| 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 |
| 41.1-a1 |
41.1-a |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5-6a^3+a^2+7a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.000898331$ |
$258775.1594$ |
2.07054 |
\( -\frac{18790953}{41} a^{5} + \frac{6125244}{41} a^{4} + \frac{114705520}{41} a^{3} - \frac{31448339}{41} a^{2} - \frac{163512831}{41} a + \frac{24344025}{41} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 1\) , \( -a^{5} + 6 a^{3} - a^{2} - 7 a + 2\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{4} + a^{3} - a - 3\) , \( 2 a^{5} + 2 a^{4} - 7 a^{3} - 5 a^{2} + 5 a + 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-a^{2}-7a+2\right){x}^{2}+\left(a^{4}+a^{3}-a-3\right){x}+2a^{5}+2a^{4}-7a^{3}-5a^{2}+5a+1$ |
| 41.1-b1 |
41.1-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5-6a^3+a^2+7a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( \frac{12961040607}{41} a^{5} + \frac{9890721675}{41} a^{4} - \frac{57231718290}{41} a^{3} - \frac{19855931193}{41} a^{2} + \frac{55985643342}{41} a - \frac{13499009721}{41} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( a^{5} - 5 a^{3} + 6 a - 1\) , \( a^{4} - 4 a^{2} + 3\) , \( 4 a^{5} + 3 a^{4} - 20 a^{3} - 8 a^{2} + 19 a - 4\) , \( 28 a^{5} + 18 a^{4} - 137 a^{3} - 58 a^{2} + 124 a - 19\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(a^{5}-5a^{3}+6a-1\right){x}^{2}+\left(4a^{5}+3a^{4}-20a^{3}-8a^{2}+19a-4\right){x}+28a^{5}+18a^{4}-137a^{3}-58a^{2}+124a-19$ |
| 41.1-b2 |
41.1-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^5-6a^3+a^2+7a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( \frac{1692444184785}{68921} a^{5} + \frac{1104206748561}{68921} a^{4} - \frac{8329796043201}{68921} a^{3} - \frac{3610036693062}{68921} a^{2} + \frac{7573650019722}{68921} a - \frac{1024189061643}{68921} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 5 a - 4\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{4} + a^{3} + 2 a^{2} - 4 a + 3\) , \( -a^{4} + 3 a^{2} - a - 1\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+5a-4\right){x}^{2}+\left(-a^{4}+a^{3}+2a^{2}-4a+3\right){x}-a^{4}+3a^{2}-a-1$ |
| 41.1-c1 |
41.1-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5-6a^3+a^2+7a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.004176285$ |
$59892.46614$ |
2.22786 |
\( \frac{12961040607}{41} a^{5} + \frac{9890721675}{41} a^{4} - \frac{57231718290}{41} a^{3} - \frac{19855931193}{41} a^{2} + \frac{55985643342}{41} a - \frac{13499009721}{41} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( -a^{5} + 5 a^{3} - 2 a^{2} - 4 a + 5\) , \( a^{5} - 5 a^{3} + a^{2} + 6 a - 1\) , \( -3 a^{5} - 2 a^{4} + 13 a^{3} + 5 a^{2} - 12 a + 4\) , \( -a^{5} - a^{4} + 5 a^{3} + 2 a^{2} - 5 a + 1\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-2a^{2}-4a+5\right){x}^{2}+\left(-3a^{5}-2a^{4}+13a^{3}+5a^{2}-12a+4\right){x}-a^{5}-a^{4}+5a^{3}+2a^{2}-5a+1$ |
| 41.1-c2 |
41.1-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^5-6a^3+a^2+7a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.001392095$ |
$59892.46614$ |
2.22786 |
\( \frac{1692444184785}{68921} a^{5} + \frac{1104206748561}{68921} a^{4} - \frac{8329796043201}{68921} a^{3} - \frac{3610036693062}{68921} a^{2} + \frac{7573650019722}{68921} a - \frac{1024189061643}{68921} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 2\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 4 a - 3\) , \( a^{3} - 3 a\) , \( 8 a^{5} + 3 a^{4} - 45 a^{3} - 2 a^{2} + 62 a - 8\) , \( -4 a^{5} + 9 a^{4} + 37 a^{3} - 30 a^{2} - 71 a + 11\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+4a-3\right){x}^{2}+\left(8a^{5}+3a^{4}-45a^{3}-2a^{2}+62a-8\right){x}-4a^{5}+9a^{4}+37a^{3}-30a^{2}-71a+11$ |
| 41.1-d1 |
41.1-d |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.1 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5-6a^3+a^2+7a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$603.9405774$ |
0.896535 |
\( -\frac{18790953}{41} a^{5} + \frac{6125244}{41} a^{4} + \frac{114705520}{41} a^{3} - \frac{31448339}{41} a^{2} - \frac{163512831}{41} a + \frac{24344025}{41} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( -a^{5} + 4 a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -a^{5} + 3 a^{4} + a^{3} - 14 a^{2} + 10 a + 7\) , \( -a^{5} + 4 a^{4} + 2 a^{3} - 17 a^{2} + 8 a + 4\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(-a^{5}+3a^{4}+a^{3}-14a^{2}+10a+7\right){x}-a^{5}+4a^{4}+2a^{3}-17a^{2}+8a+4$ |
*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.
