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
19.1-a4
19.1-a
$4$
$10$
4.4.8069.1
$4$
$[4, 0]$
19.1
\( 19 \)
\( 19^{10} \)
$11.59825$
$(a^2+a-4)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.4.2
$25$
\( 2 \)
$1$
$17.09285042$
2.378562937
\( -\frac{7453708100651023724731227}{6131066257801} a^{3} - \frac{1737250893512074261087546}{6131066257801} a^{2} + \frac{35126388668209027651898288}{6131066257801} a + \frac{6044827780477046155354082}{6131066257801} \)
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} + a - 2\) , \( 659 a^{3} - 621 a^{2} - 3152 a + 3119\) , \( 10968 a^{3} - 11477 a^{2} - 52551 a + 57385\bigr] \)
${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(659a^{3}-621a^{2}-3152a+3119\right){x}+10968a^{3}-11477a^{2}-52551a+57385$
19.1-f4
19.1-f
$4$
$10$
4.4.8069.1
$4$
$[4, 0]$
19.1
\( 19 \)
\( 19^{10} \)
$11.59825$
$(a^2+a-4)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 5$
2B , 5B.1.2
$25$
\( 2 \)
$1$
$3.286550298$
0.457341317
\( -\frac{7453708100651023724731227}{6131066257801} a^{3} - \frac{1737250893512074261087546}{6131066257801} a^{2} + \frac{35126388668209027651898288}{6131066257801} a + \frac{6044827780477046155354082}{6131066257801} \)
\( \bigl[a^{3} - 4 a + 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 763 a^{3} - 898 a^{2} - 3634 a + 4396\) , \( -17404 a^{3} + 20349 a^{2} + 83667 a - 101492\bigr] \)
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(763a^{3}-898a^{2}-3634a+4396\right){x}-17404a^{3}+20349a^{2}+83667a-101492$
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Pari/GP
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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.