| 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 |
| 7.1-a1 |
7.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{6} \) |
$5.04163$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$79.35083879$ |
0.896863012 |
\( \frac{443958861593546408990}{117649} a^{3} + \frac{915220682926034165096}{117649} a^{2} + \frac{110890939600298461445}{117649} a - \frac{215357317054416491281}{117649} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - a - 1\) , \( 32 a^{3} - 57 a^{2} - 33 a + 17\) , \( 204 a^{3} - 364 a^{2} - 182 a + 118\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(32a^{3}-57a^{2}-33a+17\right){x}+204a^{3}-364a^{2}-182a+118$ |
| 7.1-a2 |
7.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$5.04163$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$158.7016775$ |
0.896863012 |
\( \frac{124820}{7} a^{3} - \frac{238513}{7} a^{2} - \frac{74080}{7} a + \frac{71744}{7} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 3 a\) , \( -a^{3} + a^{2} + a\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(a^{2}-3a\right){x}-a^{3}+a^{2}+a$ |
| 7.1-a3 |
7.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$5.04163$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$79.35083879$ |
0.896863012 |
\( -\frac{170631953648}{49} a^{3} + \frac{301490635046}{49} a^{2} + \frac{150363436017}{49} a - \frac{96330872074}{49} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( -5 a^{3} - 4 a^{2} + 7 a + 5\) , \( -6 a^{3} - 8 a^{2} - 3\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-5a^{3}-4a^{2}+7a+5\right){x}-6a^{3}-8a^{2}-3$ |
| 7.1-a4 |
7.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{3} \) |
$5.04163$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$158.7016775$ |
0.896863012 |
\( -\frac{9690935416}{343} a^{3} - \frac{12215671997}{343} a^{2} + \frac{16657874830}{343} a + \frac{14201869411}{343} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{3} - 3 a\) , \( -a^{3} - 4 a^{2} - 6 a + 1\) , \( -5 a^{3} - 11 a^{2} - a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-a^{3}-4a^{2}-6a+1\right){x}-5a^{3}-11a^{2}-a+3$ |
| 19.1-a1 |
19.1-a |
$1$ |
$1$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$5.71187$ |
$(a^3-a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$61.31233619$ |
1.385965601 |
\( -\frac{2062477872}{19} a^{3} - \frac{4353909970}{19} a^{2} - \frac{545558007}{19} a + \frac{1028555383}{19} \) |
\( \bigl[-a^{3} + a^{2} + 4 a - 1\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - 3 a - 1\) , \( 10 a^{3} - 17 a^{2} - 10 a + 7\) , \( 47 a^{3} - 82 a^{2} - 43 a + 26\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(10a^{3}-17a^{2}-10a+7\right){x}+47a^{3}-82a^{2}-43a+26$ |
| 19.1-b1 |
19.1-b |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{5} \) |
$5.71187$ |
$(a^3-a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$1.178437935$ |
0.665964854 |
\( \frac{22217245546666612500}{6859} a^{3} + \frac{8805551373160716923}{6859} a^{2} - \frac{85379003522040468141}{6859} a - \frac{56056237584037687000}{6859} \) |
\( \bigl[a^{2} - 1\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( -a^{3} + a^{2} + 3 a\) , \( 38 a^{3} - 44 a^{2} - 62 a - 20\) , \( 401 a^{3} - 614 a^{2} - 483 a + 89\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(38a^{3}-44a^{2}-62a-20\right){x}+401a^{3}-614a^{2}-483a+89$ |
| 19.1-b2 |
19.1-b |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19 \) |
$5.71187$ |
$(a^3-a^2-4a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$736.5237097$ |
0.665964854 |
\( \frac{10334}{19} a^{3} - \frac{3313}{19} a^{2} - \frac{36107}{19} a - \frac{5371}{19} \) |
\( \bigl[a^{2} - 1\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( -a^{3} + a^{2} + 3 a\) , \( 3 a^{3} - 4 a^{2} - 7 a + 5\) , \( -3 a^{3} + 4 a^{2} + 6 a - 3\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(3a^{3}-4a^{2}-7a+5\right){x}-3a^{3}+4a^{2}+6a-3$ |
| 19.1-b3 |
19.1-b |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{3} \) |
$5.71187$ |
$(a^3-a^2-4a)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$736.5237097$ |
0.665964854 |
\( \frac{44732636264}{361} a^{3} - \frac{78909304839}{361} a^{2} - \frac{39733288819}{361} a + \frac{25358512200}{361} \) |
\( \bigl[-a^{3} + a^{2} + 4 a - 1\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( a^{2} - a - 1\) , \( -a^{3} + 7 a + 6\) , \( -5 a^{3} - 7 a^{2} + 11 a + 10\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(-a^{3}+7a+6\right){x}-5a^{3}-7a^{2}+11a+10$ |
| 19.1-b4 |
19.1-b |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
19.1 |
\( 19 \) |
\( 19^{15} \) |
$5.71187$ |
$(a^3-a^2-4a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$1.178437935$ |
0.665964854 |
\( -\frac{7261286418859671754}{16983563041} a^{3} - \frac{14970804000790893451}{16983563041} a^{2} - \frac{1817822990455263041}{16983563041} a + \frac{3520277452889246795}{16983563041} \) |
\( \bigl[1\) , \( -a^{3} + 5 a\) , \( a^{2} - 2\) , \( -38 a^{3} + 67 a^{2} + 92 a - 143\) , \( -658 a^{3} + 655 a^{2} + 2105 a - 1361\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-38a^{3}+67a^{2}+92a-143\right){x}-658a^{3}+655a^{2}+2105a-1361$ |
| 21.1-a1 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{6} \cdot 7^{3} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$1050.985704$ |
0.742422999 |
\( -\frac{4584696391003}{250047} a^{3} - \frac{1823709580711}{250047} a^{2} + \frac{5875898465972}{83349} a + \frac{11577122282035}{250047} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a - 1\) , \( 2 a^{3} - a^{2} - 5 a\) , \( 7 a^{3} - 2 a^{2} - 30 a + 11\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(2a^{3}-a^{2}-5a\right){x}+7a^{3}-2a^{2}-30a+11$ |
| 21.1-a2 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{48} \cdot 7^{6} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$2.052706454$ |
0.742422999 |
\( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a - 1\) , \( 312 a^{3} - 201 a^{2} - 1090 a + 375\) , \( -2482 a^{3} + 1995 a^{2} + 8193 a - 3631\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(312a^{3}-201a^{2}-1090a+375\right){x}-2482a^{3}+1995a^{2}+8193a-3631$ |
| 21.1-a3 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{24} \cdot 7^{12} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$32.84330326$ |
0.742422999 |
\( -\frac{49478337663774144106792217}{3909188328478827879681} a^{3} + \frac{86867577756332942729622262}{3909188328478827879681} a^{2} + \frac{19815761443619939196843979}{1303062776159609293227} a - \frac{6943686789535186694527711}{3909188328478827879681} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - 3 a - 1\) , \( -14 a^{3} - 52 a^{2} - 24 a + 2\) , \( -149 a^{3} - 293 a^{2} - 54 a + 53\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(-14a^{3}-52a^{2}-24a+2\right){x}-149a^{3}-293a^{2}-54a+53$ |
| 21.1-a4 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{12} \cdot 7^{6} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$525.4928522$ |
0.742422999 |
\( -\frac{84413562014875846901}{62523502209} a^{3} + \frac{148907375867724857512}{62523502209} a^{2} + \frac{24992949793645640572}{20841167403} a - \frac{47851964758998446149}{62523502209} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - 3 a - 1\) , \( a^{3} - 17 a^{2} - 9 a + 2\) , \( 8 a^{3} + 41 a^{2} + 11 a - 9\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(a^{3}-17a^{2}-9a+2\right){x}+8a^{3}+41a^{2}+11a-9$ |
| 21.1-a5 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{16} \cdot 7^{2} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$2.052706454$ |
0.742422999 |
\( -\frac{29879008921030774408202450507522}{2109289329} a^{3} - \frac{11842202503929076694840862982208}{2109289329} a^{2} + \frac{38274171513246722984375757845464}{703096443} a + \frac{75387595661097076378967854941161}{2109289329} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} - 3 a\) , \( -12 a^{3} + 21 a^{2} - 3 a - 34\) , \( 660 a^{3} - 1000 a^{2} - 561 a + 285\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-12a^{3}+21a^{2}-3a-34\right){x}+660a^{3}-1000a^{2}-561a+285$ |
| 21.1-a6 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{6} \cdot 7^{3} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$262.7464261$ |
0.742422999 |
\( -\frac{4031277404041470845619545}{250047} a^{3} + \frac{7111233508460767828860022}{250047} a^{2} + \frac{1193595856869520566460555}{83349} a - \frac{2285285314923636665330287}{250047} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( -1\) , \( a + 1\) , \( -80 a^{3} - 32 a^{2} + 311 a + 126\) , \( -989 a^{3} - 406 a^{2} + 3795 a + 2783\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-80a^{3}-32a^{2}+311a+126\right){x}-989a^{3}-406a^{2}+3795a+2783$ |
| 21.1-a7 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{12} \cdot 7^{24} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$2.052706454$ |
0.742422999 |
\( \frac{981621700851969745102419250048115}{101814121186119595835681841} a^{3} + \frac{1761760795662379320695876261373041}{101814121186119595835681841} a^{2} - \frac{37671995275184386601460285791065}{33938040395373198611893947} a - \frac{293036539712498872630739037955790}{101814121186119595835681841} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( -1\) , \( a + 1\) , \( -355 a^{3} - 122 a^{2} + 1271 a + 666\) , \( 7906 a^{3} + 3150 a^{2} - 31007 a - 21110\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-355a^{3}-122a^{2}+1271a+666\right){x}+7906a^{3}+3150a^{2}-31007a-21110$ |
| 21.1-a8 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$32.84330326$ |
0.742422999 |
\( -\frac{921288230848937984}{15752961} a^{3} - \frac{392913302754752516}{15752961} a^{2} + \frac{1182898550569873408}{5250987} a + \frac{2425491268899001073}{15752961} \) |
\( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( 20 a^{2} - 89\) , \( 100 a^{2} - 16 a - 384\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(20a^{2}-89\right){x}+100a^{2}-16a-384$ |
| 21.1-a9 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$525.4928522$ |
0.742422999 |
\( \frac{18710001614848}{3969} a^{3} + \frac{38571223778296}{3969} a^{2} + \frac{1558338097840}{1323} a - \frac{9074746748287}{3969} \) |
\( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( -4\) , \( 4 a^{2} - 9\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}-4{x}+4a^{2}-9$ |
| 21.1-a10 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$262.7464261$ |
0.742422999 |
\( \frac{24573261493292438142976}{63} a^{3} + \frac{50657750325995678467324}{63} a^{2} + \frac{2045949367379882389504}{21} a - \frac{11920094685054769076767}{63} \) |
\( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( -20 a^{2} + 1\) , \( 84 a^{2} + 16 a - 30\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-20a^{2}+1\right){x}+84a^{2}+16a-30$ |
| 21.1-a11 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7 \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$1050.985704$ |
0.742422999 |
\( -\frac{1064704}{63} a^{3} - \frac{2122864}{63} a^{2} - \frac{47392}{21} a + \frac{567631}{63} \) |
\( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+{x}$ |
| 21.1-a12 |
21.1-a |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{4} \cdot 7^{8} \) |
$5.78377$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{2} \) |
$1$ |
$2.052706454$ |
0.742422999 |
\( \frac{3925517460083619669474871874}{466948881} a^{3} - \frac{2723612167164915292630390720}{466948881} a^{2} - \frac{4604122185339962156855866072}{155649627} a + \frac{5657812633917689859889126471}{466948881} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 2\) , \( 59 a^{3} - 73 a^{2} - 337 a - 189\) , \( 583 a^{3} - 32 a^{2} - 2275 a - 1390\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(59a^{3}-73a^{2}-337a-189\right){x}+583a^{3}-32a^{2}-2275a-1390$ |
| 23.1-a1 |
23.1-a |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{2} \) |
$5.84992$ |
$(-2a^3+a^2+6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$84.90966535$ |
0.959691660 |
\( -\frac{7418820639147057}{529} a^{3} - \frac{2940364437579043}{529} a^{2} + \frac{28509902770920202}{529} a + \frac{18718393590733878}{529} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - 5 a - 2\) , \( a^{2} - 2\) , \( 6 a^{3} - 4 a^{2} - 31 a - 9\) , \( 10 a^{3} + 14 a^{2} - 16 a - 17\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(6a^{3}-4a^{2}-31a-9\right){x}+10a^{3}+14a^{2}-16a-17$ |
| 23.1-a2 |
23.1-a |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$5.84992$ |
$(-2a^3+a^2+6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$169.8193307$ |
0.959691660 |
\( \frac{43532247}{23} a^{3} + \frac{17211519}{23} a^{2} - \frac{167270339}{23} a - \frac{109671331}{23} \) |
\( \bigl[a^{3} - 3 a\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( a^{2} - a - 1\) , \( 3 a + 4\) , \( a^{3} - a^{2} + a + 1\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(3a+4\right){x}+a^{3}-a^{2}+a+1$ |
| 31.1-a1 |
31.1-a |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{5} \) |
$6.07231$ |
$(a^3-5a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 5 \) |
$1.022761215$ |
$2.399984655$ |
1.109729927 |
\( -\frac{180825454881646489600000}{28629151} a^{3} + \frac{318978801202372556800000}{28629151} a^{2} + \frac{160618453429727667920896}{28629151} a - \frac{102507894005169170706432}{28629151} \) |
\( \bigl[0\) , \( a^{3} - 4 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 69 a^{3} + 29 a^{2} - 257 a - 177\) , \( 387 a^{3} + 140 a^{2} - 1515 a - 1004\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(69a^{3}+29a^{2}-257a-177\right){x}+387a^{3}+140a^{2}-1515a-1004$ |
| 31.1-a2 |
31.1-a |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{3} \) |
$6.07231$ |
$(a^3-5a)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 3 \) |
$0.068184081$ |
$1499.990409$ |
1.109729927 |
\( \frac{446038016}{29791} a^{3} + \frac{34282332160}{29791} a^{2} - \frac{48219545600}{29791} a - \frac{50153631744}{29791} \) |
\( \bigl[0\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 4 a\) , \( 2 a^{3} + 2 a^{2} - 8 a - 6\) , \( -5 a^{3} - 2 a^{2} + 18 a + 11\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(2a^{3}+2a^{2}-8a-6\right){x}-5a^{3}-2a^{2}+18a+11$ |
| 31.1-a3 |
31.1-a |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{15} \) |
$6.07231$ |
$(a^3-5a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$1$ |
\( 3 \cdot 5 \) |
$0.340920405$ |
$2.399984655$ |
1.109729927 |
\( -\frac{2536784722881790079590400000}{23465261991844685929951} a^{3} - \frac{1326320955838364685202235392}{23465261991844685929951} a^{2} + \frac{561810697949918688555835392}{23465261991844685929951} a + \frac{181027024945269059059658752}{23465261991844685929951} \) |
\( \bigl[0\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 4 a\) , \( -28 a^{3} - 38 a^{2} + 52 a + 54\) , \( -103 a^{3} - 116 a^{2} + 64 a - 103\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-28a^{3}-38a^{2}+52a+54\right){x}-103a^{3}-116a^{2}+64a-103$ |
| 31.1-a4 |
31.1-a |
$4$ |
$15$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$6.07231$ |
$(a^3-5a)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 1 \) |
$0.204552243$ |
$1499.990409$ |
1.109729927 |
\( -\frac{65536}{31} a^{3} - \frac{139264}{31} a^{2} - \frac{65536}{31} a + \frac{53248}{31} \) |
\( \bigl[0\) , \( -a^{3} + 3 a\) , \( 1\) , \( 3 a^{3} - 2 a^{2} - 10 a + 5\) , \( -2 a^{3} + a^{2} + 7 a - 2\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(3a^{3}-2a^{2}-10a+5\right){x}-2a^{3}+a^{2}+7a-2$ |
| 43.1-a1 |
43.1-a |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
43.1 |
\( 43 \) |
\( - 43^{3} \) |
$6.32583$ |
$(-a^2-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.006791259$ |
$669.3900782$ |
1.233148718 |
\( \frac{1592683107521}{79507} a^{3} + \frac{629635995686}{79507} a^{2} - \frac{6120072768045}{79507} a - \frac{4012649003175}{79507} \) |
\( \bigl[a^{2} - 1\) , \( 2 a^{3} - a^{2} - 8 a\) , \( 0\) , \( -3 a^{3} + 3 a^{2} + 10 a - 7\) , \( -a^{3} - 2 a^{2} + 4 a + 9\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(2a^{3}-a^{2}-8a\right){x}^{2}+\left(-3a^{3}+3a^{2}+10a-7\right){x}-a^{3}-2a^{2}+4a+9$ |
| 43.1-a2 |
43.1-a |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
43.1 |
\( 43 \) |
\( -43 \) |
$6.32583$ |
$(-a^2-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.020373779$ |
$669.3900782$ |
1.233148718 |
\( \frac{655988}{43} a^{3} - \frac{455074}{43} a^{2} - \frac{2306280}{43} a + \frac{943113}{43} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{2} - a - 1\) , \( a\) , \( -a^{2} + a + 1\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+a{x}-a^{2}+a+1$ |
| 47.1-a1 |
47.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{2} \) |
$6.39656$ |
$(2a^3-6a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$151.5709415$ |
1.713130865 |
\( \frac{26469954681304186}{2209} a^{3} - \frac{18365448892640224}{2209} a^{2} - \frac{93137457700655592}{2209} a + \frac{38150905028515907}{2209} \) |
\( \bigl[a^{2} - 2\) , \( a\) , \( a^{2} - a - 2\) , \( -a^{2} + 3 a\) , \( 64 a^{3} + 28 a^{2} - 249 a - 166\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+a{x}^{2}+\left(-a^{2}+3a\right){x}+64a^{3}+28a^{2}-249a-166$ |
| 47.1-a2 |
47.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{3} \) |
$6.39656$ |
$(2a^3-6a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$303.1418830$ |
1.713130865 |
\( \frac{1184609061189}{103823} a^{3} + \frac{2580667051169}{103823} a^{2} + \frac{343044349773}{103823} a - \frac{608636026432}{103823} \) |
\( \bigl[a^{2} - a - 1\) , \( -2 a^{3} + a^{2} + 8 a\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 7 a^{3} - 3 a^{2} - 21 a - 4\) , \( -5 a^{3} + 2 a^{2} + 15 a + 2\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a\right){x}^{2}+\left(7a^{3}-3a^{2}-21a-4\right){x}-5a^{3}+2a^{2}+15a+2$ |
| 47.1-a3 |
47.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{6} \) |
$6.39656$ |
$(2a^3-6a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$151.5709415$ |
1.713130865 |
\( -\frac{11148830379801159533743}{10779215329} a^{3} + \frac{19666703347863657066005}{10779215329} a^{2} + \frac{9902963271520642369814}{10779215329} a - \frac{6320145061844672607397}{10779215329} \) |
\( \bigl[a^{2} - a - 1\) , \( -2 a^{3} + a^{2} + 8 a\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 32 a^{3} - 58 a^{2} - 26 a + 21\) , \( -197 a^{3} + 305 a^{2} + 233 a - 53\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a\right){x}^{2}+\left(32a^{3}-58a^{2}-26a+21\right){x}-197a^{3}+305a^{2}+233a-53$ |
| 47.1-a4 |
47.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$6.39656$ |
$(2a^3-6a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$303.1418830$ |
1.713130865 |
\( -\frac{91309792}{47} a^{3} + \frac{62772516}{47} a^{2} + \frac{321433088}{47} a - \frac{129364665}{47} \) |
\( \bigl[a^{3} - 4 a\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 3 a^{2} - 4 a + 1\) , \( 2 a^{3} - 3 a^{2} - 2 a - 1\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(2a^{3}-3a^{2}-4a+1\right){x}+2a^{3}-3a^{2}-2a-1$ |
| 47.1-b1 |
47.1-b |
$4$ |
$4$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$6.39656$ |
$(2a^3-6a-1)$ |
$0 \le r \le 2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$690.3504462$ |
0.975335910 |
\( -\frac{1527277}{47} a^{3} + \frac{1688549}{47} a^{2} + \frac{2970352}{47} a + \frac{259372}{47} \) |
\( \bigl[a^{2} - 2\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( -3 a^{3} - 4 a^{2} + 7 a + 7\) , \( -a^{3} - a^{2} + 4 a + 2\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(-3a^{3}-4a^{2}+7a+7\right){x}-a^{3}-a^{2}+4a+2$ |
| 47.1-b2 |
47.1-b |
$4$ |
$4$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{2} \) |
$6.39656$ |
$(2a^3-6a-1)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1$ |
$86.29380577$ |
0.975335910 |
\( -\frac{8324109193351}{2209} a^{3} + \frac{14771368925300}{2209} a^{2} + \frac{7178823816030}{2209} a - \frac{4647340584154}{2209} \) |
\( \bigl[a^{2} - 2\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( -13 a^{3} - 19 a^{2} + 12 a + 7\) , \( -65 a^{3} - 133 a^{2} - 13 a + 31\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(-13a^{3}-19a^{2}+12a+7\right){x}-65a^{3}-133a^{2}-13a+31$ |
| 47.1-b3 |
47.1-b |
$4$ |
$4$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( 47^{4} \) |
$6.39656$ |
$(2a^3-6a-1)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$5.393362860$ |
0.975335910 |
\( \frac{21493599447952576879551}{4879681} a^{3} - \frac{14912741973407868587654}{4879681} a^{2} - \frac{75627602524472124243387}{4879681} a + \frac{30978529535192993029238}{4879681} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - 4 a\) , \( a^{3} - 4 a - 1\) , \( 49 a^{3} - 80 a^{2} - 46 a + 12\) , \( 423 a^{3} - 748 a^{2} - 365 a + 218\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(49a^{3}-80a^{2}-46a+12\right){x}+423a^{3}-748a^{2}-365a+218$ |
| 47.1-b4 |
47.1-b |
$4$ |
$4$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$6.39656$ |
$(2a^3-6a-1)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 1 \) |
$1$ |
$10.78672572$ |
0.975335910 |
\( -\frac{6019345658438801329873}{47} a^{3} + \frac{10618215581395079746074}{47} a^{2} + \frac{5346691869248498657045}{47} a - \frac{3412298599331115695098}{47} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 2 a\) , \( -a^{3} + a^{2} + 3 a\) , \( -75 a^{3} - 29 a^{2} + 245 a + 124\) , \( 171 a^{3} + 21 a^{2} - 837 a - 601\bigr] \) |
${y}^2+a{x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(-75a^{3}-29a^{2}+245a+124\right){x}+171a^{3}+21a^{2}-837a-601$ |
| 47.1-c1 |
47.1-c |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( - 47^{2} \) |
$6.39656$ |
$(2a^3-6a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.030350985$ |
$805.5519662$ |
1.105352853 |
\( \frac{260493073}{2209} a^{3} - \frac{513538831655}{2209} a^{2} + \frac{153993847068}{2209} a + \frac{1871080196928}{2209} \) |
\( \bigl[a^{2} - a - 2\) , \( 2 a^{3} - a^{2} - 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( -13 a^{3} + 8 a^{2} + 44 a - 21\) , \( 16 a^{3} - 14 a^{2} - 57 a + 29\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(2a^{3}-a^{2}-8a+1\right){x}^{2}+\left(-13a^{3}+8a^{2}+44a-21\right){x}+16a^{3}-14a^{2}-57a+29$ |
| 47.1-c2 |
47.1-c |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
47.1 |
\( 47 \) |
\( -47 \) |
$6.39656$ |
$(2a^3-6a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.015175492$ |
$3222.207864$ |
1.105352853 |
\( -\frac{5774625}{47} a^{3} + \frac{9047691}{47} a^{2} + \frac{7169077}{47} a - \frac{2507139}{47} \) |
\( \bigl[a^{2} - a - 2\) , \( 2 a^{3} - a^{2} - 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( 2 a^{3} - 2 a^{2} - 6 a + 4\) , \( a^{3} - a^{2} - 3 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(2a^{3}-a^{2}-8a+1\right){x}^{2}+\left(2a^{3}-2a^{2}-6a+4\right){x}+a^{3}-a^{2}-3a+1$ |
| 48.1-a1 |
48.1-a |
$2$ |
$5$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{5} \) |
$6.41341$ |
$(a^3-4a), (2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$319.8556448$ |
1.446067622 |
\( -\frac{256787}{486} a^{3} - \frac{917825}{486} a^{2} - \frac{221027}{162} a + \frac{155944}{243} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -a^{3} + 5 a + 2\) , \( -a^{3} + a^{2} + 4 a\) , \( -2 a^{3} + a^{2} + 7 a + 2\) , \( 2 a^{3} - 5 a^{2} + 2\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-2a^{3}+a^{2}+7a+2\right){x}+2a^{3}-5a^{2}+2$ |
| 48.1-a2 |
48.1-a |
$2$ |
$5$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{20} \cdot 3 \) |
$6.41341$ |
$(a^3-4a), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 5 \) |
$1$ |
$0.511769031$ |
1.446067622 |
\( \frac{2808717352846811}{48} a^{3} + \frac{2225925010621747}{96} a^{2} - \frac{7195553538165561}{32} a - \frac{14172658068106189}{96} \) |
\( \bigl[-a^{3} + a^{2} + 4 a\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - a - 2\) , \( -255 a^{3} + 230 a^{2} + 886 a - 555\) , \( -3441 a^{3} + 2714 a^{2} + 11994 a - 6178\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-255a^{3}+230a^{2}+886a-555\right){x}-3441a^{3}+2714a^{2}+11994a-6178$ |
| 48.1-b1 |
48.1-b |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{12} \cdot 3^{9} \) |
$6.41341$ |
$(a^3-4a), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$41.70561828$ |
0.942755665 |
\( -\frac{4778497131611}{157464} a^{3} - \frac{1148773643641}{78732} a^{2} + \frac{761381050117}{13122} a - \frac{370433033072}{19683} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 4 a - 1\) , \( a^{2} - 1\) , \( -6 a^{3} + a^{2} + 13 a - 4\) , \( -3 a^{3} - 2 a^{2} - 7 a + 3\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(-6a^{3}+a^{2}+13a-4\right){x}-3a^{3}-2a^{2}-7a+3$ |
| 48.1-b2 |
48.1-b |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{3} \) |
$6.41341$ |
$(a^3-4a), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$41.70561828$ |
0.942755665 |
\( \frac{938631780020}{27} a^{3} + \frac{372015936188}{27} a^{2} - \frac{2404721908865}{18} a - \frac{2368257700055}{27} \) |
\( \bigl[1\) , \( a^{3} - 4 a - 2\) , \( a^{2} - a - 2\) , \( 6 a^{3} - 13 a^{2} - 2 a + 7\) , \( 40 a^{3} - 69 a^{2} - 38 a + 21\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(6a^{3}-13a^{2}-2a+7\right){x}+40a^{3}-69a^{2}-38a+21$ |
| 49.1-a1 |
49.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{9} \) |
$6.42996$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.029401742$ |
$432.7430195$ |
1.150449917 |
\( -\frac{9690935416}{343} a^{3} - \frac{12215671997}{343} a^{2} + \frac{16657874830}{343} a + \frac{14201869411}{343} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( a\) , \( -30 a^{3} + 55 a^{2} + 22 a - 15\) , \( -267 a^{3} + 469 a^{2} + 242 a - 153\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(-30a^{3}+55a^{2}+22a-15\right){x}-267a^{3}+469a^{2}+242a-153$ |
| 49.1-a2 |
49.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{8} \) |
$6.42996$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.019601161$ |
$649.1145293$ |
1.150449917 |
\( -\frac{170631953648}{49} a^{3} + \frac{301490635046}{49} a^{2} + \frac{150363436017}{49} a - \frac{96330872074}{49} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - 4 a - 1\) , \( -63 a^{3} + 23 a^{2} + 227 a - 17\) , \( 162 a^{3} - 125 a^{2} - 566 a + 280\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-63a^{3}+23a^{2}+227a-17\right){x}+162a^{3}-125a^{2}-566a+280$ |
| 49.1-a3 |
49.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{7} \) |
$6.42996$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.009800580$ |
$1298.229058$ |
1.150449917 |
\( \frac{124820}{7} a^{3} - \frac{238513}{7} a^{2} - \frac{74080}{7} a + \frac{71744}{7} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - 3 a\) , \( a^{2} - a - 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(a^{3}-a^{2}-3a\right){x}+a^{3}-a$ |
| 49.1-a4 |
49.1-a |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{12} \) |
$6.42996$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.058803484$ |
$216.3715097$ |
1.150449917 |
\( \frac{443958861593546408990}{117649} a^{3} + \frac{915220682926034165096}{117649} a^{2} + \frac{110890939600298461445}{117649} a - \frac{215357317054416491281}{117649} \) |
\( \bigl[a\) , \( -a^{3} + 3 a + 1\) , \( -a^{3} + a^{2} + 4 a\) , \( -a^{3} - 16 a^{2} + 56 a - 44\) , \( -109 a^{3} + 141 a^{2} + 199 a - 57\bigr] \) |
${y}^2+a{x}{y}+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-a^{3}-16a^{2}+56a-44\right){x}-109a^{3}+141a^{2}+199a-57$ |
| 57.1-a1 |
57.1-a |
$8$ |
$12$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3 \cdot 19^{3} \) |
$6.55267$ |
$(a^3-4a), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 1 \) |
$1$ |
$48.45722553$ |
1.095375774 |
\( -\frac{1236877195737659}{1083} a^{3} - \frac{484396917322811}{1083} a^{2} + \frac{1580740068049001}{361} a + \frac{3118669260369488}{1083} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( -59 a^{3} + 40 a^{2} + 158 a - 173\) , \( 730 a^{3} - 219 a^{2} - 2417 a + 422\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-59a^{3}+40a^{2}+158a-173\right){x}+730a^{3}-219a^{2}-2417a+422$ |
| 57.1-a2 |
57.1-a |
$8$ |
$12$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{2} \cdot 19^{6} \) |
$6.55267$ |
$(a^3-4a), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$193.8289021$ |
1.095375774 |
\( -\frac{32864500283}{61731} a^{3} - \frac{638130530}{3249} a^{2} + \frac{41046464113}{20577} a + \frac{85760775575}{61731} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( -4 a^{3} + 8 a - 8\) , \( 12 a^{3} - 13 a^{2} - 54 a + 11\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-4a^{3}+8a-8\right){x}+12a^{3}-13a^{2}-54a+11$ |
| 57.1-a3 |
57.1-a |
$8$ |
$12$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
57.1 |
\( 3 \cdot 19 \) |
\( 3^{3} \cdot 19 \) |
$6.55267$ |
$(a^3-4a), (a^3-a^2-4a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 1 \) |
$1$ |
$48.45722553$ |
1.095375774 |
\( \frac{72942210831600947693872817724388}{513} a^{3} - \frac{50608943902051469792355565222382}{513} a^{2} - \frac{85551740515727781170115992032406}{171} a + \frac{105130945456975830283712819159561}{513} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 2 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 52 a^{3} + 99 a^{2} - 225 a - 448\) , \( 1490 a^{3} + 58 a^{2} - 5543 a - 1945\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{2}-2a-1\right){x}^{2}+\left(52a^{3}+99a^{2}-225a-448\right){x}+1490a^{3}+58a^{2}-5543a-1945$ |
*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.
