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
3969.1-CMc1
3969.1-CMc
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
3969.1
\( 3^{4} \cdot 7^{2} \)
\( 3^{12} \cdot 7^{10} \)
$1.22848$
$(-2a+1), (-3a+1)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$7$
7Cs.2.1
$1$
\( 1 \)
$1$
$0.924992894$
1.068089793
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -159 a + 177\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-159a+177$
3969.1-CMb1
3969.1-CMb
$2$
$3$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
3969.1
\( 3^{4} \cdot 7^{2} \)
\( 3^{12} \cdot 7^{6} \)
$1.22848$
$(-2a+1), (-3a+1)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3, 7$
3Cs[2] , 7Cs.2.1
$1$
\( 1 \)
$1$
$1.769447752$
2.043182272
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 15 a - 28\bigr] \)
${y}^2+{y}={x}^{3}+15a-28$
3969.1-CMb2
3969.1-CMb
$2$
$3$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
3969.1
\( 3^{4} \cdot 7^{2} \)
\( 3^{4} \cdot 7^{6} \)
$1.22848$
$(-2a+1), (-3a+1)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-27$
$\mathrm{U}(1)$
✓
✓
$7$
7Cs.2.1
$1$
\( 1 \)
$1$
$1.769447752$
2.043182272
\( -12288000 \)
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -79 a + 50\) , \( -177 a + 303\bigr] \)
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-79a+50\right){x}-177a+303$
3969.1-CMa1
3969.1-CMa
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
3969.1
\( 3^{4} \cdot 7^{2} \)
\( 3^{6} \cdot 7^{4} \)
$1.22848$
$(-2a+1), (-3a+1)$
0
$\Z/3\Z$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3$
3B.1.1[2]
$1$
\( 3 \)
$1$
$4.238849958$
1.631534109
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -2\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-2$
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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.