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-a2
41.3-a
$4$
$10$
\(\Q(\zeta_{7})^+\)
$3$
$[3, 0]$
41.3
\( 41 \)
\( 41^{5} \)
$1.16154$
$(3a^2-a-3)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.1.2
$1$
\( 5 \)
$1$
$2.715654732$
0.484938345
\( \frac{653981503916958524755}{115856201} a^{2} - \frac{1469483101129546552831}{115856201} a + \frac{524452446825637320235}{115856201} \)
\( \bigl[1\) , \( a^{2} - a - 3\) , \( 0\) , \( -71 a^{2} + 146 a - 43\) , \( -456 a^{2} + 1027 a - 381\bigr] \)
${y}^2+{x}{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-71a^{2}+146a-43\right){x}-456a^{2}+1027a-381$
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
\( \frac{653981503916958524755}{115856201} a^{2} - \frac{1469483101129546552831}{115856201} a + \frac{524452446825637320235}{115856201} \)
\( \bigl[a^{2} - 1\) , \( a^{2} - a - 1\) , \( a^{2} - 1\) , \( -267 a^{2} + 195 a + 96\) , \( -707 a^{2} + 3324 a - 255\bigr] \)
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-267a^{2}+195a+96\right){x}-707a^{2}+3324a-255$
2009.3-a1
2009.3-a
$4$
$10$
\(\Q(\zeta_{7})^+\)
$3$
$[3, 0]$
2009.3
\( 7^{2} \cdot 41 \)
\( 7^{6} \cdot 41^{5} \)
$2.22194$
$(-a^2-a+2), (3a^2-a-3)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2, 5$
2B , 5B.4.2
$1$
\( 2 \)
$1$
$19.74415128$
1.410296520
\( \frac{653981503916958524755}{115856201} a^{2} - \frac{1469483101129546552831}{115856201} a + \frac{524452446825637320235}{115856201} \)
\( \bigl[a^{2} - 2\) , \( a + 1\) , \( 0\) , \( -957 a^{2} + 2101 a - 741\) , \( 25291 a^{2} - 56714 a + 20237\bigr] \)
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-957a^{2}+2101a-741\right){x}+25291a^{2}-56714a+20237$
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*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.