| 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 |
| 41.5-a1 |
41.5-a |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.5 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+2a^3-4a^2-6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.000898331$ |
$258775.1594$ |
2.07054 |
\( -\frac{1959802}{41} a^{5} + \frac{7305801}{41} a^{4} - \frac{1686142}{41} a^{3} - \frac{25057762}{41} a^{2} + \frac{36500036}{41} a - \frac{14311424}{41} \) |
\( \bigl[a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 3 a - 1\) , \( -a^{4} + a^{3} + 4 a^{2} - 2 a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\) , \( 17 a^{5} + 13 a^{4} - 68 a^{3} - 31 a^{2} + 59 a - 8\) , \( 42 a^{5} + 54 a^{4} - 196 a^{3} - 123 a^{2} + 199 a - 27\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+3a-1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-2a-1\right){x}^{2}+\left(17a^{5}+13a^{4}-68a^{3}-31a^{2}+59a-8\right){x}+42a^{5}+54a^{4}-196a^{3}-123a^{2}+199a-27$ |
| 41.5-b1 |
41.5-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.5 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^4+2a^3-4a^2-6a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( -\frac{1824869065509}{68921} a^{5} + \frac{491296262031}{68921} a^{4} + \frac{11308085774361}{68921} a^{3} - \frac{2685847365072}{68921} a^{2} - \frac{16561220551596}{68921} a + \frac{2497339386342}{68921} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( 4 a^{5} + 6 a^{4} - 20 a^{3} - 14 a^{2} + 21 a - 3\) , \( 4 a^{5} + 6 a^{4} - 20 a^{3} - 14 a^{2} + 21 a - 3\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+5a-1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){y}={x}^{3}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}^{2}+\left(4a^{5}+6a^{4}-20a^{3}-14a^{2}+21a-3\right){x}+4a^{5}+6a^{4}-20a^{3}-14a^{2}+21a-3$ |
| 41.5-b2 |
41.5-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.5 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+2a^3-4a^2-6a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( -\frac{20534525352}{41} a^{5} - \frac{13900894542}{41} a^{4} + \frac{101732772825}{41} a^{3} + \frac{44959774491}{41} a^{2} - \frac{93107150055}{41} a + \frac{12580316820}{41} \) |
\( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 6 a^{3} - 3 a^{2} + 7 a\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a\) , \( -4 a^{5} + 2 a^{4} + 23 a^{3} - 8 a^{2} - 32 a + 6\) , \( -32 a^{5} + 10 a^{4} + 198 a^{3} - 51 a^{2} - 290 a + 44\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a\right){y}={x}^{3}+\left(a^{5}+a^{4}-6a^{3}-3a^{2}+7a\right){x}^{2}+\left(-4a^{5}+2a^{4}+23a^{3}-8a^{2}-32a+6\right){x}-32a^{5}+10a^{4}+198a^{3}-51a^{2}-290a+44$ |
| 41.5-c1 |
41.5-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.5 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^4+2a^3-4a^2-6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.001392095$ |
$59892.46614$ |
2.22786 |
\( -\frac{1824869065509}{68921} a^{5} + \frac{491296262031}{68921} a^{4} + \frac{11308085774361}{68921} a^{3} - \frac{2685847365072}{68921} a^{2} - \frac{16561220551596}{68921} a + \frac{2497339386342}{68921} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 2 a - 1\) , \( a^{4} - 3 a^{2} + 1\) , \( 17 a^{5} + 11 a^{4} - 72 a^{3} - 20 a^{2} + 60 a - 11\) , \( 34 a^{5} + 34 a^{4} - 144 a^{3} - 84 a^{2} + 133 a - 13\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+2a-1\right){x}^{2}+\left(17a^{5}+11a^{4}-72a^{3}-20a^{2}+60a-11\right){x}+34a^{5}+34a^{4}-144a^{3}-84a^{2}+133a-13$ |
| 41.5-c2 |
41.5-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.5 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+2a^3-4a^2-6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.004176285$ |
$59892.46614$ |
2.22786 |
\( -\frac{20534525352}{41} a^{5} - \frac{13900894542}{41} a^{4} + \frac{101732772825}{41} a^{3} + \frac{44959774491}{41} a^{2} - \frac{93107150055}{41} a + \frac{12580316820}{41} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 1\) , \( -a^{5} + 6 a^{3} - 2 a^{2} - 9 a + 4\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{5} + a^{4} - 5 a^{3} - 5 a^{2} + 4 a + 2\) , \( -4 a^{5} - 2 a^{4} + 20 a^{3} + 6 a^{2} - 19 a + 3\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+5a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-2a^{2}-9a+4\right){x}^{2}+\left(a^{5}+a^{4}-5a^{3}-5a^{2}+4a+2\right){x}-4a^{5}-2a^{4}+20a^{3}+6a^{2}-19a+3$ |
| 41.5-d1 |
41.5-d |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.5 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+2a^3-4a^2-6a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$603.9405774$ |
0.896535 |
\( -\frac{1959802}{41} a^{5} + \frac{7305801}{41} a^{4} - \frac{1686142}{41} a^{3} - \frac{25057762}{41} a^{2} + \frac{36500036}{41} a - \frac{14311424}{41} \) |
\( \bigl[a^{5} - 4 a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{5} - a^{4} - 4 a^{3} + 6 a^{2} + a - 7\) , \( a^{5} - 5 a^{3} + a^{2} + 6 a - 2\) , \( 6 a^{5} - 6 a^{4} - 32 a^{3} + 31 a^{2} + 35 a - 26\) , \( 4 a^{5} - 5 a^{4} - 20 a^{3} + 25 a^{2} + 18 a - 20\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+2a^{2}+2a-4\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+6a^{2}+a-7\right){x}^{2}+\left(6a^{5}-6a^{4}-32a^{3}+31a^{2}+35a-26\right){x}+4a^{5}-5a^{4}-20a^{3}+25a^{2}+18a-20$ |
*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.
