| 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 |
| 41.4-a1 |
41.4-a |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41^{5} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$6510.954223$ |
1.94068 |
\( -\frac{111795332208180804}{115856201} a^{5} + \frac{54644123139141956}{115856201} a^{4} + \frac{641562038225938648}{115856201} a^{3} - \frac{260220358529917204}{115856201} a^{2} - \frac{616867665124371048}{115856201} a + \frac{73976389687605349}{115856201} \) |
\( \bigl[a\) , \( -a^{5} + 6 a^{3} + a^{2} - 6 a - 4\) , \( a^{2} + a - 2\) , \( -2 a^{5} + 13 a^{3} - 9 a^{2} - 11 a + 7\) , \( -4 a^{5} + 3 a^{4} + 10 a^{3} - 3 a^{2} - 7 a - 2\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{5}+6a^{3}+a^{2}-6a-4\right){x}^{2}+\left(-2a^{5}+13a^{3}-9a^{2}-11a+7\right){x}-4a^{5}+3a^{4}+10a^{3}-3a^{2}-7a-2$ |
| 41.4-a2 |
41.4-a |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41^{10} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$813.8692778$ |
1.94068 |
\( -\frac{1745139273055731148197952}{13422659310152401} a^{5} + \frac{3373227824416182197653522}{13422659310152401} a^{4} + \frac{8846215934494906178049104}{13422659310152401} a^{3} - \frac{18077979504508233993782986}{13422659310152401} a^{2} - \frac{5322436363201050502227782}{13422659310152401} a + \frac{14959775970217750942840429}{13422659310152401} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + 5 a^{2} + 11 a - 1\) , \( -a^{5} + 2 a^{4} + 5 a^{3} - 9 a^{2} - 2 a + 3\) , \( a^{5} - a^{4} - 6 a^{3} + 6 a^{2} + 6 a - 4\) , \( 36 a^{5} - 20 a^{4} - 199 a^{3} + 92 a^{2} + 184 a - 31\) , \( -102 a^{5} + 51 a^{4} + 576 a^{3} - 221 a^{2} - 549 a + 38\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+5a^{2}+11a-1\right){x}{y}+\left(a^{5}-a^{4}-6a^{3}+6a^{2}+6a-4\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+5a^{3}-9a^{2}-2a+3\right){x}^{2}+\left(36a^{5}-20a^{4}-199a^{3}+92a^{2}+184a-31\right){x}-102a^{5}+51a^{4}+576a^{3}-221a^{2}-549a+38$ |
| 41.4-b1 |
41.4-b |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41 \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$4678.254988$ |
1.39442 |
\( \frac{132422900}{41} a^{5} - \frac{141704392}{41} a^{4} - \frac{798145247}{41} a^{3} + \frac{689842493}{41} a^{2} + \frac{906291495}{41} a - \frac{237335216}{41} \) |
\( \bigl[2 a^{5} - 2 a^{4} - 11 a^{3} + 10 a^{2} + 9 a - 4\) , \( -a^{5} + a^{4} + 5 a^{3} - 4 a^{2} - 3 a - 1\) , \( 3 a^{5} - 2 a^{4} - 17 a^{3} + 10 a^{2} + 17 a - 2\) , \( -6 a^{5} + 5 a^{4} + 34 a^{3} - 23 a^{2} - 33 a + 4\) , \( -6 a^{5} + 3 a^{4} + 36 a^{3} - 15 a^{2} - 43 a + 5\bigr] \) |
${y}^2+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+9a-4\right){x}{y}+\left(3a^{5}-2a^{4}-17a^{3}+10a^{2}+17a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-4a^{2}-3a-1\right){x}^{2}+\left(-6a^{5}+5a^{4}+34a^{3}-23a^{2}-33a+4\right){x}-6a^{5}+3a^{4}+36a^{3}-15a^{2}-43a+5$ |
| 41.4-b2 |
41.4-b |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( - 41^{2} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$584.7818735$ |
1.39442 |
\( -\frac{46785115689974520}{1681} a^{5} + \frac{25523470799745310}{1681} a^{4} + \frac{267495305548152369}{1681} a^{3} - \frac{124251905123044313}{1681} a^{2} - \frac{253745891578685595}{1681} a + \frac{46114837230979857}{1681} \) |
\( \bigl[2 a^{5} - 2 a^{4} - 11 a^{3} + 10 a^{2} + 9 a - 4\) , \( -a^{5} + a^{4} + 5 a^{3} - 4 a^{2} - 3 a - 1\) , \( 3 a^{5} - 2 a^{4} - 17 a^{3} + 10 a^{2} + 17 a - 2\) , \( 19 a^{5} - 10 a^{4} - 106 a^{3} + 47 a^{2} + 92 a - 16\) , \( 55 a^{5} - 29 a^{4} - 312 a^{3} + 137 a^{2} + 286 a - 40\bigr] \) |
${y}^2+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+9a-4\right){x}{y}+\left(3a^{5}-2a^{4}-17a^{3}+10a^{2}+17a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-4a^{2}-3a-1\right){x}^{2}+\left(19a^{5}-10a^{4}-106a^{3}+47a^{2}+92a-16\right){x}+55a^{5}-29a^{4}-312a^{3}+137a^{2}+286a-40$ |
| 41.4-c1 |
41.4-c |
$1$ |
$1$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41 \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1351.437236$ |
1.61126 |
\( \frac{125711789726}{41} a^{5} - \frac{61449523288}{41} a^{4} - \frac{721420682984}{41} a^{3} + \frac{292627885035}{41} a^{2} + \frac{693639106205}{41} a - \frac{83187425920}{41} \) |
\( \bigl[a^{5} - 6 a^{3} + a^{2} + 8 a\) , \( -2 a^{5} + 2 a^{4} + 11 a^{3} - 10 a^{2} - 10 a + 4\) , \( a + 1\) , \( 5 a^{5} - a^{4} - 27 a^{3} + 8 a^{2} + 26 a - 1\) , \( -a^{5} + 4 a^{4} + 10 a^{3} - 14 a^{2} - 14 a + 4\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}+a^{2}+8a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a^{5}+2a^{4}+11a^{3}-10a^{2}-10a+4\right){x}^{2}+\left(5a^{5}-a^{4}-27a^{3}+8a^{2}+26a-1\right){x}-a^{5}+4a^{4}+10a^{3}-14a^{2}-14a+4$ |
| 41.4-d1 |
41.4-d |
$1$ |
$1$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41^{7} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1358.459986$ |
1.61963 |
\( \frac{2193085400250940456}{194754273881} a^{5} - \frac{9040335834589460238}{194754273881} a^{4} + \frac{8220320611440950985}{194754273881} a^{3} + \frac{6678172413832281891}{194754273881} a^{2} - \frac{9786903378511537191}{194754273881} a + \frac{1033336974108669203}{194754273881} \) |
\( \bigl[2 a^{5} - a^{4} - 12 a^{3} + 6 a^{2} + 13 a - 2\) , \( -2 a^{5} + 2 a^{4} + 11 a^{3} - 11 a^{2} - 9 a + 5\) , \( a^{5} - 6 a^{3} + 8 a + 2\) , \( -2 a^{5} + 3 a^{4} + 10 a^{3} - 16 a^{2} - 7 a + 14\) , \( -6 a^{5} + 16 a^{4} + 27 a^{3} - 92 a^{2} - 3 a + 86\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-12a^{3}+6a^{2}+13a-2\right){x}{y}+\left(a^{5}-6a^{3}+8a+2\right){y}={x}^{3}+\left(-2a^{5}+2a^{4}+11a^{3}-11a^{2}-9a+5\right){x}^{2}+\left(-2a^{5}+3a^{4}+10a^{3}-16a^{2}-7a+14\right){x}-6a^{5}+16a^{4}+27a^{3}-92a^{2}-3a+86$ |
| 41.4-e1 |
41.4-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( - 41^{2} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$1.477785976$ |
$5.976442953$ |
2.55877 |
\( \frac{36649377214722269688376981404391}{1681} a^{5} + \frac{4804213580822606147447044113221}{1681} a^{4} - \frac{173008694646261741480385301974650}{1681} a^{3} + \frac{34446773860453626004806361131099}{1681} a^{2} + \frac{146707785669385287322288443886585}{1681} a - \frac{17197513544561278871243743300424}{1681} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + 6 a^{2} + 10 a - 3\) , \( -a^{5} + 2 a^{4} + 5 a^{3} - 10 a^{2} - a + 6\) , \( a^{5} - 6 a^{3} + a^{2} + 7 a - 1\) , \( 78 a^{5} - 247 a^{4} + 122 a^{3} + 129 a^{2} - 104 a + 23\) , \( 2279 a^{5} - 9254 a^{4} + 8032 a^{3} + 6740 a^{2} - 9382 a + 993\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+6a^{2}+10a-3\right){x}{y}+\left(a^{5}-6a^{3}+a^{2}+7a-1\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+5a^{3}-10a^{2}-a+6\right){x}^{2}+\left(78a^{5}-247a^{4}+122a^{3}+129a^{2}-104a+23\right){x}+2279a^{5}-9254a^{4}+8032a^{3}+6740a^{2}-9382a+993$ |
| 41.4-e2 |
41.4-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41 \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$0.738892988$ |
$23.90577181$ |
2.55877 |
\( \frac{1661681620471232}{41} a^{5} + \frac{213704000156457}{41} a^{4} - \frac{7841722250015960}{41} a^{3} + \frac{1577558381739890}{41} a^{2} + \frac{6635361185868735}{41} a - \frac{778052671995566}{41} \) |
\( \bigl[2 a^{5} - a^{4} - 12 a^{3} + 5 a^{2} + 14 a - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 7 a + 4\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 11 a^{2} + 9 a - 5\) , \( -30 a^{5} + 3 a^{4} + 169 a^{3} - 71 a^{2} - 156 a + 28\) , \( -115 a^{5} - 28 a^{4} + 635 a^{3} - 157 a^{2} - 561 a + 63\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+14a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-7a+4\right){x}^{2}+\left(-30a^{5}+3a^{4}+169a^{3}-71a^{2}-156a+28\right){x}-115a^{5}-28a^{4}+635a^{3}-157a^{2}-561a+63$ |
| 41.4-e3 |
41.4-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( - 41^{6} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.492595325$ |
$4356.826913$ |
2.55877 |
\( \frac{2361415517723691336}{4750104241} a^{5} + \frac{312405531641926157}{4750104241} a^{4} - \frac{11145383118536595673}{4750104241} a^{3} + \frac{2208505528549032067}{4750104241} a^{2} + \frac{9451172168707485868}{4750104241} a - \frac{1100146408697507811}{4750104241} \) |
\( \bigl[2 a^{5} - a^{4} - 12 a^{3} + 5 a^{2} + 14 a - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 7 a + 4\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 11 a^{2} + 9 a - 5\) , \( -5 a^{5} - 2 a^{4} + 29 a^{3} - a^{2} - 31 a - 2\) , \( -10 a^{5} - 2 a^{4} + 55 a^{3} - 13 a^{2} - 49 a + 5\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+14a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-7a+4\right){x}^{2}+\left(-5a^{5}-2a^{4}+29a^{3}-a^{2}-31a-2\right){x}-10a^{5}-2a^{4}+55a^{3}-13a^{2}-49a+5$ |
| 41.4-e4 |
41.4-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( 41^{3} \) |
$102.13324$ |
$(a^5-a^4-6a^3+4a^2+7a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$0.246297662$ |
$17427.30765$ |
2.55877 |
\( \frac{540166855}{68921} a^{5} - \frac{112536867}{68921} a^{4} - \frac{2583214205}{68921} a^{3} + \frac{1175827261}{68921} a^{2} + \frac{1962941452}{68921} a - \frac{284998027}{68921} \) |
\( \bigl[2 a^{5} - a^{4} - 12 a^{3} + 5 a^{2} + 14 a - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 7 a + 4\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 11 a^{2} + 9 a - 5\) , \( -5 a^{5} + 3 a^{4} + 29 a^{3} - 16 a^{2} - 31 a + 8\) , \( -a^{5} + 6 a^{3} - a^{2} - 6 a - 2\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+14a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-7a+4\right){x}^{2}+\left(-5a^{5}+3a^{4}+29a^{3}-16a^{2}-31a+8\right){x}-a^{5}+6a^{3}-a^{2}-6a-2$ |
*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.
