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
13.1-a2
13.1-a
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
13.1
\( 13 \)
\( 13^{3} \)
$12.25734$
$(-a^2+2a+2)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3Cs.1.1
$1$
\( 3 \)
$0.359736041$
$441.3551652$
2.126649835
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \)
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -2 a^{3} + 2 a^{2} + 9 a - 5\) , \( -14 a^{3} + 17 a^{2} + 64 a - 35\bigr] \)
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-2a^{3}+2a^{2}+9a-5\right){x}-14a^{3}+17a^{2}+64a-35$
13.1-b1
13.1-b
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
13.1
\( 13 \)
\( 13^{3} \)
$12.25734$
$(-a^2+2a+2)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3Cs.1.1
$1$
\( 3 \)
$0.277561838$
$403.6980252$
1.500860066
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \)
\( \bigl[a^{3} - 4 a - 2\) , \( a\) , \( a^{3} - 5 a - 1\) , \( 9 a^{3} - 18 a^{2} - 22 a + 14\) , \( -46 a^{3} + 74 a^{2} + 104 a - 66\bigr] \)
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+a{x}^{2}+\left(9a^{3}-18a^{2}-22a+14\right){x}-46a^{3}+74a^{2}+104a-66$
169.1-c3
169.1-c
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
169.1
\( 13^{2} \)
\( 13^{9} \)
$16.89038$
$(-a^2+2a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3Cs
$1$
\( 2 \)
$1$
$224.5790519$
4.512158279
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \)
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( 1\) , \( a^{3} - 5 a - 1\) , \( 2 a^{3} + 33 a^{2} - 33 a - 163\) , \( 28 a^{3} + 139 a^{2} - 246 a - 773\bigr] \)
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+{x}^{2}+\left(2a^{3}+33a^{2}-33a-163\right){x}+28a^{3}+139a^{2}-246a-773$
169.1-e1
169.1-e
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
169.1
\( 13^{2} \)
\( 13^{9} \)
$16.89038$
$(-a^2+2a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3Cs
$1$
\( 2 \)
$1$
$61.02843627$
1.226160506
\( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \)
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a + 1\) , \( 0\) , \( 22 a^{3} - 6 a^{2} - 103 a - 77\) , \( -14 a^{3} - 59 a^{2} + 152 a + 245\bigr] \)
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(22a^{3}-6a^{2}-103a-77\right){x}-14a^{3}-59a^{2}+152a+245$
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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.