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-a3
13.1-a
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
13.1
\( 13 \)
\( 13 \)
$12.25734$
$(-a^2+2a+2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$9$
\( 1 \)
$1.079208123$
$5.448829201$
2.126649835
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \)
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a\) , \( 0\) , \( 21 a^{3} + 8 a^{2} - 26 a - 9\) , \( 562 a^{3} + 1305 a^{2} + 277 a - 581\bigr] \)
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+a{x}^{2}+\left(21a^{3}+8a^{2}-26a-9\right){x}+562a^{3}+1305a^{2}+277a-581$
13.1-b3
13.1-b
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
13.1
\( 13 \)
\( 13 \)
$12.25734$
$(-a^2+2a+2)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 1 \)
$0.832685514$
$403.6980252$
1.500860066
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \)
\( \bigl[1\) , \( -a^{3} + a^{2} + 4 a\) , \( 1\) , \( -220 a^{3} + 274 a^{2} + 1002 a - 570\) , \( 2785 a^{3} - 3453 a^{2} - 12765 a + 7022\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-220a^{3}+274a^{2}+1002a-570\right){x}+2785a^{3}-3453a^{2}-12765a+7022$
169.1-c2
169.1-c
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
169.1
\( 13^{2} \)
\( 13^{7} \)
$16.89038$
$(-a^2+2a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B
$9$
\( 2 \)
$1$
$24.95322799$
4.512158279
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \)
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 6 a - 2\) , \( a^{2} - a - 3\) , \( -417 a^{3} + 568 a^{2} + 1989 a - 1102\) , \( -7135 a^{3} + 13736 a^{2} + 39490 a - 22620\bigr] \)
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{3}-6a-2\right){x}^{2}+\left(-417a^{3}+568a^{2}+1989a-1102\right){x}-7135a^{3}+13736a^{2}+39490a-22620$
169.1-e2
169.1-e
$3$
$9$
4.4.9909.1
$4$
$[4, 0]$
169.1
\( 13^{2} \)
\( 13^{7} \)
$16.89038$
$(-a^2+2a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B
$9$
\( 2 \)
$1$
$6.780937363$
1.226160506
\( \frac{54952780378700013073}{13} a^{3} - \frac{66356600984288283278}{13} a^{2} - \frac{249589735669688957882}{13} a + \frac{136526345323237627463}{13} \)
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( a + 1\) , \( -487 a^{3} + 447 a^{2} + 2665 a - 1404\) , \( 17336 a^{3} - 23043 a^{2} - 72419 a + 40406\bigr] \)
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-487a^{3}+447a^{2}+2665a-1404\right){x}+17336a^{3}-23043a^{2}-72419a+40406$
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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.