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-a2
27.2-a
$2$
$3$
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
$1$
\( 3 \)
$0.068663543$
$802.8036549$
4.698443578
\( \frac{2272}{3} a^{3} + \frac{1171}{3} a^{2} - 5297 a - 3398 \)
\( \bigl[a + 1\) , \( \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 6 a + 1\) , \( a^{2} - 3\) , \( -\frac{5}{3} a^{3} + \frac{1}{3} a^{2} + 10 a - 2\) , \( -\frac{7}{3} a^{3} + \frac{8}{3} a^{2} + 17 a - 22\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-6a+1\right){x}^{2}+\left(-\frac{5}{3}a^{3}+\frac{1}{3}a^{2}+10a-2\right){x}-\frac{7}{3}a^{3}+\frac{8}{3}a^{2}+17a-22$
27.2-f1
27.2-f
$2$
$3$
4.4.19821.1
$4$
$[4, 0]$
27.2
\( 3^{3} \)
\( - 3^{3} \)
$18.99423$
$(-1/3a^3-1/3a^2+3a+2)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 1 \)
$1.962322637$
$594.7379778$
3.684264377
\( \frac{2272}{3} a^{3} + \frac{1171}{3} a^{2} - 5297 a - 3398 \)
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( 0\bigr] \)
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){x}^{2}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){x}$
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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.