| 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 |
| 41.2-a1 |
41.2-a |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.2 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+a^3-5a^2-3a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.000898331$ |
$258775.1594$ |
2.07054 |
\( -\frac{7678148}{41} a^{5} - \frac{7305801}{41} a^{4} + \frac{31443377}{41} a^{3} + \frac{15419812}{41} a^{2} - \frac{29392241}{41} a + \frac{4246630}{41} \) |
\( \bigl[a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 4 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} - 4 a^{4} + 6 a^{3} + 14 a^{2} - 13 a - 2\) , \( -a^{5} - 3 a^{4} + 6 a^{3} + 9 a^{2} - 12 a + 3\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-4a^{2}-2a+2\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+4a-1\right){x}^{2}+\left(-a^{5}-4a^{4}+6a^{3}+14a^{2}-13a-2\right){x}-a^{5}-3a^{4}+6a^{3}+9a^{2}-12a+3$ |
| 41.2-b1 |
41.2-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.2 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^4+a^3-5a^2-3a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( \frac{164991566655}{68921} a^{5} - \frac{491296262031}{68921} a^{4} - \frac{18167532093}{68921} a^{3} + \frac{1025969866218}{68921} a^{2} - \frac{709759186668}{68921} a + \frac{83606184531}{68921} \) |
\( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( -a^{5} + 4 a^{3} - a^{2} - 3 a + 1\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -a^{5} + 4 a^{3} - 3 a - 1\) , \( -1\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-a^{5}+4a^{3}-3a-1\right){x}-1$ |
| 41.2-b2 |
41.2-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.2 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+a^3-5a^2-3a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( -\frac{17389718577}{41} a^{5} + \frac{13900894542}{41} a^{4} + \frac{106655548392}{41} a^{3} - \frac{82884018420}{41} a^{2} - \frac{152815374306}{41} a + \frac{108203083038}{41} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 8 a + 6\) , \( a^{5} - 5 a^{3} + 2 a^{2} + 5 a - 3\) , \( -3 a^{5} - a^{4} + 19 a^{3} - 4 a^{2} - 27 a + 15\) , \( a^{5} - 8 a^{4} + 8 a^{3} + 14 a^{2} - 20 a + 6\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){x}{y}+\left(a^{5}-5a^{3}+2a^{2}+5a-3\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-8a+6\right){x}^{2}+\left(-3a^{5}-a^{4}+19a^{3}-4a^{2}-27a+15\right){x}+a^{5}-8a^{4}+8a^{3}+14a^{2}-20a+6$ |
| 41.2-c1 |
41.2-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.2 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^4+a^3-5a^2-3a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.001392095$ |
$59892.46614$ |
2.22786 |
\( \frac{164991566655}{68921} a^{5} - \frac{491296262031}{68921} a^{4} - \frac{18167532093}{68921} a^{3} + \frac{1025969866218}{68921} a^{2} - \frac{709759186668}{68921} a + \frac{83606184531}{68921} \) |
\( \bigl[a^{4} - 3 a^{2} + 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 2 a^{2} - a + 1\) , \( a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 9 a + 1\) , \( 3 a^{5} - a^{4} - 20 a^{3} + 5 a^{2} + 31 a - 5\bigr] \) |
${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+2a^{2}-a+1\right){x}^{2}+\left(-a^{5}-a^{4}+6a^{3}+3a^{2}-9a+1\right){x}+3a^{5}-a^{4}-20a^{3}+5a^{2}+31a-5$ |
| 41.2-c2 |
41.2-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.2 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+a^3-5a^2-3a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.004176285$ |
$59892.46614$ |
2.22786 |
\( -\frac{17389718577}{41} a^{5} + \frac{13900894542}{41} a^{4} + \frac{106655548392}{41} a^{3} - \frac{82884018420}{41} a^{2} - \frac{152815374306}{41} a + \frac{108203083038}{41} \) |
\( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 4\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( -2 a^{4} - a^{3} + 8 a^{2} + a - 3\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a\bigr] \) |
${y}^2+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+4\right){x}^{2}+\left(-2a^{4}-a^{3}+8a^{2}+a-3\right){x}+a^{4}-a^{3}-4a^{2}+3a$ |
| 41.2-d1 |
41.2-d |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.2 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^4+a^3-5a^2-3a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$603.9405774$ |
0.896535 |
\( -\frac{7678148}{41} a^{5} - \frac{7305801}{41} a^{4} + \frac{31443377}{41} a^{3} + \frac{15419812}{41} a^{2} - \frac{29392241}{41} a + \frac{4246630}{41} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 1\) , \( -a^{5} - a^{4} + 5 a^{3} + 4 a^{2} - 5 a - 1\) , \( a^{2} + a - 1\) , \( -a^{5} + 7 a^{3} - 7 a + 1\) , \( 2 a^{5} - 2 a^{4} - 2 a^{3} + 3 a^{2} - 2 a\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+5a^{3}+4a^{2}-5a-1\right){x}^{2}+\left(-a^{5}+7a^{3}-7a+1\right){x}+2a^{5}-2a^{4}-2a^{3}+3a^{2}-2a$ |
*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.
