}{6561} a^{3} - \frac{172716011140}{2187} a^{2} - \frac{1691366984851}{6561} a + \frac{1008964616863}{6561} \) |
\( \bigl[a\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{3} - a^{2} - 5 a + 2\) , \( 14 a^{3} - 7 a^{2} - 74 a - 21\) , \( -51 a^{3} + 22 a^{2} + 272 a + 91\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-5a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(14a^{3}-7a^{2}-74a-21\right){x}-51a^{3}+22a^{2}+272a+91$ |
| 27.2-a1 |
27.2-a |
$1$ |
$1$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{9} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$271.4803164$ |
2.868514567 |
\( \frac{14503525715}{2187} a^{3} - \frac{41251915609}{2187} a^{2} + \frac{1193857693}{729} a + \frac{7853184944}{2187} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 7 a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} + 3 a^{2} - 2\) , \( -a^{3} + 3 a^{2} + 3 a - 1\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(-a^{3}+3a^{2}-2\right){x}-a^{3}+3a^{2}+3a-1$ |
| 27.2-b1 |
27.2-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{10} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$79.69139537$ |
1.684070001 |
\( -\frac{20994713603610368}{6561} a^{3} + \frac{9609430392906430}{6561} a^{2} + \frac{36729750016168565}{2187} a + \frac{38756374822321297}{6561} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{2} + 2 a + 4\) , \( a^{2} - 2\) , \( 520 a^{3} - 257 a^{2} - 2751 a - 962\) , \( -9030 a^{3} + 4048 a^{2} + 47165 a + 16587\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(520a^{3}-257a^{2}-2751a-962\right){x}-9030a^{3}+4048a^{2}+47165a+16587$ |
| 27.2-b2 |
27.2-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{10} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$19.92284884$ |
1.684070001 |
\( \frac{205009524018879162112}{81} a^{3} + \frac{343680100615261674146}{81} a^{2} - \frac{105218620425845600431}{81} a - \frac{76598687220433178993}{81} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( -a^{3} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -22 a^{3} + 24 a^{2} + 119 a - 63\) , \( -28559 a^{3} + 39215 a^{2} + 128182 a - 76453\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-22a^{3}+24a^{2}+119a-63\right){x}-28559a^{3}+39215a^{2}+128182a-76453$ |
| 27.2-b3 |
27.2-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{8} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$318.7655815$ |
1.684070001 |
\( \frac{23686808200}{81} a^{3} + \frac{39848573761}{81} a^{2} - \frac{3947954770}{27} a - \frac{8747944163}{81} \) |
\( \bigl[1\) , \( -a^{3} + 5 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -4 a^{3} - 8 a^{2} + 58 a - 23\) , \( -59 a^{3} + 129 a^{2} + 109 a - 86\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-4a^{3}-8a^{2}+58a-23\right){x}-59a^{3}+129a^{2}+109a-86$ |
| 27.2-b4 |
27.2-b |
$4$ |
$4$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{4} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1275.062326$ |
1.684070001 |
\( \frac{35693}{9} a^{3} + \frac{41195}{9} a^{2} - 5565 a - \frac{2488}{9} \) |
\( \bigl[a + 1\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a\) , \( 23 a^{3} - 13 a^{2} - 120 a - 31\) , \( -139 a^{3} + 65 a^{2} + 728 a + 249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(23a^{3}-13a^{2}-120a-31\right){x}-139a^{3}+65a^{2}+728a+249$ |
| 27.2-c1 |
27.2-c |
$2$ |
$2$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{20} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$43.69579399$ |
0.461698377 |
\( -\frac{961716530229309650}{43046721} a^{3} + \frac{440577528310595239}{43046721} a^{2} + \frac{1682310848703461210}{14348907} a + \frac{1774039952467206793}{43046721} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 5 a - 2\) , \( -11 a^{3} + 10 a^{2} + 42 a - 30\) , \( -415 a^{3} + 552 a^{2} + 1836 a - 1097\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-11a^{3}+10a^{2}+42a-30\right){x}-415a^{3}+552a^{2}+1836a-1097$ |
| 27.2-c2 |
27.2-c |
$2$ |
$2$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$12.76851$ |
$(a+1), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$87.39158798$ |
0.461698377 |
\( \frac{53277286649}{6561} a^{3} + \frac{87738695951}{6561} a^{2} - \frac{10090121675}{2187} a - \frac{19970060308}{6561} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( -a^{3} + 5 a + 2\) , \( a + 1\) , \( 3 a^{3} - 10 a^{2} + 6 a + 3\) , \( -12 a^{3} + 41 a^{2} - 20 a - 15\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(3a^{3}-10a^{2}+6a+3\right){x}-12a^{3}+41a^{2}-20a-15$ |
| 27.3-a1 |
27.3-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$157.8457147$ |
2.223771938 |
\( -\frac{193000}{729} a^{3} - \frac{216595}{243} a^{2} + \frac{167570}{729} a + \frac{1308079}{729} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a\) , \( a^{2} - 2\) , \( 5 a^{3} - 8 a^{2} - 21 a + 20\) , \( -6 a^{3} + 7 a^{2} + 28 a - 10\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(5a^{3}-8a^{2}-21a+20\right){x}-6a^{3}+7a^{2}+28a-10$ |
| 27.3-a2 |
27.3-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$5.846137584$ |
2.223771938 |
\( -\frac{23133374770600}{729} a^{3} - \frac{38781018923176}{729} a^{2} + \frac{3957644514448}{243} a + \frac{8643418515383}{729} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 2 a^{2} - 4 a + 5\) , \( a + 1\) , \( -686 a^{3} + 941 a^{2} + 3077 a - 1829\) , \( -16100 a^{3} + 22112 a^{2} + 72246 a - 43093\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+5\right){x}^{2}+\left(-686a^{3}+941a^{2}+3077a-1829\right){x}-16100a^{3}+22112a^{2}+72246a-43093$ |
| 27.3-b1 |
27.3-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{15} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$114.8079824$ |
1.213083785 |
\( \frac{10706918413258}{729} a^{3} - \frac{4900654674128}{243} a^{2} - \frac{48056868655580}{729} a + \frac{28663984181117}{729} \) |
\( \bigl[a^{3} - 6 a - 2\) , \( a^{3} - 5 a - 1\) , \( 1\) , \( -129 a^{3} + 167 a^{2} + 603 a - 350\) , \( -1245 a^{3} + 1722 a^{2} + 5516 a - 3297\bigr] \) |
${y}^2+\left(a^{3}-6a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-129a^{3}+167a^{2}+603a-350\right){x}-1245a^{3}+1722a^{2}+5516a-3297$ |
| 27.3-b2 |
27.3-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{12} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$229.6159649$ |
1.213083785 |
\( -\frac{1854092}{27} a^{3} + \frac{787504}{9} a^{2} + \frac{8414068}{27} a - \frac{3896581}{27} \) |
\( \bigl[a^{3} - 6 a - 2\) , \( a^{3} - 5 a - 1\) , \( 1\) , \( -9 a^{3} + 12 a^{2} + 43 a - 20\) , \( -21 a^{3} + 29 a^{2} + 95 a - 56\bigr] \) |
${y}^2+\left(a^{3}-6a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-9a^{3}+12a^{2}+43a-20\right){x}-21a^{3}+29a^{2}+95a-56$ |
| 27.3-b3 |
27.3-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{21} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$114.8079824$ |
1.213083785 |
\( -\frac{456425695791973076}{387420489} a^{3} + \frac{69673390319987974}{129140163} a^{2} + \frac{2395433686628442655}{387420489} a + \frac{842031922181564789}{387420489} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 6 a - 1\) , \( -3 a^{3} + 29 a - 10\) , \( -4 a^{3} + 2 a^{2} + 34 a - 23\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-3a^{3}+29a-10\right){x}-4a^{3}+2a^{2}+34a-23$ |
| 27.3-b4 |
27.3-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{12} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$229.6159649$ |
1.213083785 |
\( -\frac{870157184}{19683} a^{3} + \frac{575012737}{6561} a^{2} + \frac{1508587678}{19683} a + \frac{287871479}{19683} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 6 a - 1\) , \( 2 a^{3} - 5 a^{2} + 4 a + 5\) , \( 8 a^{3} - 18 a^{2} - 2 a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(2a^{3}-5a^{2}+4a+5\right){x}+8a^{3}-18a^{2}-2a$ |
| 27.3-c1 |
27.3-c |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.030281455$ |
$496.8989375$ |
2.543803197 |
\( -\frac{1640215}{729} a^{3} + \frac{2483000}{729} a^{2} + \frac{2388430}{243} a - \frac{4353061}{729} \) |
\( \bigl[a^{2} - 2\) , \( a\) , \( a^{3} - 5 a - 1\) , \( -6 a^{3} + 21 a^{2} - 5 a - 5\) , \( -3 a^{3} + 14 a^{2} - 5 a - 4\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+a{x}^{2}+\left(-6a^{3}+21a^{2}-5a-5\right){x}-3a^{3}+14a^{2}-5a-4$ |
| 27.3-c2 |
27.3-c |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.090844366$ |
$165.6329791$ |
2.543803197 |
\( -\frac{165708334003432}{729} a^{3} + \frac{75874242565736}{243} a^{2} + \frac{743493901552784}{729} a - \frac{443503509186329}{729} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -4 a^{3} + 4 a^{2} + 21 a - 7\) , \( 17 a^{3} - 29 a^{2} - 55 a + 28\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(-4a^{3}+4a^{2}+21a-7\right){x}+17a^{3}-29a^{2}-55a+28$ |
| 27.3-d1 |
27.3-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{6} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$268.7710361$ |
1.419943888 |
\( -\frac{1854092}{27} a^{3} + \frac{787504}{9} a^{2} + \frac{8414068}{27} a - \frac{3896581}{27} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - a^{2} - 6 a + 1\) , \( a^{3} - 5 a - 2\) , \( 7 a^{3} - 5 a^{2} - 36 a - 4\) , \( 11 a^{3} + a^{2} - 41 a - 10\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+1\right){x}^{2}+\left(7a^{3}-5a^{2}-36a-4\right){x}+11a^{3}+a^{2}-41a-10$ |
| 27.3-d2 |
27.3-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{27} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$134.3855180$ |
1.419943888 |
\( -\frac{456425695791973076}{387420489} a^{3} + \frac{69673390319987974}{129140163} a^{2} + \frac{2395433686628442655}{387420489} a + \frac{842031922181564789}{387420489} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( a^{3} - 6 a - 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( -15 a^{3} + 18 a^{2} + 68 a - 39\) , \( -41 a^{3} + 51 a^{2} + 184 a - 96\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(-15a^{3}+18a^{2}+68a-39\right){x}-41a^{3}+51a^{2}+184a-96$ |
| 27.3-d3 |
27.3-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{18} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$268.7710361$ |
1.419943888 |
\( -\frac{870157184}{19683} a^{3} + \frac{575012737}{6561} a^{2} + \frac{1508587678}{19683} a + \frac{287871479}{19683} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 1\) , \( a^{3} - 6 a - 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( -2 a^{2} + 3 a + 6\) , \( -2 a^{3} + 4 a^{2} + 7 a - 10\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+1\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(-2a^{2}+3a+6\right){x}-2a^{3}+4a^{2}+7a-10$ |
| 27.3-d4 |
27.3-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{9} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$134.3855180$ |
1.419943888 |
\( \frac{10706918413258}{729} a^{3} - \frac{4900654674128}{243} a^{2} - \frac{48056868655580}{729} a + \frac{28663984181117}{729} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a + 1\) , \( a^{3} - a^{2} - 5 a + 1\) , \( 99 a^{3} - 44 a^{2} - 520 a - 188\) , \( -423 a^{3} + 195 a^{2} + 2219 a + 774\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-5a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(99a^{3}-44a^{2}-520a-188\right){x}-423a^{3}+195a^{2}+2219a+774$ |
| 27.4-a1 |
27.4-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$5.846137584$ |
2.223771938 |
\( -\frac{165708334003432}{729} a^{3} + \frac{75874242565736}{243} a^{2} + \frac{743493901552784}{729} a - \frac{443503509186329}{729} \) |
\( \bigl[a\) , \( -a^{3} + 5 a + 1\) , \( a^{2} - a - 2\) , \( -43 a^{3} + 17 a^{2} + 224 a + 79\) , \( 9061 a^{3} - 4153 a^{2} - 47554 a - 16716\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-43a^{3}+17a^{2}+224a+79\right){x}+9061a^{3}-4153a^{2}-47554a-16716$ |
| 27.4-a2 |
27.4-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$157.8457147$ |
2.223771938 |
\( -\frac{1640215}{729} a^{3} + \frac{2483000}{729} a^{2} + \frac{2388430}{243} a - \frac{4353061}{729} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 2 a^{3} - 13 a - 5\) , \( -123 a^{3} + 57 a^{2} + 644 a + 226\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(2a^{3}-13a-5\right){x}-123a^{3}+57a^{2}+644a+226$ |
| 27.4-b1 |
27.4-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{15} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$114.8079824$ |
1.213083785 |
\( \frac{1482677801584}{729} a^{3} + \frac{2512367807542}{729} a^{2} - \frac{233838734596}{243} a - \frac{535484476187}{729} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - 3\) , \( 2233 a^{3} - 1024 a^{2} - 11720 a - 4120\) , \( -80805 a^{3} + 37004 a^{2} + 424082 a + 149069\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(2233a^{3}-1024a^{2}-11720a-4120\right){x}-80805a^{3}+37004a^{2}+424082a+149069$ |
| 27.4-b2 |
27.4-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{12} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$229.6159649$ |
1.213083785 |
\( -\frac{347972}{27} a^{3} - \frac{160448}{27} a^{2} + \frac{131396}{9} a + \frac{151639}{27} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - 3\) , \( 138 a^{3} - 64 a^{2} - 725 a - 255\) , \( -840 a^{3} + 385 a^{2} + 4407 a + 1547\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(138a^{3}-64a^{2}-725a-255\right){x}-840a^{3}+385a^{2}+4407a+1547$ |
| 27.4-b3 |
27.4-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{21} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$114.8079824$ |
1.213083785 |
\( -\frac{134100317163431879}{387420489} a^{3} + \frac{381505841995441033}{387420489} a^{2} - \frac{11109878268940945}{129140163} a - \frac{72670323349939784}{387420489} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{2} - 3\) , \( 14 a^{3} - 12 a^{2} - 74 a - 6\) , \( 11 a^{3} - 9 a^{2} - 63 a - 17\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(14a^{3}-12a^{2}-74a-6\right){x}+11a^{3}-9a^{2}-63a-17$ |
| 27.4-b4 |
27.4-b |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{12} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$229.6159649$ |
1.213083785 |
\( -\frac{1987317215}{19683} a^{3} + \frac{1132436188}{19683} a^{2} + \frac{3307103306}{6561} a + \frac{3445116583}{19683} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{2} - 3\) , \( 14 a^{3} - 12 a^{2} - 69 a + 4\) , \( 27 a^{3} - 17 a^{2} - 139 a - 30\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(14a^{3}-12a^{2}-69a+4\right){x}+27a^{3}-17a^{2}-139a-30$ |
| 27.4-c1 |
27.4-c |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.090844366$ |
$165.6329791$ |
2.543803197 |
\( -\frac{23133374770600}{729} a^{3} - \frac{38781018923176}{729} a^{2} + \frac{3957644514448}{243} a + \frac{8643418515383}{729} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 6 a + 3\) , \( a^{2} - a - 2\) , \( -a^{2} - 3 a - 1\) , \( 18 a^{3} - 10 a^{2} - 89 a - 31\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+6a+3\right){x}^{2}+\left(-a^{2}-3a-1\right){x}+18a^{3}-10a^{2}-89a-31$ |
| 27.4-c2 |
27.4-c |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{14} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.030281455$ |
$496.8989375$ |
2.543803197 |
\( -\frac{193000}{729} a^{3} - \frac{216595}{243} a^{2} + \frac{167570}{729} a + \frac{1308079}{729} \) |
\( \bigl[a^{3} - 6 a - 2\) , \( a^{3} - a^{2} - 4 a\) , \( a^{3} - a^{2} - 5 a + 2\) , \( -25 a^{3} + 11 a^{2} + 132 a + 47\) , \( -62 a^{3} + 29 a^{2} + 325 a + 111\bigr] \) |
${y}^2+\left(a^{3}-6a-2\right){x}{y}+\left(a^{3}-a^{2}-5a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(-25a^{3}+11a^{2}+132a+47\right){x}-62a^{3}+29a^{2}+325a+111$ |
| 27.4-d1 |
27.4-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{6} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$268.7710361$ |
1.419943888 |
\( -\frac{347972}{27} a^{3} - \frac{160448}{27} a^{2} + \frac{131396}{9} a + \frac{151639}{27} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( a^{2} - a - 2\) , \( 4 a^{3} - 6 a^{2} - 14 a + 4\) , \( 29 a^{3} - 45 a^{2} - 114 a + 69\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a\right){x}^{2}+\left(4a^{3}-6a^{2}-14a+4\right){x}+29a^{3}-45a^{2}-114a+69$ |
| 27.4-d2 |
27.4-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{27} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$134.3855180$ |
1.419943888 |
\( -\frac{134100317163431879}{387420489} a^{3} + \frac{381505841995441033}{387420489} a^{2} - \frac{11109878268940945}{129140163} a - \frac{72670323349939784}{387420489} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( -4 a^{3} + 12 a - 8\) , \( -15 a^{3} + 2 a^{2} + 34 a - 27\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(-4a^{3}+12a-8\right){x}-15a^{3}+2a^{2}+34a-27$ |
| 27.4-d3 |
27.4-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{18} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$268.7710361$ |
1.419943888 |
\( -\frac{1987317215}{19683} a^{3} + \frac{1132436188}{19683} a^{2} + \frac{3307103306}{6561} a + \frac{3445116583}{19683} \) |
\( \bigl[a\) , \( -a^{2} + 2 a + 4\) , \( 0\) , \( a^{3} - 3 a + 2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(a^{3}-3a+2\right){x}$ |
| 27.4-d4 |
27.4-d |
$4$ |
$6$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{9} \) |
$12.76851$ |
$(a+1), (a^3-a^2-5a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$134.3855180$ |
1.419943888 |
\( \frac{1482677801584}{729} a^{3} + \frac{2512367807542}{729} a^{2} - \frac{233838734596}{243} a - \frac{535484476187}{729} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 5 a + 2\) , \( a\) , \( 30 a^{3} - 85 a^{2} + 4 a + 18\) , \( -124 a^{3} + 352 a^{2} - 31 a - 67\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+2\right){x}^{2}+\left(30a^{3}-85a^{2}+4a+18\right){x}-124a^{3}+352a^{2}-31a-67$ |
| 27.5-a1 |
27.5-a |
$2$ |
$3$ |
4.4.8957.1 |
$4$ |
$[4, 0]$ |
27.5 |
\( 3^{3} \) |
\( - 3^{9} \) |
$12.76851$ |
$(a^3-a^2-5a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$277.4567899$ |
2.931663164 |
\( 106430 a^{3} - 290970 a^{2} - 12384 a + 76945 \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 6 a + 3\) , \( a^{3} - 5 a - 2\) , \( -3 a^{3} + 2 a^{2} + 16 a + 3\) , \( -a^{3} + a^{2} + 7 a + 3\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(-a^{3}+6a+3\right){x}^{2}+\left(-3a^{3}+2a^{2}+16a+3\right){x}-a^{3}+a^{2}+7a+3$ |
*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.
