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
21.1-a3
21.1-a
$6$
$8$
4.4.9909.1
$4$
$[4, 0]$
21.1
\( 3 \cdot 7 \)
\( 3 \cdot 7^{32} \)
$13.01460$
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$4$
\( 2 \)
$8.512674650$
$5.444064395$
3.724468981
\( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \)
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -13750 a^{3} + 16605 a^{2} + 62452 a - 34158\) , \( 1313123 a^{3} - 1585648 a^{2} - 5964098 a + 3262387\bigr] \)
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16605a^{2}+62452a-34158\right){x}+1313123a^{3}-1585648a^{2}-5964098a+3262387$
21.1-d3
21.1-d
$6$
$8$
4.4.9909.1
$4$
$[4, 0]$
21.1
\( 3 \cdot 7 \)
\( 3 \cdot 7^{32} \)
$13.01460$
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$16$
\( 2 \)
$6.618237813$
$1.380280330$
2.936599577
\( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \)
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -84 a^{3} + 146 a^{2} + 299 a - 174\) , \( -188 a^{3} + 858 a^{2} + 1498 a - 961\bigr] \)
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-84a^{3}+146a^{2}+299a-174\right){x}-188a^{3}+858a^{2}+1498a-961$
Download
displayed columns for
results
to
Text
Pari/GP
SageMath
Magma
Oscar
CSV
*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.