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
20.2-a1
20.2-a
$2$
$3$
\(\Q(\sqrt{19}) \)
$2$
$[2, 0]$
20.2
\( 2^{2} \cdot 5 \)
\( 2^{8} \cdot 5^{2} \)
$1.64741$
$(-3a+13), (-2a+9)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B.1.1
$1$
\( 2 \cdot 3 \)
$1.605083656$
$12.81909334$
3.146928843
\( \frac{612}{25} a - \frac{3656}{25} \)
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 92 a - 364\) , \( 2556 a - 11098\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(92a-364\right){x}+2556a-11098$
20.2-a2
20.2-a
$2$
$3$
\(\Q(\sqrt{19}) \)
$2$
$[2, 0]$
20.2
\( 2^{2} \cdot 5 \)
\( 2^{8} \cdot 5^{6} \)
$1.64741$
$(-3a+13), (-2a+9)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B.1.2
$1$
\( 2 \)
$4.815250970$
$1.424343704$
3.146928843
\( \frac{842348523228}{15625} a - \frac{3671711480264}{15625} \)
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 10262 a - 44694\) , \( 1172044 a - 5108778\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10262a-44694\right){x}+1172044a-5108778$
20.2-b1
20.2-b
$2$
$3$
\(\Q(\sqrt{19}) \)
$2$
$[2, 0]$
20.2
\( 2^{2} \cdot 5 \)
\( 2^{8} \cdot 5^{2} \)
$1.64741$
$(-3a+13), (-2a+9)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B
$1$
\( 2 \)
$0.254665897$
$14.05008960$
1.641735093
\( \frac{612}{25} a - \frac{3656}{25} \)
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 93 a - 361\) , \( -2664 a + 11669\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(93a-361\right){x}-2664a+11669$
20.2-b2
20.2-b
$2$
$3$
\(\Q(\sqrt{19}) \)
$2$
$[2, 0]$
20.2
\( 2^{2} \cdot 5 \)
\( 2^{8} \cdot 5^{6} \)
$1.64741$
$(-3a+13), (-2a+9)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B
$1$
\( 2 \cdot 3 \)
$0.084888632$
$14.05008960$
1.641735093
\( \frac{842348523228}{15625} a - \frac{3671711480264}{15625} \)
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 10263 a - 44691\) , \( -1185972 a + 5169589\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(10263a-44691\right){x}-1185972a+5169589$
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.