| 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 |
| 41.3-a1 |
41.3-a |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41^{10} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$813.8692778$ |
1.94068 |
\( \frac{3683207165193102481237192}{13422659310152401} a^{5} - \frac{1642339292420493726158390}{13422659310152401} a^{4} - \frac{20474623287319134176284544}{13422659310152401} a^{3} + \frac{229842362061702235031886}{327381934393961} a^{2} + \frac{20619800140158338304997594}{13422659310152401} a - \frac{2484108563337772728271739}{13422659310152401} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 2\) , \( 2 a^{5} - a^{4} - 12 a^{3} + 5 a^{2} + 13 a - 1\) , \( a^{5} - 6 a^{3} + 8 a + 2\) , \( -49 a^{5} + 35 a^{4} + 281 a^{3} - 160 a^{2} - 275 a + 11\) , \( 136 a^{5} - 103 a^{4} - 781 a^{3} + 472 a^{2} + 745 a - 62\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-2\right){x}{y}+\left(a^{5}-6a^{3}+8a+2\right){y}={x}^{3}+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+13a-1\right){x}^{2}+\left(-49a^{5}+35a^{4}+281a^{3}-160a^{2}-275a+11\right){x}+136a^{5}-103a^{4}-781a^{3}+472a^{2}+745a-62$ |
| 41.3-a2 |
41.3-a |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41^{5} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$6510.954223$ |
1.94068 |
\( \frac{7619330091674956}{115856201} a^{5} - \frac{22560326971775052}{115856201} a^{4} - \frac{16506025526903560}{115856201} a^{3} + \frac{2434179943733724}{2825761} a^{2} - \frac{80280553523802984}{115856201} a + \frac{7994447419048309}{115856201} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 2\) , \( 2 a^{5} - a^{4} - 12 a^{3} + 5 a^{2} + 13 a - 1\) , \( a^{5} - 6 a^{3} + 8 a + 2\) , \( a^{5} + 5 a^{4} - 14 a^{3} - 20 a^{2} + 40 a - 9\) , \( 12 a^{5} - 24 a^{4} - 45 a^{3} + 110 a^{2} - 43 a - 8\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-2\right){x}{y}+\left(a^{5}-6a^{3}+8a+2\right){y}={x}^{3}+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+13a-1\right){x}^{2}+\left(a^{5}+5a^{4}-14a^{3}-20a^{2}+40a-9\right){x}+12a^{5}-24a^{4}-45a^{3}+110a^{2}-43a-8$ |
| 41.3-b1 |
41.3-b |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41 \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$4678.254988$ |
1.39442 |
\( -\frac{716692}{41} a^{5} - \frac{23636420}{41} a^{4} + \frac{7907999}{41} a^{3} + 3338087 a^{2} - \frac{149688851}{41} a + \frac{15615439}{41} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + 6 a^{2} + 10 a - 3\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 6 a + 5\) , \( 0\) , \( 2 a^{5} + a^{4} - 10 a^{3} - 3 a^{2} + 10 a + 5\) , \( 0\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+6a^{2}+10a-3\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-6a+5\right){x}^{2}+\left(2a^{5}+a^{4}-10a^{3}-3a^{2}+10a+5\right){x}$ |
| 41.3-b2 |
41.3-b |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( - 41^{2} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$584.7818735$ |
1.39442 |
\( \frac{4227054751862476}{1681} a^{5} - \frac{8907862174714698}{1681} a^{4} - \frac{12146939919480105}{1681} a^{3} + \frac{1004240536533933}{41} a^{2} - \frac{27544926363068101}{1681} a + \frac{2659697336777508}{1681} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + 5 a^{2} + 10 a\) , \( a^{4} - 6 a^{2} + a + 6\) , \( 2 a^{5} - a^{4} - 12 a^{3} + 6 a^{2} + 14 a - 2\) , \( -19 a^{5} + 17 a^{4} + 104 a^{3} - 82 a^{2} - 82 a + 23\) , \( -22 a^{5} + 24 a^{4} + 107 a^{3} - 112 a^{2} - 37 a + 12\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+5a^{2}+10a\right){x}{y}+\left(2a^{5}-a^{4}-12a^{3}+6a^{2}+14a-2\right){y}={x}^{3}+\left(a^{4}-6a^{2}+a+6\right){x}^{2}+\left(-19a^{5}+17a^{4}+104a^{3}-82a^{2}-82a+23\right){x}-22a^{5}+24a^{4}+107a^{3}-112a^{2}-37a+12$ |
| 41.3-c1 |
41.3-c |
$1$ |
$1$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41 \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1351.437236$ |
1.61126 |
\( -\frac{8584162099}{41} a^{5} + \frac{25376641760}{41} a^{4} + \frac{18654917222}{41} a^{3} - 2738133595 a^{2} + \frac{90181405656}{41} a - \frac{8977485322}{41} \) |
\( \bigl[a^{5} - 6 a^{3} + a^{2} + 7 a - 1\) , \( -a^{4} + 4 a^{2} - 2 a - 2\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 11 a^{2} + 10 a - 6\) , \( -a^{5} - a^{4} + 6 a^{3} + 4 a^{2} - 9 a - 2\) , \( 4 a^{5} - 18 a^{3} + 8 a^{2} + 13 a - 8\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}+a^{2}+7a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+10a-6\right){y}={x}^{3}+\left(-a^{4}+4a^{2}-2a-2\right){x}^{2}+\left(-a^{5}-a^{4}+6a^{3}+4a^{2}-9a-2\right){x}+4a^{5}-18a^{3}+8a^{2}+13a-8$ |
| 41.3-d1 |
41.3-d |
$1$ |
$1$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41^{7} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1358.459986$ |
1.61963 |
\( -\frac{208407593862926594044}{194754273881} a^{5} + \frac{141658003656526461242}{194754273881} a^{4} + \frac{1229066730164612970543}{194754273881} a^{3} - \frac{16335768573744324071}{4750104241} a^{2} - \frac{1301096988038281036921}{194754273881} a + \frac{157850815853926536686}{194754273881} \) |
\( \bigl[a^{2} - 1\) , \( -a^{5} + 7 a^{3} - a^{2} - 9 a + 2\) , \( 0\) , \( -a^{5} + 2 a^{4} + 7 a^{3} - 8 a^{2} - 10 a + 9\) , \( 7 a^{5} - 32 a^{3} + 11 a^{2} + 23 a - 4\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(-a^{5}+7a^{3}-a^{2}-9a+2\right){x}^{2}+\left(-a^{5}+2a^{4}+7a^{3}-8a^{2}-10a+9\right){x}+7a^{5}-32a^{3}+11a^{2}+23a-4$ |
| 41.3-e1 |
41.3-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41^{3} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$0.246297662$ |
$17427.30765$ |
2.55877 |
\( -\frac{1132366102}{68921} a^{5} + \frac{1515352091}{68921} a^{4} + \frac{6136409687}{68921} a^{3} - \frac{199753741}{1681} a^{2} - \frac{4705520957}{68921} a + \frac{6304165628}{68921} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + 6 a^{2} + 10 a - 2\) , \( -3 a^{5} + 2 a^{4} + 17 a^{3} - 10 a^{2} - 15 a + 2\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 10 a^{2} + 10 a - 3\) , \( 2 a^{4} + 3 a^{3} - 5 a^{2} - 6 a\) , \( -7 a^{5} + 9 a^{4} + 47 a^{3} - 35 a^{2} - 55 a + 6\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+6a^{2}+10a-2\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+10a-3\right){y}={x}^{3}+\left(-3a^{5}+2a^{4}+17a^{3}-10a^{2}-15a+2\right){x}^{2}+\left(2a^{4}+3a^{3}-5a^{2}-6a\right){x}-7a^{5}+9a^{4}+47a^{3}-35a^{2}-55a+6$ |
| 41.3-e2 |
41.3-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( - 41^{6} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.492595325$ |
$4356.826913$ |
2.55877 |
\( -\frac{3868146036397569720}{4750104241} a^{5} + \frac{7289390284716297722}{4750104241} a^{4} + \frac{20185766230579865977}{4750104241} a^{3} - \frac{980914258788784182}{115856201} a^{2} - \frac{12396489983066410677}{4750104241} a + \frac{33384677007660926967}{4750104241} \) |
\( \bigl[2 a^{5} - a^{4} - 11 a^{3} + 6 a^{2} + 10 a - 2\) , \( -3 a^{5} + 2 a^{4} + 17 a^{3} - 10 a^{2} - 15 a + 2\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 10 a^{2} + 10 a - 3\) , \( -70 a^{5} + 52 a^{4} + 413 a^{3} - 240 a^{2} - 431 a + 50\) , \( -248 a^{5} + 166 a^{4} + 1478 a^{3} - 780 a^{2} - 1603 a + 192\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-11a^{3}+6a^{2}+10a-2\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+10a-3\right){y}={x}^{3}+\left(-3a^{5}+2a^{4}+17a^{3}-10a^{2}-15a+2\right){x}^{2}+\left(-70a^{5}+52a^{4}+413a^{3}-240a^{2}-431a+50\right){x}-248a^{5}+166a^{4}+1478a^{3}-780a^{2}-1603a+192$ |
| 41.3-e3 |
41.3-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( 41 \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$0.738892988$ |
$23.90577181$ |
2.55877 |
\( -\frac{2734619766578096}{41} a^{5} + \frac{5127716296916528}{41} a^{4} + \frac{14279351126657144}{41} a^{3} - 689869752856215 a^{2} - \frac{8804507911543798}{41} a + \frac{23455352094983401}{41} \) |
\( \bigl[a^{5} - 6 a^{3} + 8 a + 2\) , \( -a^{5} + 2 a^{4} + 5 a^{3} - 9 a^{2} - 2 a + 2\) , \( a^{2} - 2\) , \( -114 a^{5} + 45 a^{4} + 697 a^{3} - 169 a^{2} - 813 a - 140\) , \( -618 a^{5} + 161 a^{4} + 3799 a^{3} - 440 a^{2} - 4532 a - 1239\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}+8a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+5a^{3}-9a^{2}-2a+2\right){x}^{2}+\left(-114a^{5}+45a^{4}+697a^{3}-169a^{2}-813a-140\right){x}-618a^{5}+161a^{4}+3799a^{3}-440a^{2}-4532a-1239$ |
| 41.3-e4 |
41.3-e |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( - 41^{2} \) |
$102.13324$ |
$(a^3-a^2-4a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \) |
$1.477785976$ |
$5.976442953$ |
2.55877 |
\( -\frac{59985851336154346732878490305700}{1681} a^{5} + \frac{113019715254194002660867511068820}{1681} a^{4} + \frac{313027539374854203747394355382504}{1681} a^{3} - \frac{15208937025256991952350710659544}{41} a^{2} - \frac{192239175684393217825484451013707}{1681} a + \frac{517593336136737221171637388394509}{1681} \) |
\( \bigl[3 a^{5} - 2 a^{4} - 17 a^{3} + 10 a^{2} + 17 a - 2\) , \( 2 a^{5} - a^{4} - 11 a^{3} + 4 a^{2} + 9 a + 1\) , \( a^{5} - a^{4} - 5 a^{3} + 6 a^{2} + 4 a - 3\) , \( 183 a^{5} - 297 a^{4} - 948 a^{3} + 1709 a^{2} + 549 a - 1692\) , \( 2440 a^{5} - 5092 a^{4} - 12258 a^{3} + 28901 a^{2} + 5669 a - 26972\bigr] \) |
${y}^2+\left(3a^{5}-2a^{4}-17a^{3}+10a^{2}+17a-2\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+6a^{2}+4a-3\right){y}={x}^{3}+\left(2a^{5}-a^{4}-11a^{3}+4a^{2}+9a+1\right){x}^{2}+\left(183a^{5}-297a^{4}-948a^{3}+1709a^{2}+549a-1692\right){x}+2440a^{5}-5092a^{4}-12258a^{3}+28901a^{2}+5669a-26972$ |
*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.
