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
54.1-a1
54.1-a
$4$
$27$
\(\Q(\sqrt{-2}) \)
$2$
$[0, 1]$
54.1
\( 2 \cdot 3^{3} \)
\( 2^{3} \cdot 3^{3} \)
$0.68514$
$(a), (-a-1)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3Cs.1.1
$1$
\( 1 \)
$1$
$8.206562124$
0.644768414
\( \frac{6935}{4} a - \frac{9251}{2} \)
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -a - 1\) , \( -1\bigr] \)
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-1\right){x}-1$
54.1-a2
54.1-a
$4$
$27$
\(\Q(\sqrt{-2}) \)
$2$
$[0, 1]$
54.1
\( 2 \cdot 3^{3} \)
\( 2 \cdot 3^{5} \)
$0.68514$
$(a), (-a-1)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B.1.1
$1$
\( 1 \)
$1$
$8.206562124$
0.644768414
\( -\frac{269}{2} a + 1619 \)
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -a - 1\) , \( 0\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}$
54.1-a3
54.1-a
$4$
$27$
\(\Q(\sqrt{-2}) \)
$2$
$[0, 1]$
54.1
\( 2 \cdot 3^{3} \)
\( 2^{27} \cdot 3^{11} \)
$0.68514$
$(a), (-a-1)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3B.1.2
$1$
\( 1 \)
$1$
$0.911840236$
0.644768414
\( \frac{241123607}{16384} a + \frac{59710933}{8192} \)
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -51 a + 69\) , \( 62 a + 339\bigr] \)
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-51a+69\right){x}+62a+339$
54.1-a4
54.1-a
$4$
$27$
\(\Q(\sqrt{-2}) \)
$2$
$[0, 1]$
54.1
\( 2 \cdot 3^{3} \)
\( 2^{9} \cdot 3^{9} \)
$0.68514$
$(a), (-a-1)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$3$
3Cs.1.1
$1$
\( 3 \)
$1$
$2.735520708$
0.644768414
\( -\frac{2628365}{32} a + \frac{183347}{16} \)
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -11 a + 4\) , \( -26\bigr] \)
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-11a+4\right){x}-26$
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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.