| 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 |
| 41.6-a1 |
41.6-a |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.6 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5-a^4+5a^3+3a^2-5a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.000898331$ |
$258775.1594$ |
2.07054 |
\( \frac{13444954}{41} a^{5} - \frac{11843590}{41} a^{4} - \frac{81028162}{41} a^{3} + \frac{68125115}{41} a^{2} + \frac{114760190}{41} a - \frac{87740634}{41} \) |
\( \bigl[a^{4} - 3 a^{2}\) , \( a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + 3 a - 5\) , \( a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 2 a - 2\) , \( -2 a^{5} + 13 a^{3} - 3 a^{2} - 21 a + 8\) , \( -a^{4} - a^{3} + 3 a^{2} + 2 a - 1\bigr] \) |
${y}^2+\left(a^{4}-3a^{2}\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+2a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+5a^{2}+3a-5\right){x}^{2}+\left(-2a^{5}+13a^{3}-3a^{2}-21a+8\right){x}-a^{4}-a^{3}+3a^{2}+2a-1$ |
| 41.6-b1 |
41.6-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.6 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5-a^4+5a^3+3a^2-5a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( \frac{21474379287}{41} a^{5} - \frac{6745914900}{41} a^{4} - \frac{134652336687}{41} a^{3} + \frac{34557144345}{41} a^{2} + \frac{199103938866}{41} a - \frac{30078826092}{41} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{5} - 6 a^{3} + 8 a - 1\) , \( a + 1\) , \( a^{5} - 11 a^{3} + 26 a - 3\) , \( -4 a^{5} + 14 a^{4} + 3 a^{3} - 53 a^{2} + 49 a - 8\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{5}-6a^{3}+8a-1\right){x}^{2}+\left(a^{5}-11a^{3}+26a-3\right){x}-4a^{5}+14a^{4}+3a^{3}-53a^{2}+49a-8$ |
| 41.6-b2 |
41.6-b |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.6 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(-a^5-a^4+5a^3+3a^2-5a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$798.2883122$ |
1.18504 |
\( -\frac{358871381307}{68921} a^{5} + \frac{885653883603}{68921} a^{4} + \frac{855141724629}{68921} a^{3} - \frac{3410190653688}{68921} a^{2} + \frac{2127495387744}{68921} a - \frac{244988831367}{68921} \) |
\( \bigl[a^{5} - 4 a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 5\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 4 a^{5} + 7 a^{4} - 15 a^{3} - 27 a^{2} + 9 a + 20\) , \( 6 a^{5} + 11 a^{4} - 21 a^{3} - 39 a^{2} + 10 a + 22\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+2a^{2}+3a-4\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+5\right){x}^{2}+\left(4a^{5}+7a^{4}-15a^{3}-27a^{2}+9a+20\right){x}+6a^{5}+11a^{4}-21a^{3}-39a^{2}+10a+22$ |
| 41.6-c1 |
41.6-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.6 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5-a^4+5a^3+3a^2-5a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.004176285$ |
$59892.46614$ |
2.22786 |
\( \frac{21474379287}{41} a^{5} - \frac{6745914900}{41} a^{4} - \frac{134652336687}{41} a^{3} + \frac{34557144345}{41} a^{2} + \frac{199103938866}{41} a - \frac{30078826092}{41} \) |
\( \bigl[a^{5} - 4 a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{5} - a^{4} - 6 a^{3} + 6 a^{2} + 7 a - 7\) , \( a^{3} - 3 a\) , \( -a^{5} + 3 a^{4} + 11 a^{3} - 9 a^{2} - 22 a + 5\) , \( 6 a^{5} + 2 a^{4} - 31 a^{3} - 2 a^{2} + 37 a - 6\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+2a^{2}+3a-4\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+6a^{2}+7a-7\right){x}^{2}+\left(-a^{5}+3a^{4}+11a^{3}-9a^{2}-22a+5\right){x}+6a^{5}+2a^{4}-31a^{3}-2a^{2}+37a-6$ |
| 41.6-c2 |
41.6-c |
$2$ |
$3$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.6 |
\( 41 \) |
\( - 41^{3} \) |
$82.02839$ |
$(-a^5-a^4+5a^3+3a^2-5a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.001392095$ |
$59892.46614$ |
2.22786 |
\( -\frac{358871381307}{68921} a^{5} + \frac{885653883603}{68921} a^{4} + \frac{855141724629}{68921} a^{3} - \frac{3410190653688}{68921} a^{2} + \frac{2127495387744}{68921} a - \frac{244988831367}{68921} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{5} - 6 a^{3} + a^{2} + 7 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} + 2 a^{3} - a^{2} + a + 3\) , \( a^{5} + a^{4} - 6 a^{3} - 2 a^{2} + 8 a - 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{5}-6a^{3}+a^{2}+7a-1\right){x}^{2}+\left(-a^{5}+2a^{3}-a^{2}+a+3\right){x}+a^{5}+a^{4}-6a^{3}-2a^{2}+8a-1$ |
| 41.6-d1 |
41.6-d |
$1$ |
$1$ |
\(\Q(\zeta_{21})^+\) |
$6$ |
$[6, 0]$ |
41.6 |
\( 41 \) |
\( -41 \) |
$82.02839$ |
$(-a^5-a^4+5a^3+3a^2-5a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$603.9405774$ |
0.896535 |
\( \frac{13444954}{41} a^{5} - \frac{11843590}{41} a^{4} - \frac{81028162}{41} a^{3} + \frac{68125115}{41} a^{2} + \frac{114760190}{41} a - \frac{87740634}{41} \) |
\( \bigl[a^{3} - 2 a + 1\) , \( -a^{5} + 4 a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{4} - 3 a^{2} + 1\) , \( -2 a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 3\) , \( a^{5} - 3 a^{4} + 5 a^{2} - 4 a + 1\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+3\right){x}^{2}+\left(-2a^{4}+2a^{3}+4a^{2}-6a+3\right){x}+a^{5}-3a^{4}+5a^{2}-4a+1$ |
*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.
