| 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 |
| 8.1-a1 |
8.1-a |
$2$ |
$3$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$6.10679$ |
$(a^3-a^2-4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$4.308867838$ |
0.735898204 |
\( 49366998461727605 a^{3} - \frac{17200987518233623}{2} a^{2} - \frac{1636561189853588101}{8} a - \frac{478255480551161905}{4} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a\) , \( a^{2} - a - 1\) , \( 21 a^{3} - 17 a^{2} - 68 a - 26\) , \( 51 a^{3} - 31 a^{2} - 177 a - 87\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(21a^{3}-17a^{2}-68a-26\right){x}+51a^{3}-31a^{2}-177a-87$ |
| 8.1-a2 |
8.1-a |
$2$ |
$3$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{3} \) |
$6.10679$ |
$(a^3-a^2-4a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$349.0182948$ |
0.735898204 |
\( \frac{237573}{2} a^{3} - \frac{39803}{2} a^{2} - \frac{985431}{2} a - 289273 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a\) , \( a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 2 a^{2} - 2 a + 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(a^{3}-2a^{2}-3a+4\right){x}+a^{3}-2a^{2}-2a+2$ |
| 8.2-a1 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{4} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$91.38443470$ |
0.867070267 |
\( 8986661761218306693 a^{3} + 12242887281724452580 a^{2} - 7024786750071806011 a - 7608271795805318134 \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a^{3} - 4 a\) , \( -2 a^{3} - 9 a^{2} - 10 a + 2\) , \( 32 a^{3} + 44 a^{2} - 34 a - 39\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(-2a^{3}-9a^{2}-10a+2\right){x}+32a^{3}+44a^{2}-34a-39$ |
| 8.2-a2 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$182.7688694$ |
0.867070267 |
\( -601 a^{3} + 1330 a^{2} + 1931 a - 2180 \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -a^{3} - a^{2} + 4 a + 5\) , \( -a^{3} - a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-a^{3}-a^{2}+4a+5\right){x}-a^{3}-a^{2}+4a+1$ |
| 8.2-a3 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{8} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$365.5377388$ |
0.867070267 |
\( -2732527394 a^{3} + 1558959970 a^{2} + 13692770763 a + 7728253510 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 2\) , \( 0\) , \( 9 a^{3} - 25 a^{2} + 14 a + 4\) , \( -30 a^{3} + 72 a^{2} + 25 a - 54\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-3a-2\right){x}^{2}+\left(9a^{3}-25a^{2}+14a+4\right){x}-30a^{3}+72a^{2}+25a-54$ |
| 8.2-a4 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{10} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$91.38443470$ |
0.867070267 |
\( -63795770348435003481 a^{3} + 11114208719071878840 a^{2} + 264361023260378305267 a + 154509478780590380994 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 3 a - 2\) , \( 0\) , \( 154 a^{3} - 405 a^{2} + 39 a + 154\) , \( -2510 a^{3} + 6386 a^{2} + 264 a - 3160\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-3a-2\right){x}^{2}+\left(154a^{3}-405a^{2}+39a+154\right){x}-2510a^{3}+6386a^{2}+264a-3160$ |
| 8.2-a5 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$365.5377388$ |
0.867070267 |
\( 2973144 a^{3} - 5002655 a^{2} - 8485568 a + 8881408 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 3 a^{3} - 2 a^{2} - 9 a - 6\) , \( 6 a^{3} - 3 a^{2} - 22 a - 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a^{3}-2a^{2}-9a-6\right){x}+6a^{3}-3a^{2}-22a-13$ |
| 8.2-a6 |
8.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$6.10679$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$45.69221735$ |
0.867070267 |
\( 34259927531436 a^{3} - 57543520518160 a^{2} - 97932989693671 a + 100819215025068 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( 18 a^{3} - 37 a^{2} - 14 a + 4\) , \( 63 a^{3} - 160 a^{2} - 5 a + 73\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(18a^{3}-37a^{2}-14a+4\right){x}+63a^{3}-160a^{2}-5a+73$ |
| 11.1-a1 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{8} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$120.9845885$ |
1.147921305 |
\( -\frac{1271382434328458944}{214358881} a^{3} + \frac{3189239753697071944}{214358881} a^{2} + \frac{274621437566108395}{214358881} a - \frac{1685647152341252737}{214358881} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -1\) , \( a^{3} - 4 a\) , \( 5 a^{3} - 14 a^{2} + 3 a + 1\) , \( -16 a^{3} + 40 a^{2} - 7 a - 14\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}-{x}^{2}+\left(5a^{3}-14a^{2}+3a+1\right){x}-16a^{3}+40a^{2}-7a-14$ |
| 11.1-a2 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$241.9691770$ |
1.147921305 |
\( \frac{143460}{11} a^{3} - \frac{313574}{11} a^{2} - \frac{467081}{11} a + \frac{520237}{11} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + 2 a^{2} - a - 1\) , \( a^{3} + 3 a^{2} - 4\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(a^{3}+2a^{2}-a-1\right){x}+a^{3}+3a^{2}-4$ |
| 11.1-a3 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$483.9383541$ |
1.147921305 |
\( \frac{11189839269}{121} a^{3} + \frac{15244624290}{121} a^{2} - \frac{8746346853}{121} a - \frac{9472934785}{121} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{2} - a - 1\) , \( -2 a^{2} - a + 1\) , \( -2 a^{2} + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}-a{x}^{2}+\left(-2a^{2}-a+1\right){x}-2a^{2}+1$ |
| 11.1-a4 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$60.49229427$ |
1.147921305 |
\( \frac{1702129469435805324}{11} a^{3} + \frac{2318878776895144329}{11} a^{2} - \frac{1330538175521684262}{11} a - \frac{1441053861737062367}{11} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{2} - a - 1\) , \( -5 a^{3} - 27 a^{2} + 4 a + 16\) , \( -36 a^{3} - 102 a^{2} + 35 a + 61\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}-a{x}^{2}+\left(-5a^{3}-27a^{2}+4a+16\right){x}-36a^{3}-102a^{2}+35a+61$ |
| 11.1-a5 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{2} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$120.9845885$ |
1.147921305 |
\( -\frac{218889110914391240}{121} a^{3} + \frac{37981189225429456}{121} a^{2} + \frac{907177163042977213}{121} a + \frac{530680189689696129}{121} \) |
\( \bigl[1\) , \( -a^{3} + 3 a\) , \( a^{2} - 1\) , \( -68 a^{3} + 158 a^{2} + 160 a - 355\) , \( 927 a^{3} - 1319 a^{2} - 2859 a + 1883\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-68a^{3}+158a^{2}+160a-355\right){x}+927a^{3}-1319a^{2}-2859a+1883$ |
| 11.1-a6 |
11.1-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$6.35478$ |
$(-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$483.9383541$ |
1.147921305 |
\( -\frac{261546261312}{14641} a^{3} + \frac{36220688425}{14641} a^{2} + \frac{1091148881914}{14641} a + \frac{668876936913}{14641} \) |
\( \bigl[1\) , \( -a^{3} + 3 a\) , \( a^{2} - 1\) , \( -3 a^{3} + 8 a^{2} + 10 a - 20\) , \( 16 a^{3} - 23 a^{2} - 54 a + 33\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-3a^{3}+8a^{2}+10a-20\right){x}+16a^{3}-23a^{2}-54a+33$ |
| 16.1-a1 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{45} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$8.132353688$ |
0.694449760 |
\( -\frac{9448889797}{4096} a^{3} + \frac{16023247993}{4096} a^{2} + \frac{27103130219}{4096} a - \frac{14003335939}{2048} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -27 a^{3} + 35 a^{2} + 72 a - 62\) , \( -75 a^{3} + 102 a^{2} + 199 a - 190\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-27a^{3}+35a^{2}+72a-62\right){x}-75a^{3}+102a^{2}+199a-190$ |
| 16.1-a2 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{15} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$658.7206488$ |
0.694449760 |
\( \frac{13351}{16} a^{3} + \frac{5977}{8} a^{2} + \frac{6795}{16} a + \frac{823}{4} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -2 a^{3} + 7 a + 3\) , \( -2 a^{3} + 2 a^{2} + 7 a - 1\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-2a^{3}+7a+3\right){x}-2a^{3}+2a^{2}+7a-1$ |
| 16.1-a3 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1317.441297$ |
0.694449760 |
\( \frac{820372365}{64} a^{3} + \frac{1284429089}{64} a^{2} - \frac{333707999}{32} a - \frac{786579399}{64} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a - 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -11 a^{3} - 17 a^{2} + 2 a + 4\) , \( 56 a^{3} + 79 a^{2} - 40 a - 48\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11a^{3}-17a^{2}+2a+4\right){x}+56a^{3}+79a^{2}-40a-48$ |
| 16.1-a4 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{45} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$8.132353688$ |
0.694449760 |
\( -\frac{11596855953503300897687859}{68719476736} a^{3} + \frac{2020351217336280011001041}{68719476736} a^{2} + \frac{24027899520071110551507553}{34359738368} a + \frac{28086881208189762512317305}{68719476736} \) |
\( \bigl[1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{3} - 4 a\) , \( 248 a^{3} + 94 a^{2} - 178 a - 86\) , \( -61 a^{3} + 1602 a^{2} - 240 a - 924\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(248a^{3}+94a^{2}-178a-86\right){x}-61a^{3}+1602a^{2}-240a-924$ |
| 16.1-a5 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{6} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$329.3603244$ |
0.694449760 |
\( \frac{26337901600044709}{8} a^{3} + \frac{35881160809650025}{8} a^{2} - \frac{10294045696327409}{4} a - \frac{22298133469852751}{8} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - 2\) , \( 110 a^{3} - 287 a^{2} + 7 a + 128\) , \( -927 a^{3} + 2286 a^{2} + 326 a - 1304\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(110a^{3}-287a^{2}+7a+128\right){x}-927a^{3}+2286a^{2}+326a-1304$ |
| 16.1-a6 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{15} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$658.7206488$ |
0.694449760 |
\( -\frac{14270157697541427}{4096} a^{3} + \frac{35796422109839569}{4096} a^{2} + \frac{1541200524057185}{2048} a - \frac{18919901006348423}{4096} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 1\) , \( 0\) , \( 42 a^{3} - 3 a^{2} - 178 a - 115\) , \( 260 a^{3} - 47 a^{2} - 1076 a - 623\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(42a^{3}-3a^{2}-178a-115\right){x}+260a^{3}-47a^{2}-1076a-623$ |
| 16.1-a7 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{36} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$16.26470737$ |
0.694449760 |
\( \frac{5353573950671463885}{262144} a^{3} - \frac{8992096210769676335}{262144} a^{2} - \frac{7651439252430264991}{131072} a + \frac{15754080188450938489}{262144} \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 2 a + 1\) , \( 30 a^{3} - 52 a^{2} - 65 a - 13\) , \( 91 a^{3} - 127 a^{2} - 206 a - 56\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(30a^{3}-52a^{2}-65a-13\right){x}+91a^{3}-127a^{2}-206a-56$ |
| 16.1-a8 |
16.1-a |
$8$ |
$12$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{18} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$36$ |
\( 1 \) |
$1$ |
$4.066176844$ |
0.694449760 |
\( \frac{810521103884523357631151609357}{512} a^{3} - \frac{1361386248893928824308169088175}{512} a^{2} - \frac{1158413764279677545759724509727}{256} a + \frac{2385137145792622768763692460537}{512} \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - 2 a + 1\) , \( 230 a^{3} - 692 a^{2} - 65 a + 307\) , \( 4699 a^{3} - 10439 a^{2} - 1286 a + 5304\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(230a^{3}-692a^{2}-65a+307\right){x}+4699a^{3}-10439a^{2}-1286a+5304$ |
| 16.1-b1 |
16.1-b |
$4$ |
$14$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{23} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.3 |
$49$ |
\( 2 \cdot 7 \) |
$1$ |
$0.460733588$ |
1.499429526 |
\( \frac{414403249587808247093}{128} a^{3} - \frac{696044271483193936579}{128} a^{2} - \frac{592282302218182028565}{64} a + \frac{76217710069831417535}{8} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + a - 3\) , \( a^{2} - a - 1\) , \( 1375 a^{3} - 489 a^{2} - 5280 a - 3053\) , \( 42544 a^{3} - 11186 a^{2} - 169964 a - 97423\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+a-3\right){x}^{2}+\left(1375a^{3}-489a^{2}-5280a-3053\right){x}+42544a^{3}-11186a^{2}-169964a-97423$ |
| 16.1-b2 |
16.1-b |
$4$ |
$14$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{17} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/14\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.1 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1106.221345$ |
1.499429526 |
\( -\frac{1466589303347}{16384} a^{3} + \frac{255514832849}{16384} a^{2} + \frac{3038659044705}{8192} a + \frac{3551992972921}{16384} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a\) , \( a\) , \( -4 a^{3} + 7 a^{2} + 11 a - 12\) , \( 3 a^{3} - 3 a^{2} - 7 a + 6\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-4a^{3}+7a^{2}+11a-12\right){x}+3a^{3}-3a^{2}-7a+6$ |
| 16.1-b3 |
16.1-b |
$4$ |
$14$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{13} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/14\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.1 |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$1106.221345$ |
1.499429526 |
\( \frac{620979}{128} a^{3} - \frac{68657}{128} a^{2} - \frac{1152721}{64} a - \frac{1119033}{128} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 3 a^{3} - 3 a^{2} - 10 a + 3\) , \( -a^{3} + a^{2} + 3 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(3a^{3}-3a^{2}-10a+3\right){x}-a^{3}+a^{2}+3a-1$ |
| 16.1-b4 |
16.1-b |
$4$ |
$14$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( - 2^{43} \) |
$6.65950$ |
$(-a), (a^3-a^2-4a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.3 |
$49$ |
\( 2 \cdot 7 \) |
$1$ |
$0.460733588$ |
1.499429526 |
\( \frac{14221956074310291}{16384} a^{3} + \frac{19421003373675559}{16384} a^{2} - \frac{5542858545784381}{8192} a - \frac{1512226792940977}{2048} \) |
\( \bigl[1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{2} - a - 1\) , \( 559 a^{3} - 1437 a^{2} - 22 a + 670\) , \( 15729 a^{3} - 39584 a^{2} - 3004 a + 20573\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(559a^{3}-1437a^{2}-22a+670\right){x}+15729a^{3}-39584a^{2}-3004a+20573$ |
| 16.2-a1 |
16.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( 2^{10} \) |
$6.65950$ |
$(-a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$928.5469574$ |
1.101275426 |
\( 2973144 a^{3} - 5002655 a^{2} - 8485568 a + 8881408 \) |
\( \bigl[a^{3} - 4 a\) , \( -a + 1\) , \( a^{3} - a^{2} - 2 a + 2\) , \( -6 a^{3} + 12 a^{2} + 12 a - 24\) , \( 9 a^{3} - 10 a^{2} - 32 a + 7\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-2a+2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a^{3}+12a^{2}+12a-24\right){x}+9a^{3}-10a^{2}-32a+7$ |
| 16.2-a2 |
16.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{4} \) |
$6.65950$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$14.50854621$ |
1.101275426 |
\( 8986661761218306693 a^{3} + 12242887281724452580 a^{2} - 7024786750071806011 a - 7608271795805318134 \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + 1\) , \( a^{2} - a - 2\) , \( -76 a^{3} + 192 a^{2} + 10 a - 99\) , \( 264 a^{3} - 660 a^{2} - 64 a + 352\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-76a^{3}+192a^{2}+10a-99\right){x}+264a^{3}-660a^{2}-64a+352$ |
| 16.2-a3 |
16.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{11} \) |
$6.65950$ |
$(-a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$928.5469574$ |
1.101275426 |
\( -601 a^{3} + 1330 a^{2} + 1931 a - 2180 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( -2 a^{3} + 6 a + 2\) , \( -a^{3} + 3 a + 1\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-2a^{3}+6a+2\right){x}-a^{3}+3a+1$ |
| 16.2-a4 |
16.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{11} \) |
$6.65950$ |
$(-a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$464.2734787$ |
1.101275426 |
\( 34259927531436 a^{3} - 57543520518160 a^{2} - 97932989693671 a + 100819215025068 \) |
\( \bigl[a\) , \( -a^{2} + a + 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 36 a^{3} - 6 a^{2} - 152 a - 92\) , \( -24 a^{3} - 4 a^{2} + 80 a + 48\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(36a^{3}-6a^{2}-152a-92\right){x}-24a^{3}-4a^{2}+80a+48$ |
| 16.2-a5 |
16.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{10} \) |
$6.65950$ |
$(-a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$7.254273105$ |
1.101275426 |
\( -63795770348435003481 a^{3} + 11114208719071878840 a^{2} + 264361023260378305267 a + 154509478780590380994 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a\) , \( -4 a^{3} + 16 a^{2} - 4 a - 67\) , \( -8 a^{3} + 53 a^{2} - 21 a - 199\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(-4a^{3}+16a^{2}-4a-67\right){x}-8a^{3}+53a^{2}-21a-199$ |
| 16.2-a6 |
16.2-a |
$6$ |
$8$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
16.2 |
\( 2^{4} \) |
\( 2^{8} \) |
$6.65950$ |
$(-a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$116.0683696$ |
1.101275426 |
\( -2732527394 a^{3} + 1558959970 a^{2} + 13692770763 a + 7728253510 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a\) , \( -4 a^{3} + a^{2} + 6 a - 7\) , \( -9 a^{2} - 7 a + 9\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(-4a^{3}+a^{2}+6a-7\right){x}-9a^{2}-7a+9$ |
| 22.1-a1 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{4} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1596.650924$ |
0.841626760 |
\( -\frac{201814041699}{1936} a^{3} + \frac{35160513793}{1936} a^{2} + \frac{418144402265}{968} a + \frac{488783259193}{1936} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} - 2 a^{2} + 4 a + 4\) , \( -a^{3} + 4 a\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{3}-2a^{2}+4a+4\right){x}-a^{3}+4a$ |
| 22.1-a2 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{2} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$199.5813655$ |
0.841626760 |
\( -\frac{2581686241508983459}{44} a^{3} + \frac{449769625560286565}{44} a^{2} + \frac{5349078887234667531}{22} a + \frac{6252687182257486625}{44} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 4 a^{3} - 22 a^{2} - a + 14\) , \( 6 a^{3} - 65 a^{2} + 6 a + 36\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(4a^{3}-22a^{2}-a+14\right){x}+6a^{3}-65a^{2}+6a+36$ |
| 22.1-a3 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{16} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$399.1627311$ |
0.841626760 |
\( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( -20 a^{3} + 35 a^{2} + 56 a - 64\) , \( 299 a^{3} - 501 a^{2} - 856 a + 876\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(-20a^{3}+35a^{2}+56a-64\right){x}+299a^{3}-501a^{2}-856a+876$ |
| 22.1-a4 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{48} \cdot 11^{6} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$4.927934952$ |
0.841626760 |
\( -\frac{1502268147076192175039019336779}{498650091216506454016} a^{3} + \frac{261719896030458847171340772313}{498650091216506454016} a^{2} + \frac{3112599543413066339294706687465}{249325045608253227008} a + \frac{3638403917729845697875388025793}{498650091216506454016} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( 255 a^{3} - 345 a^{2} - 799 a + 451\) , \( -6627 a^{3} + 11701 a^{2} + 18463 a - 21549\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(255a^{3}-345a^{2}-799a+451\right){x}-6627a^{3}+11701a^{2}+18463a-21549$ |
| 22.1-a5 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{6} \cdot 11^{3} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$9.855869904$ |
0.841626760 |
\( \frac{174550817373650134723}{85184} a^{3} - \frac{293183090937363721025}{85184} a^{2} - \frac{249471689806794754865}{42592} a + \frac{513654285309827502679}{85184} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( -13 a^{3} + 17 a^{2} + 48 a - 41\) , \( -75 a^{3} + 116 a^{2} + 245 a - 240\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(-13a^{3}+17a^{2}+48a-41\right){x}-75a^{3}+116a^{2}+245a-240$ |
| 22.1-a6 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{2} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$798.3254622$ |
0.841626760 |
\( \frac{2490151}{44} a^{3} - \frac{3891025}{44} a^{2} - \frac{3683329}{22} a + \frac{6322871}{44} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 3 a^{2} - 7 a + 4\) , \( a^{3} - a^{2} - 4 a\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(2a^{3}-3a^{2}-7a+4\right){x}+a^{3}-a^{2}-4a$ |
| 22.1-a7 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{4} \cdot 11^{8} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$399.1627311$ |
0.841626760 |
\( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -29 a^{3} - 37 a^{2} + 29 a + 24\) , \( 247 a^{3} + 345 a^{2} - 177 a - 202\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(-29a^{3}-37a^{2}+29a+24\right){x}+247a^{3}+345a^{2}-177a-202$ |
| 22.1-a8 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{12} \cdot 11^{6} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$19.71173980$ |
0.841626760 |
\( -\frac{20157112611959390547}{7256313856} a^{3} + \frac{50708561159662371377}{7256313856} a^{2} + \frac{1953876284050968001}{3628156928} a - \frac{26455281535659018279}{7256313856} \) |
\( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( -65 a^{3} + 111 a^{2} + 209 a - 254\) , \( -552 a^{3} + 882 a^{2} + 1709 a - 1680\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-65a^{3}+111a^{2}+209a-254\right){x}-552a^{3}+882a^{2}+1709a-1680$ |
| 22.1-a9 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{6} \cdot 11^{3} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$36$ |
\( 2 \) |
$1$ |
$2.463967476$ |
0.841626760 |
\( -\frac{5006065775663608963358467}{85184} a^{3} + \frac{12557622075673248231038785}{85184} a^{2} + \frac{540663280751702976925425}{42592} a - \frac{6637226716732893866394967}{85184} \) |
\( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( 210 a^{3} - 374 a^{2} - 201 a - 214\) , \( 1496 a^{3} - 6224 a^{2} + 4609 a + 4096\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(210a^{3}-374a^{2}-201a-214\right){x}+1496a^{3}-6224a^{2}+4609a+4096$ |
| 22.1-a10 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{12} \cdot 11^{24} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$4.927934952$ |
0.841626760 |
\( \frac{60659117288902836130412254108283}{40344505040107975043939700736} a^{3} - \frac{86300110386401779908460150637033}{40344505040107975043939700736} a^{2} - \frac{97109108201994369690495482301689}{20172252520053987521969850368} a + \frac{184521640869151181066006363788207}{40344505040107975043939700736} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -115 a^{3} + 187 a^{2} + 362 a - 350\) , \( 926 a^{3} - 1541 a^{2} - 2691 a + 2732\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-115a^{3}+187a^{2}+362a-350\right){x}+926a^{3}-1541a^{2}-2691a+2732$ |
| 22.1-a11 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{24} \cdot 11^{12} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$19.71173980$ |
0.841626760 |
\( -\frac{2125946353940568480801651}{52654090776777588736} a^{3} + \frac{2559556155064329492768145}{52654090776777588736} a^{2} + \frac{2498155651166365388592929}{26327045388388794368} a + \frac{2080764888610005913150009}{52654090776777588736} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -30 a^{3} + 42 a^{2} + 102 a - 90\) , \( -111 a^{3} + 178 a^{2} + 344 a - 362\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-30a^{3}+42a^{2}+102a-90\right){x}-111a^{3}+178a^{2}+344a-362$ |
| 22.1-a12 |
22.1-a |
$12$ |
$24$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{8} \cdot 11^{4} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1596.650924$ |
0.841626760 |
\( \frac{1878243643917}{3748096} a^{3} + \frac{1364709508625}{3748096} a^{2} - \frac{1139759245535}{1874048} a - \frac{14164000839}{3748096} \) |
\( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -10 a^{3} + 12 a^{2} + 32 a - 15\) , \( 16 a^{3} - 29 a^{2} - 45 a + 56\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-10a^{3}+12a^{2}+32a-15\right){x}+16a^{3}-29a^{2}-45a+56$ |
| 22.1-b1 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2 \cdot 11^{20} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.679157918$ |
1.610989990 |
\( \frac{1573390889429092537195325472495367149}{1345499989865120018402} a^{3} - \frac{2642735286096795180305969741051658561}{1345499989865120018402} a^{2} - \frac{2248723264231462392984836360638223253}{672749994932560009201} a + \frac{4630049775522972966851131993588823229}{1345499989865120018402} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 574 a^{3} - 1057 a^{2} - 549 a - 385\) , \( 14427 a^{3} - 29786 a^{2} - 11937 a + 4219\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(574a^{3}-1057a^{2}-549a-385\right){x}+14427a^{3}-29786a^{2}-11937a+4219$ |
| 22.1-b2 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( - 2^{5} \cdot 11^{4} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$424.4736992$ |
1.610989990 |
\( \frac{19492220557608647265}{468512} a^{3} + \frac{26555029218814954293}{468512} a^{2} - \frac{7618439534802966603}{234256} a - \frac{16502468789812849155}{468512} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 9 a^{3} + 8 a^{2} - 59 a - 35\) , \( -37 a^{3} + 17 a^{2} + 136 a + 71\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(9a^{3}+8a^{2}-59a-35\right){x}-37a^{3}+17a^{2}+136a+71$ |
| 22.1-b3 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{10} \cdot 11^{2} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1697.894796$ |
1.610989990 |
\( \frac{189687996477}{123904} a^{3} + \frac{252879329025}{123904} a^{2} - \frac{75995630703}{61952} a - \frac{153356764887}{123904} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 4 a^{3} - 7 a^{2} - 9 a + 5\) , \( -a^{3} + 2 a^{2} + a - 3\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(4a^{3}-7a^{2}-9a+5\right){x}-a^{3}+2a^{2}+a-3$ |
| 22.1-b4 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2 \cdot 11^{5} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 5 \) |
$1$ |
$2.716631675$ |
1.610989990 |
\( -\frac{776940933840351053858986959}{322102} a^{3} + \frac{135355112947607678458237413}{322102} a^{2} + \frac{1609768948326117754076482158}{161051} a + \frac{1881703725424533772239476133}{322102} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -25 a^{3} + 26 a^{2} + 84 a - 75\) , \( -255 a^{3} - 134 a^{2} + 413 a - 111\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(-25a^{3}+26a^{2}+84a-75\right){x}-255a^{3}-134a^{2}+413a-111$ |
| 22.1-b5 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{5} \cdot 11 \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$1697.894796$ |
1.610989990 |
\( -\frac{38015163}{352} a^{3} + \frac{90874953}{352} a^{2} + \frac{7617321}{176} a - \frac{43556751}{352} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 3 a - 1\) , \( -2 a - 2\) , \( 28 a^{3} - 47 a^{2} - 81 a + 81\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-2a-2\right){x}+28a^{3}-47a^{2}-81a+81$ |
| 22.1-b6 |
22.1-b |
$8$ |
$20$ |
4.4.2777.1 |
$4$ |
$[4, 0]$ |
22.1 |
\( 2 \cdot 11 \) |
\( 2^{2} \cdot 11^{10} \) |
$6.92994$ |
$(-a), (-a^3+2a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 5$ |
2Cs, 5B.1.2 |
$25$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.716631675$ |
1.610989990 |
\( -\frac{64272348420343413978591}{103749698404} a^{3} + \frac{156108841633210925549601}{103749698404} a^{2} + \frac{11245678214829850896543}{51874849202} a - \frac{76993471097806369350591}{103749698404} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{2} - a - 1\) , \( -833 a^{3} - 1088 a^{2} + 652 a + 590\) , \( -39688 a^{3} - 53892 a^{2} + 31054 a + 33219\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-833a^{3}-1088a^{2}+652a+590\right){x}-39688a^{3}-53892a^{2}+31054a+33219$ |
*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.
