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
139.3-a1
139.3-a
$1$
$1$
4.4.725.1
$4$
$[4, 0]$
139.3
\( 139 \)
\( 139 \)
$4.45846$
$(-a^3+a^2+3)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 1 \)
$1$
$26.07113072$
0.968257487
\( \frac{4931689084}{139} a^{3} - \frac{11272518940}{139} a^{2} + \frac{396236521}{139} a + \frac{3292988433}{139} \)
\( \bigl[1\) , \( a^{3} - a^{2} - 2 a\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -a^{2}\) , \( -a^{3} - a^{2} + a\bigr] \)
${y}^2+{x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a\right){x}^{2}-a^{2}{x}-a^{3}-a^{2}+a$
139.3-b1
139.3-b
$2$
$7$
4.4.725.1
$4$
$[4, 0]$
139.3
\( 139 \)
\( 139^{7} \)
$4.45846$
$(-a^3+a^2+3)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$7$
7B.1.3
$49$
\( 1 \)
$1$
$0.475474198$
0.865274752
\( \frac{15524334380384952878789090}{1002544368429379} a^{3} - \frac{36570245179369727318794324}{1002544368429379} a^{2} + \frac{3003941181725277973894315}{1002544368429379} a + \frac{11451131736082354617257880}{1002544368429379} \)
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - a + 2\) , \( -16 a^{3} - 18 a^{2} + 148 a - 106\) , \( -75 a^{3} - 188 a^{2} + 945 a - 576\bigr] \)
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{3}-a^{2}-a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(-16a^{3}-18a^{2}+148a-106\right){x}-75a^{3}-188a^{2}+945a-576$
139.3-b2
139.3-b
$2$
$7$
4.4.725.1
$4$
$[4, 0]$
139.3
\( 139 \)
\( 139 \)
$4.45846$
$(-a^3+a^2+3)$
0
$\Z/7\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$7$
7B.1.1
$1$
\( 1 \)
$1$
$1141.613550$
0.865274752
\( -\frac{44113}{139} a^{3} + \frac{90532}{139} a^{2} + \frac{31128}{139} a - \frac{54250}{139} \)
\( \bigl[a^{3} - 2 a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a^{2} - a + 2\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( a^{3} + a^{2} - a - 1\bigr] \)
${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{3}-a^{2}-a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a\right){x}^{2}+\left(-a^{3}+2a^{2}+3a-1\right){x}+a^{3}+a^{2}-a-1$
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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.