| 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 |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5+a^4-6a^3-4a^2+9a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.000898331$ |
$258775.1594$ |
2.07054 |
\( \frac{358438}{41} a^{5} - \frac{6125244}{41} a^{4} + \frac{5527520}{41} a^{3} + \frac{13015824}{41} a^{2} - \frac{12861139}{41} a + \frac{1612826}{41} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a - 2\) , \( a^{5} - 5 a^{3} + 2 a^{2} + 5 a - 3\) , \( a^{5} - a^{4} - 6 a^{3} + 4 a^{2} + 6 a - 3\) , \( -a^{5} - a^{4} + 4 a^{3} + 3 a^{2} - 2 a - 1\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{5}-5a^{3}+2a^{2}+5a-3\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+5a-2\right){x}^{2}+\left(a^{5}-a^{4}-6a^{3}+4a^{2}+6a-3\right){x}-a^{5}-a^{4}+4a^{3}+3a^{2}-2a-1$ |
| 41.3-b1 |
41.3-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^5+a^4-6a^3-4a^2+9a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( \frac{1298086563213}{68921} a^{5} - \frac{1104206748561}{68921} a^{4} - \frac{7953510945933}{68921} a^{3} + \frac{6600567441060}{68921} a^{2} + \frac{11370963467700}{68921} a - \frac{8686212609186}{68921} \) |
\( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{5} - 4 a^{3} + a^{2} + a - 2\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( -a^{5} - 2 a^{4} + 6 a^{3} + 7 a^{2} - 10 a\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 10 a\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a\right){y}={x}^{3}+\left(a^{5}-4a^{3}+a^{2}+a-2\right){x}^{2}+\left(-a^{5}-2a^{4}+6a^{3}+7a^{2}-10a\right){x}-a^{5}-a^{4}+6a^{3}+3a^{2}-10a$ |
| 41.3-b2 |
41.3-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5+a^4-6a^3-4a^2+9a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( \frac{5806060965}{41} a^{5} - \frac{9890721675}{41} a^{4} - \frac{17446647213}{41} a^{3} + \frac{38623032765}{41} a^{2} - \frac{19621562553}{41} a + \frac{2132617005}{41} \) |
\( \bigl[a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 3 a - 1\) , \( a^{5} - 6 a^{3} + a^{2} + 9 a - 3\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 2\) , \( 15 a^{5} + 15 a^{4} - 66 a^{3} - 37 a^{2} + 68 a - 10\) , \( 41 a^{5} + 38 a^{4} - 177 a^{3} - 90 a^{2} + 170 a - 26\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+3a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+5a-2\right){y}={x}^{3}+\left(a^{5}-6a^{3}+a^{2}+9a-3\right){x}^{2}+\left(15a^{5}+15a^{4}-66a^{3}-37a^{2}+68a-10\right){x}+41a^{5}+38a^{4}-177a^{3}-90a^{2}+170a-26$ |
| 41.3-c1 |
41.3-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(a^5+a^4-6a^3-4a^2+9a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.001392095$ |
$59892.46614$ |
2.22786 |
\( \frac{1298086563213}{68921} a^{5} - \frac{1104206748561}{68921} a^{4} - \frac{7953510945933}{68921} a^{3} + \frac{6600567441060}{68921} a^{2} + \frac{11370963467700}{68921} a - \frac{8686212609186}{68921} \) |
\( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 7 a - 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( 3 a^{5} + 5 a^{4} - 15 a^{3} - 11 a^{2} + 17 a - 1\) , \( 9 a^{5} + 3 a^{4} - 33 a^{3} - 8 a^{2} + 25 a - 4\bigr] \) |
${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a+1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+6a^{3}+3a^{2}-7a-1\right){x}^{2}+\left(3a^{5}+5a^{4}-15a^{3}-11a^{2}+17a-1\right){x}+9a^{5}+3a^{4}-33a^{3}-8a^{2}+25a-4$ |
| 41.3-c2 |
41.3-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5+a^4-6a^3-4a^2+9a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.004176285$ |
$59892.46614$ |
2.22786 |
\( \frac{5806060965}{41} a^{5} - \frac{9890721675}{41} a^{4} - \frac{17446647213}{41} a^{3} + \frac{38623032765}{41} a^{2} - \frac{19621562553}{41} a + \frac{2132617005}{41} \) |
\( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( -a^{4} + a^{3} + 5 a^{2} - 2 a - 5\) , \( a^{5} - 4 a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{5} + 2 a^{4} - 7 a^{3} - 8 a^{2} + 11 a + 5\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 9 a - 1\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{5}-4a^{3}+2a^{2}+3a-3\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-2a-5\right){x}^{2}+\left(a^{5}+2a^{4}-7a^{3}-8a^{2}+11a+5\right){x}-a^{5}-a^{4}+6a^{3}+3a^{2}-9a-1$ |
| 41.3-d1 |
41.3-d |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.3 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(a^5+a^4-6a^3-4a^2+9a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$603.9405774$ |
0.896535 |
\( \frac{358438}{41} a^{5} - \frac{6125244}{41} a^{4} + \frac{5527520}{41} a^{3} + \frac{13015824}{41} a^{2} - \frac{12861139}{41} a + \frac{1612826}{41} \) |
\( \bigl[a^{4} - 3 a^{2} + a + 1\) , \( -a^{5} + 6 a^{3} - 8 a\) , \( a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( 6 a^{5} - 4 a^{4} - 36 a^{3} + 23 a^{2} + 49 a - 30\) , \( 37 a^{5} - 35 a^{4} - 231 a^{3} + 201 a^{2} + 333 a - 260\bigr] \) |
${y}^2+\left(a^{4}-3a^{2}+a+1\right){x}{y}+\left(a^{5}-5a^{3}+2a^{2}+6a-3\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-8a\right){x}^{2}+\left(6a^{5}-4a^{4}-36a^{3}+23a^{2}+49a-30\right){x}+37a^{5}-35a^{4}-231a^{3}+201a^{2}+333a-260$ |
*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.
