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-a1
19.1-a
$2$
$3$
3.3.1129.1
$3$
$[3, 0]$
19.1
\( 19 \)
\( - 19^{3} \)
$4.90469$
$(-a^2-a+4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$2.851714474$
$15.31159601$
3.898529149
\( -\frac{11923591168}{6859} a^{2} + \frac{28591403008}{6859} a + \frac{14920290304}{6859} \)
\( \bigl[0\) , \( a^{2} - 2 a - 4\) , \( a^{2} - 5\) , \( -10 a^{2} + 22 a + 19\) , \( -54 a^{2} + 145 a + 21\bigr] \)
${y}^2+\left(a^{2}-5\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-10a^{2}+22a+19\right){x}-54a^{2}+145a+21$
19.1-a2
19.1-a
$2$
$3$
3.3.1129.1
$3$
$[3, 0]$
19.1
\( 19 \)
\( -19 \)
$4.90469$
$(-a^2-a+4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.950571491$
$45.93478804$
3.898529149
\( \frac{11005952}{19} a^{2} - \frac{4861952}{19} a - \frac{74928128}{19} \)
\( \bigl[0\) , \( a^{2} - 2 a - 4\) , \( a^{2} - 4\) , \( 10 a^{2} - 7 a - 60\) , \( 19 a^{2} - 11 a - 126\bigr] \)
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(10a^{2}-7a-60\right){x}+19a^{2}-11a-126$
19.1-b1
19.1-b
$2$
$3$
3.3.1129.1
$3$
$[3, 0]$
19.1
\( 19 \)
\( -19 \)
$4.90469$
$(-a^2-a+4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.035446905$
$391.8577597$
1.240169548
\( \frac{11005952}{19} a^{2} - \frac{4861952}{19} a - \frac{74928128}{19} \)
\( \bigl[0\) , \( a^{2} - a - 4\) , \( a^{2} - 5\) , \( -3 a^{2} - 10 a - 1\) , \( 11 a^{2} + 27 a + 4\bigr] \)
${y}^2+\left(a^{2}-5\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-3a^{2}-10a-1\right){x}+11a^{2}+27a+4$
19.1-b2
19.1-b
$2$
$3$
3.3.1129.1
$3$
$[3, 0]$
19.1
\( 19 \)
\( - 19^{3} \)
$4.90469$
$(-a^2-a+4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.106340716$
$130.6192532$
1.240169548
\( -\frac{11923591168}{6859} a^{2} + \frac{28591403008}{6859} a + \frac{14920290304}{6859} \)
\( \bigl[0\) , \( a^{2} - a - 4\) , \( a + 1\) , \( -2 a^{2} + 4 a + 6\) , \( a^{2} - 3 a - 3\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-2a^{2}+4a+6\right){x}+a^{2}-3a-3$
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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.