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 |
1681.5-a1 |
1681.5-a |
$4$ |
$10$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
1681.5 |
\( 41^{2} \) |
\( - 41^{16} \) |
$2.15691$ |
$(3a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$5.387838465$ |
$0.826702528$ |
1.908917005 |
\( \frac{174656123697499408263335}{13422659310152401} a^{2} + \frac{182915726357803972950650}{13422659310152401} a - \frac{115913951592431810832436}{13422659310152401} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - a - 1\) , \( a^{2} - 1\) , \( -1272 a^{2} - 1375 a - 299\) , \( 45529 a^{2} + 47686 a - 16832\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-1272a^{2}-1375a-299\right){x}+45529a^{2}+47686a-16832$ |
1681.5-a2 |
1681.5-a |
$4$ |
$10$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
1681.5 |
\( 41^{2} \) |
\( 41^{11} \) |
$2.15691$ |
$(3a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$2.693919232$ |
$1.653405056$ |
1.908917005 |
\( \frac{653981503916958524755}{115856201} a^{2} - \frac{1469483101129546552831}{115856201} a + \frac{524452446825637320235}{115856201} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - a - 1\) , \( a^{2} - 1\) , \( -267 a^{2} + 195 a + 96\) , \( -707 a^{2} + 3324 a - 255\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-267a^{2}+195a+96\right){x}-707a^{2}+3324a-255$ |
1681.5-a3 |
1681.5-a |
$4$ |
$10$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
1681.5 |
\( 41^{2} \) |
\( 41^{7} \) |
$2.15691$ |
$(3a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.538783846$ |
$8.267025280$ |
1.908917005 |
\( \frac{1703161}{41} a^{2} - \frac{968480}{41} a - \frac{3784992}{41} \) |
\( \bigl[a\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( -6 a^{2} + 15 a - 7\) , \( -3 a^{2} + 2 a - 2\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-6a^{2}+15a-7\right){x}-3a^{2}+2a-2$ |
1681.5-a4 |
1681.5-a |
$4$ |
$10$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
1681.5 |
\( 41^{2} \) |
\( - 41^{8} \) |
$2.15691$ |
$(3a^2-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$1.077567693$ |
$4.133512640$ |
1.908917005 |
\( -\frac{6655766653200}{1681} a^{2} + \frac{3693705667625}{1681} a + \frac{14955417009784}{1681} \) |
\( \bigl[a\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 29 a^{2} - 75 a - 2\) , \( 29 a^{2} - 64 a - 45\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(29a^{2}-75a-2\right){x}+29a^{2}-64a-45$ |
1681.5-b1 |
1681.5-b |
$1$ |
$1$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
1681.5 |
\( 41^{2} \) |
\( 41^{8} \) |
$2.15691$ |
$(3a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7Ns.2.1[3] |
$1$ |
\( 1 \) |
$1$ |
$9.867867490$ |
1.409695355 |
\( 6041 a^{2} - 2135 a - 11897 \) |
\( \bigl[a\) , \( 1\) , \( a^{2} + a - 1\) , \( 4 a^{2} - 9 a - 3\) , \( 16 a^{2} - 35 a + 3\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+{x}^{2}+\left(4a^{2}-9a-3\right){x}+16a^{2}-35a+3$ |
1681.5-c1 |
1681.5-c |
$1$ |
$1$ |
\(\Q(\zeta_{7})^+\) |
$3$ |
$[3, 0]$ |
1681.5 |
\( 41^{2} \) |
\( 41^{2} \) |
$2.15691$ |
$(3a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7Ns.3.1[3] |
$1$ |
\( 1 \) |
$0.017805060$ |
$242.1147169$ |
1.847514482 |
\( 6041 a^{2} - 2135 a - 11897 \) |
\( \bigl[1\) , \( -a\) , \( a^{2} - 1\) , \( a^{2} - a - 2\) , \( -a^{2} + 2\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-1\right){y}={x}^{3}-a{x}^{2}+\left(a^{2}-a-2\right){x}-a^{2}+2$ |
*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.