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
27.2-a1
27.2-a
$2$
$3$
4.4.19821.1
$4$
$[4, 0]$
27.2
\( 3^{3} \)
\( - 3^{3} \)
$18.99423$
$(-1/3a^3-1/3a^2+3a+2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$9$
\( 1 \)
$0.205990629$
$89.20040611$
4.698443578
\( -\frac{2659993414}{3} a^{3} + \frac{3059637059}{3} a^{2} + 7384788197 a - 8211399121 \)
\( \bigl[a^{2} - 4\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 1\) , \( 0\) , \( -\frac{14}{3} a^{3} - \frac{11}{3} a^{2} + 45 a + 19\) , \( -\frac{28}{3} a^{3} + \frac{20}{3} a^{2} + 54 a + 17\bigr] \)
${y}^2+\left(a^{2}-4\right){x}{y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+3a+1\right){x}^{2}+\left(-\frac{14}{3}a^{3}-\frac{11}{3}a^{2}+45a+19\right){x}-\frac{28}{3}a^{3}+\frac{20}{3}a^{2}+54a+17$
27.2-f2
27.2-f
$2$
$3$
4.4.19821.1
$4$
$[4, 0]$
27.2
\( 3^{3} \)
\( - 3^{9} \)
$18.99423$
$(-1/3a^3-1/3a^2+3a+2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$1$
\( 3 \)
$5.886967913$
$7.342444170$
3.684264377
\( -\frac{2659993414}{3} a^{3} + \frac{3059637059}{3} a^{2} + 7384788197 a - 8211399121 \)
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{19}{3} a^{3} - \frac{19}{3} a^{2} + 33 a + 9\) , \( -\frac{67}{3} a^{3} - \frac{124}{3} a^{2} + 59 a + 22\bigr] \)
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){x}^{2}+\left(-\frac{19}{3}a^{3}-\frac{19}{3}a^{2}+33a+9\right){x}-\frac{67}{3}a^{3}-\frac{124}{3}a^{2}+59a+22$
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Pari/GP
SageMath
Magma
Oscar
CSV
*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.