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-a1
13.1-a
$3$
$9$
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
$1$
\( 1 \)
$1.079208123$
$441.3551652$
2.126649835
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \)
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( 0\) , \( -17 a^{3} - 3 a^{2} + 34 a - 30\) , \( 45 a^{3} + 106 a^{2} + 25 a - 23\bigr] \)
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-17a^{3}-3a^{2}+34a-30\right){x}+45a^{3}+106a^{2}+25a-23$
13.1-b2
13.1-b
$3$
$9$
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
$9$
\( 1 \)
$0.832685514$
$4.983926238$
1.500860066
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \)
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - a - 2\) , \( -130 a^{3} + 234 a^{2} + 353 a - 228\) , \( 477 a^{3} - 959 a^{2} - 995 a + 687\bigr] \)
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-130a^{3}+234a^{2}+353a-228\right){x}+477a^{3}-959a^{2}-995a+687$
169.1-c1
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
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \)
\( \bigl[a^{2} - 2\) , \( -a^{3} + 4 a + 1\) , \( a^{2} - 3\) , \( -8355 a^{3} + 10036 a^{2} + 37871 a - 20711\) , \( 462835 a^{3} - 558413 a^{2} - 2101506 a + 1149416\bigr] \)
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(-8355a^{3}+10036a^{2}+37871a-20711\right){x}+462835a^{3}-558413a^{2}-2101506a+1149416$
169.1-e3
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
\( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \)
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( a + 1\) , \( -152 a^{3} - 223 a^{2} + 70 a + 36\) , \( -2946 a^{3} - 7182 a^{2} - 1520 a + 3011\bigr] \)
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-152a^{3}-223a^{2}+70a+36\right){x}-2946a^{3}-7182a^{2}-1520a+3011$
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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.