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
13689.3-CMc1
13689.3-CMc
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
13689.3
\( 3^{4} \cdot 13^{2} \)
\( 3^{6} \cdot 13^{8} \)
$1.67414$
$(-2a+1), (4a-3)$
$0 \le r \le 2$
$\Z/3\Z$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3$
3B.1.1[2]
$4$
\( 3 \)
$1$
$1.466622992$
2.258013812
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -4 a - 40\bigr] \)
${y}^2+a{y}={x}^{3}-4a-40$
13689.3-CMb1
13689.3-CMb
$2$
$3$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
13689.3
\( 3^{4} \cdot 13^{2} \)
\( 3^{6} \cdot 13^{6} \)
$1.67414$
$(-2a+1), (4a-3)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3$
3Cs[2]
$1$
\( 1 \)
$1$
$2.248928844$
2.596839347
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -9 a + 13\bigr] \)
${y}^2+{y}={x}^{3}-9a+13$
13689.3-CMb2
13689.3-CMb
$2$
$3$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
13689.3
\( 3^{4} \cdot 13^{2} \)
\( 3^{10} \cdot 13^{6} \)
$1.67414$
$(-2a+1), (4a-3)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-27$
$\mathrm{U}(1)$
✓
✓
$1$
\( 3 \)
$1$
$0.749642948$
2.596839347
\( -12288000 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 210 a - 450\) , \( -2277 a + 3352\bigr] \)
${y}^2+{y}={x}^{3}+\left(210a-450\right){x}-2277a+3352$
13689.3-CMa1
13689.3-CMa
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
13689.3
\( 3^{4} \cdot 13^{2} \)
\( 3^{6} \cdot 13^{2} \)
$1.67414$
$(-2a+1), (4a-3)$
$2$
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$1$
\( 3 \)
$0.032488570$
$5.287984402$
2.380517548
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -a\bigr] \)
${y}^2+a{y}={x}^{3}-a$
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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.