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
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
19.1
\( 19 \)
\( - 19^{2} \)
$30.73650$
$(-a^4+2a^3+4a^2-5a-4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 2 \)
$0.019623793$
$2766.708886$
2.11887782
\( \frac{218695}{361} a^{4} - \frac{376550}{361} a^{3} - \frac{959505}{361} a^{2} + \frac{1058864}{361} a + \frac{910976}{361} \)
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 4\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a - 4\) , \( 4 a^{4} - 5 a^{3} - 17 a^{2} + 12 a + 16\) , \( 2 a^{4} - 2 a^{3} - 10 a^{2} + 6 a + 9\bigr] \)
${y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}{y}+\left(-a^{4}+a^{3}+5a^{2}-3a-4\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+4\right){x}^{2}+\left(4a^{4}-5a^{3}-17a^{2}+12a+16\right){x}+2a^{4}-2a^{3}-10a^{2}+6a+9$
171.1-a1
171.1-a
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
171.1
\( 3^{2} \cdot 19 \)
\( - 3^{6} \cdot 19^{2} \)
$38.28941$
$(-a^4+a^3+4a^2-2a-2), (-a^4+2a^3+4a^2-5a-4)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$1$
\( 2^{2} \)
$0.023336836$
$2343.039936$
4.26786965
\( \frac{218695}{361} a^{4} - \frac{376550}{361} a^{3} - \frac{959505}{361} a^{2} + \frac{1058864}{361} a + \frac{910976}{361} \)
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 2\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 4\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 3\) , \( 4 a^{4} - 3 a^{3} - 17 a^{2} + 8 a + 16\) , \( -3 a^{4} + 6 a^{3} + 16 a^{2} - 15 a - 16\bigr] \)
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-2\right){x}{y}+\left(-a^{4}+2a^{3}+4a^{2}-6a-3\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+5a^{2}-6a-4\right){x}^{2}+\left(4a^{4}-3a^{3}-17a^{2}+8a+16\right){x}-3a^{4}+6a^{3}+16a^{2}-15a-16$
361.1-e1
361.1-e
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
361.1
\( 19^{2} \)
\( - 19^{8} \)
$41.26005$
$(-a^4+2a^3+4a^2-5a-4)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$1$
\( 2 \)
$1$
$238.2728102$
1.85979024
\( \frac{218695}{361} a^{4} - \frac{376550}{361} a^{3} - \frac{959505}{361} a^{2} + \frac{1058864}{361} a + \frac{910976}{361} \)
\( \bigl[a^{3} - 4 a\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 4 a\) , \( a^{2} - 1\) , \( 153 a^{4} - 190 a^{3} - 721 a^{2} + 477 a + 657\) , \( 104 a^{4} - 129 a^{3} - 490 a^{2} + 324 a + 445\bigr] \)
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+4a\right){x}^{2}+\left(153a^{4}-190a^{3}-721a^{2}+477a+657\right){x}+104a^{4}-129a^{3}-490a^{2}+324a+445$
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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.