| 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 |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5+a^4+6a^3-5a^2-8a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.000898331$ |
$258775.1594$ |
2.07054 |
\( \frac{14625511}{41} a^{5} + \frac{11843590}{41} a^{4} - \frac{68962113}{41} a^{3} - \frac{40054650}{41} a^{2} + \frac{54505985}{41} a - \frac{7378940}{41} \) |
\( \bigl[a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 2 a - 2\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( a^{4} - 4 a^{2} + 2\) , \( 3 a^{5} + a^{4} - 15 a^{3} - 3 a^{2} + 17 a + 1\) , \( 2 a^{5} - 10 a^{3} + 2 a^{2} + 10 a - 2\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+2a-2\right){x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(3a^{5}+a^{4}-15a^{3}-3a^{2}+17a+1\right){x}+2a^{5}-10a^{3}+2a^{2}+10a-2$ |
| 41.4-b1 |
41.4-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5+a^4+6a^3-5a^2-8a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( -\frac{2317236930}{41} a^{5} + \frac{6745914900}{41} a^{4} + \frac{942380973}{41} a^{3} - \frac{15400001988}{41} a^{2} + \frac{10454504706}{41} a - \frac{1224790551}{41} \) |
\( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( a^{4} - 5 a^{2} + a + 3\) , \( a^{2} + a - 2\) , \( 3 a^{5} - 14 a^{3} + 15 a + 1\) , \( 5 a^{5} - 28 a^{3} + 2 a^{2} + 38 a - 5\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+a+3\right){x}^{2}+\left(3a^{5}-14a^{3}+15a+1\right){x}+5a^{5}-28a^{3}+2a^{2}+38a-5$ |
| 41.4-b2 |
41.4-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(-a^5+a^4+6a^3-5a^2-8a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( -\frac{971781867837}{68921} a^{5} - \frac{885653883603}{68921} a^{4} + \frac{4138247022237}{68921} a^{3} + \frac{2079537404544}{68921} a^{2} - \frac{3801129136902}{68921} a + \frac{507608403966}{68921} \) |
\( \bigl[a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( -a^{4} - a^{3} + 4 a^{2} + 2 a - 1\) , \( a + 1\) , \( 2 a^{5} - 10 a^{3} + 3 a^{2} + 10 a - 5\) , \( 2 a^{5} - 10 a^{3} + 3 a^{2} + 11 a - 7\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+2a^{2}+6a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+4a^{2}+2a-1\right){x}^{2}+\left(2a^{5}-10a^{3}+3a^{2}+10a-5\right){x}+2a^{5}-10a^{3}+3a^{2}+11a-7$ |
| 41.4-c1 |
41.4-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5+a^4+6a^3-5a^2-8a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.004176285$ |
$59892.46614$ |
2.22786 |
\( -\frac{2317236930}{41} a^{5} + \frac{6745914900}{41} a^{4} + \frac{942380973}{41} a^{3} - \frac{15400001988}{41} a^{2} + \frac{10454504706}{41} a - \frac{1224790551}{41} \) |
\( \bigl[a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - a + 2\) , \( a^{5} - 4 a^{3} + 2 a^{2} + 2 a - 4\) , \( 3 a^{5} + 2 a^{4} - 11 a^{3} - 4 a^{2} + 8 a + 1\) , \( 6 a^{5} + 6 a^{4} - 24 a^{3} - 14 a^{2} + 19 a - 2\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+2a^{2}+6a-3\right){x}{y}+\left(a^{5}-4a^{3}+2a^{2}+2a-4\right){y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-a+2\right){x}^{2}+\left(3a^{5}+2a^{4}-11a^{3}-4a^{2}+8a+1\right){x}+6a^{5}+6a^{4}-24a^{3}-14a^{2}+19a-2$ |
| 41.4-c2 |
41.4-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(-a^5+a^4+6a^3-5a^2-8a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.001392095$ |
$59892.46614$ |
2.22786 |
\( -\frac{971781867837}{68921} a^{5} - \frac{885653883603}{68921} a^{4} + \frac{4138247022237}{68921} a^{3} + \frac{2079537404544}{68921} a^{2} - \frac{3801129136902}{68921} a + \frac{507608403966}{68921} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} + a - 1\) , \( -3 a^{5} - a^{4} + 18 a^{3} + 5 a^{2} - 20 a + 3\) , \( 17 a^{5} + 11 a^{4} - 82 a^{3} - 31 a^{2} + 79 a - 11\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-3a^{5}-a^{4}+18a^{3}+5a^{2}-20a+3\right){x}+17a^{5}+11a^{4}-82a^{3}-31a^{2}+79a-11$ |
| 41.4-d1 |
41.4-d |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.4 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5+a^4+6a^3-5a^2-8a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$603.9405774$ |
0.896535 |
\( \frac{14625511}{41} a^{5} + \frac{11843590}{41} a^{4} - \frac{68962113}{41} a^{3} - \frac{40054650}{41} a^{2} + \frac{54505985}{41} a - \frac{7378940}{41} \) |
\( \bigl[a^{5} - 5 a^{3} + a^{2} + 6 a - 2\) , \( a^{5} - a^{4} - 6 a^{3} + 5 a^{2} + 7 a - 4\) , \( a^{2} + a - 2\) , \( -6 a^{5} + 36 a^{3} - 3 a^{2} - 53 a + 7\) , \( -14 a^{5} + 3 a^{4} + 86 a^{3} - 18 a^{2} - 126 a + 18\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+7a-4\right){x}^{2}+\left(-6a^{5}+36a^{3}-3a^{2}-53a+7\right){x}-14a^{5}+3a^{4}+86a^{3}-18a^{2}-126a+18$ |
*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.
