| 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 |
| 441.1-a1 |
441.1-a |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{3} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$38.42360582$ |
1.737131522 |
\( -4253184708638 a^{3} - 8767948339232 a^{2} - 1062381721538 a + 2063135777685 \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a + 1\) , \( 36 a^{3} + 7 a^{2} - 144 a - 76\) , \( -129 a^{3} - 77 a^{2} + 493 a + 402\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(36a^{3}+7a^{2}-144a-76\right){x}-129a^{3}-77a^{2}+493a+402$ |
| 441.1-a2 |
441.1-a |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{9} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$38.42360582$ |
1.737131522 |
\( -2461 a^{3} - 4706 a^{2} - 417 a + 1053 \) |
\( \bigl[1\) , \( a^{2} - a - 1\) , \( a^{2} - a - 1\) , \( -3 a^{3} - 3 a^{2} + a + 1\) , \( -6 a^{3} - 12 a^{2} - a + 3\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-3a^{3}-3a^{2}+a+1\right){x}-6a^{3}-12a^{2}-a+3$ |
| 441.1-b1 |
441.1-b |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{9} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 2 \cdot 3 \) |
$1$ |
$1.162524344$ |
1.419058838 |
\( -4253184708638 a^{3} - 8767948339232 a^{2} - 1062381721538 a + 2063135777685 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 4 a + 2\) , \( 1\) , \( 22 a^{3} - 9 a^{2} - 84 a - 9\) , \( -71 a^{3} + 84 a^{2} + 221 a - 283\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(22a^{3}-9a^{2}-84a-9\right){x}-71a^{3}+84a^{2}+221a-283$ |
| 441.1-b2 |
441.1-b |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{3} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$94.16447188$ |
1.419058838 |
\( -2461 a^{3} - 4706 a^{2} - 417 a + 1053 \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 4 a + 2\) , \( 1\) , \( -3 a^{3} + a^{2} + 11 a + 1\) , \( 4 a^{3} - 3 a^{2} - 14 a + 7\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(-3a^{3}+a^{2}+11a+1\right){x}+4a^{3}-3a^{2}-14a+7$ |
| 441.1-c1 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{18} \cdot 7^{30} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{3} \) |
$1$ |
$1.866360265$ |
1.350050597 |
\( \frac{981621700851969745102419250048115}{101814121186119595835681841} a^{3} + \frac{1761760795662379320695876261373041}{101814121186119595835681841} a^{2} - \frac{37671995275184386601460285791065}{33938040395373198611893947} a - \frac{293036539712498872630739037955790}{101814121186119595835681841} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( -1677 a^{3} + 939 a^{2} + 5859 a - 2255\) , \( -47583 a^{3} + 34144 a^{2} + 170350 a - 68930\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-1677a^{3}+939a^{2}+5859a-2255\right){x}-47583a^{3}+34144a^{2}+170350a-68930$ |
| 441.1-c2 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{30} \cdot 7^{18} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$14.93088212$ |
1.350050597 |
\( -\frac{49478337663774144106792217}{3909188328478827879681} a^{3} + \frac{86867577756332942729622262}{3909188328478827879681} a^{2} + \frac{19815761443619939196843979}{1303062776159609293227} a - \frac{6943686789535186694527711}{3909188328478827879681} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( -72 a^{3} + 29 a^{2} + 314 a - 195\) , \( -816 a^{3} + 684 a^{2} + 2568 a - 873\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-72a^{3}+29a^{2}+314a-195\right){x}-816a^{3}+684a^{2}+2568a-873$ |
| 441.1-c3 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{54} \cdot 7^{12} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{3} \) |
$1$ |
$1.866360265$ |
1.350050597 |
\( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( 653 a^{3} - 161 a^{2} - 2351 a - 135\) , \( -7041 a^{3} + 1756 a^{2} + 24858 a + 3620\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(653a^{3}-161a^{2}-2351a-135\right){x}-7041a^{3}+1756a^{2}+24858a+3620$ |
| 441.1-c4 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{18} \cdot 7^{12} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$59.72352848$ |
1.350050597 |
\( -\frac{84413562014875846901}{62523502209} a^{3} + \frac{148907375867724857512}{62523502209} a^{2} + \frac{24992949793645640572}{20841167403} a - \frac{47851964758998446149}{62523502209} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( -17 a^{3} - 16 a^{2} + 134 a - 70\) , \( 67 a^{3} + 106 a^{2} - 600 a + 193\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-17a^{3}-16a^{2}+134a-70\right){x}+67a^{3}+106a^{2}-600a+193$ |
| 441.1-c5 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{9} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$59.72352848$ |
1.350050597 |
\( -\frac{4584696391003}{250047} a^{3} - \frac{1823709580711}{250047} a^{2} + \frac{5875898465972}{83349} a + \frac{11577122282035}{250047} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a\) , \( a^{2} - a - 2\) , \( 8 a^{3} - a^{2} - 26 a - 10\) , \( 20 a^{3} - 7 a^{2} - 73 a + 3\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(8a^{3}-a^{2}-26a-10\right){x}+20a^{3}-7a^{2}-73a+3$ |
| 441.1-c6 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{9} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$14.93088212$ |
1.350050597 |
\( -\frac{4031277404041470845619545}{250047} a^{3} + \frac{7111233508460767828860022}{250047} a^{2} + \frac{1193595856869520566460555}{83349} a - \frac{2285285314923636665330287}{250047} \) |
\( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a + 1\) , \( a^{2} - a - 1\) , \( -1003 a^{3} - 481 a^{2} + 3698 a + 2361\) , \( 63410 a^{3} + 24210 a^{2} - 246112 a - 160817\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-1003a^{3}-481a^{2}+3698a+2361\right){x}+63410a^{3}+24210a^{2}-246112a-160817$ |
| 441.1-c7 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{10} \cdot 7^{14} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{3} \) |
$1$ |
$1.866360265$ |
1.350050597 |
\( \frac{3925517460083619669474871874}{466948881} a^{3} - \frac{2723612167164915292630390720}{466948881} a^{2} - \frac{4604122185339962156855866072}{155649627} a + \frac{5657812633917689859889126471}{466948881} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 2\) , \( a + 1\) , \( -94 a^{3} + 331 a^{2} + 424 a - 1554\) , \( -12101 a^{3} + 1847 a^{2} + 46771 a + 1669\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-94a^{3}+331a^{2}+424a-1554\right){x}-12101a^{3}+1847a^{2}+46771a+1669$ |
| 441.1-c8 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{10} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$59.72352848$ |
1.350050597 |
\( \frac{18710001614848}{3969} a^{3} + \frac{38571223778296}{3969} a^{2} + \frac{1558338097840}{1323} a - \frac{9074746748287}{3969} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 2\) , \( a + 1\) , \( 16 a^{3} + a^{2} - 76 a - 54\) , \( -105 a^{3} - 65 a^{2} + 341 a + 227\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(16a^{3}+a^{2}-76a-54\right){x}-105a^{3}-65a^{2}+341a+227$ |
| 441.1-c9 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{8} \cdot 7^{7} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$14.93088212$ |
1.350050597 |
\( \frac{24573261493292438142976}{63} a^{3} + \frac{50657750325995678467324}{63} a^{2} + \frac{2045949367379882389504}{21} a - \frac{11920094685054769076767}{63} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 2\) , \( a + 1\) , \( -39 a^{3} - 139 a^{2} - 66 a - 4\) , \( -992 a^{3} - 1675 a^{2} + 343 a + 721\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-39a^{3}-139a^{2}-66a-4\right){x}-992a^{3}-1675a^{2}+343a+721$ |
| 441.1-c10 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{8} \cdot 7^{7} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$59.72352848$ |
1.350050597 |
\( -\frac{1064704}{63} a^{3} - \frac{2122864}{63} a^{2} - \frac{47392}{21} a + \frac{567631}{63} \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 2\) , \( a + 1\) , \( a^{3} + a^{2} - 6 a - 4\) , \( -2 a^{3} - a^{2} + 5 a + 3\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{3}+a^{2}-6a-4\right){x}-2a^{3}-a^{2}+5a+3$ |
| 441.1-c11 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{22} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$16$ |
\( 2^{3} \) |
$1$ |
$1.866360265$ |
1.350050597 |
\( -\frac{29879008921030774408202450507522}{2109289329} a^{3} - \frac{11842202503929076694840862982208}{2109289329} a^{2} + \frac{38274171513246722984375757845464}{703096443} a + \frac{75387595661097076378967854941161}{2109289329} \) |
\( \bigl[-a^{3} + a^{2} + 4 a - 1\) , \( -a^{3} + 3 a + 1\) , \( a^{2} - a - 1\) , \( 69 a^{3} + 13 a^{2} - 484 a - 342\) , \( -1378 a^{3} - 6629 a^{2} + 2521 a + 4423\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(69a^{3}+13a^{2}-484a-342\right){x}-1378a^{3}-6629a^{2}+2521a+4423$ |
| 441.1-c12 |
441.1-c |
$12$ |
$24$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{14} \cdot 7^{10} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$14.93088212$ |
1.350050597 |
\( -\frac{921288230848937984}{15752961} a^{3} - \frac{392913302754752516}{15752961} a^{2} + \frac{1182898550569873408}{5250987} a + \frac{2425491268899001073}{15752961} \) |
\( \bigl[-a^{3} + a^{2} + 4 a - 1\) , \( -a^{3} + 3 a + 1\) , \( a^{2} - a - 1\) , \( -16 a^{3} - 227 a^{2} - 69 a + 33\) , \( -517 a^{3} - 2691 a^{2} - 516 a + 708\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-16a^{3}-227a^{2}-69a+33\right){x}-517a^{3}-2691a^{2}-516a+708$ |
| 441.1-d1 |
441.1-d |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{14} \cdot 7^{9} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$36.66173317$ |
1.657477246 |
\( \frac{20039971932517}{6561} a^{3} - \frac{13829210189696}{6561} a^{2} - \frac{23471717262752}{2187} a + \frac{28832925148397}{6561} \) |
\( \bigl[a^{2} - 2\) , \( -2 a^{3} + a^{2} + 6 a\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( -54 a^{3} + 54 a^{2} + 191 a - 164\) , \( 713 a^{3} - 298 a^{2} - 2585 a + 332\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{3}+a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+6a\right){x}^{2}+\left(-54a^{3}+54a^{2}+191a-164\right){x}+713a^{3}-298a^{2}-2585a+332$ |
| 441.1-d2 |
441.1-d |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{10} \cdot 7^{9} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$73.32346635$ |
1.657477246 |
\( -\frac{34745989}{81} a^{3} - \frac{11143897}{81} a^{2} + \frac{44198156}{27} a + \frac{77795461}{81} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 3 a - 1\) , \( a^{3} - 4 a^{2}\) , \( -6 a^{3} + 10 a^{2} + 4 a - 4\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(a^{3}-4a^{2}\right){x}-6a^{3}+10a^{2}+4a-4$ |
| 441.1-e1 |
441.1-e |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{3} \cdot 7^{2} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.054368851$ |
$297.1433400$ |
2.921532588 |
\( -4253184708638 a^{3} - 8767948339232 a^{2} - 1062381721538 a + 2063135777685 \) |
\( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -1\) , \( a^{3} - 3 a\) , \( 11 a^{3} - 8 a^{2} - 38 a + 13\) , \( -79 a^{3} + 53 a^{2} + 281 a - 112\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}-{x}^{2}+\left(11a^{3}-8a^{2}-38a+13\right){x}-79a^{3}+53a^{2}+281a-112$ |
| 441.1-e2 |
441.1-e |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{9} \cdot 7^{2} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.018122950$ |
$891.4300201$ |
2.921532588 |
\( -2461 a^{3} - 4706 a^{2} - 417 a + 1053 \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a\) , \( a\) , \( -2 a + 1\) , \( -a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-2a+1\right){x}-a+1$ |
| 441.1-f1 |
441.1-f |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{3} \cdot 7^{2} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.047655794$ |
$326.3833127$ |
2.812795988 |
\( -2461 a^{3} - 4706 a^{2} - 417 a + 1053 \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( 0\) , \( a^{2} - a\) , \( a^{3} - 3 a - 2\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(a^{2}-a\right){x}+a^{3}-3a-2$ |
| 441.1-f2 |
441.1-f |
$2$ |
$3$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{9} \cdot 7^{2} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.142967383$ |
$108.7944375$ |
2.812795988 |
\( -4253184708638 a^{3} - 8767948339232 a^{2} - 1062381721538 a + 2063135777685 \) |
\( \bigl[a^{3} - 4 a\) , \( 2 a^{3} - a^{2} - 7 a\) , \( 0\) , \( -8 a^{3} - 32 a^{2} - 18 a + 10\) , \( 81 a^{3} + 178 a^{2} + 16 a - 40\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(2a^{3}-a^{2}-7a\right){x}^{2}+\left(-8a^{3}-32a^{2}-18a+10\right){x}+81a^{3}+178a^{2}+16a-40$ |
| 441.1-g1 |
441.1-g |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{10} \cdot 7^{3} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$67.10358446$ |
1.516876791 |
\( -\frac{34745989}{81} a^{3} - \frac{11143897}{81} a^{2} + \frac{44198156}{27} a + \frac{77795461}{81} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 4 a\) , \( 1\) , \( 3 a^{3} - 13 a - 5\) , \( 3 a^{3} - 9 a - 5\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(3a^{3}-13a-5\right){x}+3a^{3}-9a-5$ |
| 441.1-g2 |
441.1-g |
$2$ |
$2$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{14} \cdot 7^{3} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$33.55179223$ |
1.516876791 |
\( \frac{20039971932517}{6561} a^{3} - \frac{13829210189696}{6561} a^{2} - \frac{23471717262752}{2187} a + \frac{28832925148397}{6561} \) |
\( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 4 a\) , \( 1\) , \( 3 a^{3} - 20 a^{2} - 13 a\) , \( 46 a^{3} - 18 a^{2} - 27 a - 3\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(3a^{3}-20a^{2}-13a\right){x}+46a^{3}-18a^{2}-27a-3$ |
| 441.1-h1 |
441.1-h |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{12} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$37.69645068$ |
1.704256832 |
\( \frac{443958861593546408990}{117649} a^{3} + \frac{915220682926034165096}{117649} a^{2} + \frac{110890939600298461445}{117649} a - \frac{215357317054416491281}{117649} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 1\) , \( 56 a^{3} - 113 a^{2} - 86 a + 14\) , \( 427 a^{3} - 517 a^{2} - 397 a + 115\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(56a^{3}-113a^{2}-86a+14\right){x}+427a^{3}-517a^{2}-397a+115$ |
| 441.1-h2 |
441.1-h |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{9} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$75.39290136$ |
1.704256832 |
\( -\frac{9690935416}{343} a^{3} - \frac{12215671997}{343} a^{2} + \frac{16657874830}{343} a + \frac{14201869411}{343} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - a^{2} - 2 a + 2\) , \( a^{2} - a - 1\) , \( -2 a^{3} + 31 a^{2} - 59 a + 9\) , \( 251 a^{3} - 384 a^{2} - 357 a + 169\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+2\right){x}^{2}+\left(-2a^{3}+31a^{2}-59a+9\right){x}+251a^{3}-384a^{2}-357a+169$ |
| 441.1-h3 |
441.1-h |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{8} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$37.69645068$ |
1.704256832 |
\( -\frac{170631953648}{49} a^{3} + \frac{301490635046}{49} a^{2} + \frac{150363436017}{49} a - \frac{96330872074}{49} \) |
\( \bigl[a + 1\) , \( 2 a^{3} - a^{2} - 7 a - 1\) , \( a^{2} - 2\) , \( -137 a^{3} + 85 a^{2} + 483 a - 166\) , \( -1133 a^{3} + 792 a^{2} + 3986 a - 1653\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{3}-a^{2}-7a-1\right){x}^{2}+\left(-137a^{3}+85a^{2}+483a-166\right){x}-1133a^{3}+792a^{2}+3986a-1653$ |
| 441.1-h4 |
441.1-h |
$4$ |
$6$ |
4.4.1957.1 |
$4$ |
$[4, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{7} \) |
$8.46231$ |
$(a^3-4a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$75.39290136$ |
1.704256832 |
\( \frac{124820}{7} a^{3} - \frac{238513}{7} a^{2} - \frac{74080}{7} a + \frac{71744}{7} \) |
\( \bigl[a + 1\) , \( 2 a^{3} - a^{2} - 7 a - 1\) , \( a^{2} - 2\) , \( -7 a^{3} + 5 a^{2} + 23 a - 11\) , \( -26 a^{3} + 16 a^{2} + 92 a - 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{3}-a^{2}-7a-1\right){x}^{2}+\left(-7a^{3}+5a^{2}+23a-11\right){x}-26a^{3}+16a^{2}+92a-31$ |
*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.
