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
27.1-a1
27.1-a
$2$
$3$
5.5.36497.1
$5$
$[5, 0]$
27.1
\( 3^{3} \)
\( 3^{5} \)
$23.73578$
$(a^2-1)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 1 \)
$1$
$2129.057958$
1.23827296
\( -180015 a^{4} + 286223 a^{3} + 657933 a^{2} - 631336 a - 439430 \)
\( \bigl[a^{2} - 2\) , \( -a^{4} + 2 a^{3} + 3 a^{2} - 6 a - 1\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( -a^{2} + 3\) , \( -1\bigr] \)
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+3a^{2}-6a-1\right){x}^{2}+\left(-a^{2}+3\right){x}-1$
27.1-b1
27.1-b
$2$
$3$
5.5.36497.1
$5$
$[5, 0]$
27.1
\( 3^{3} \)
\( 3^{5} \)
$23.73578$
$(a^2-1)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 3 \)
$0.001419619$
$16840.33282$
1.87709116
\( -180015 a^{4} + 286223 a^{3} + 657933 a^{2} - 631336 a - 439430 \)
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\) , \( -a^{4} + a^{3} + 3 a^{2} - a + 1\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\) , \( -5 a^{4} + 6 a^{3} + 16 a^{2} - 9 a - 5\) , \( -3 a^{4} + 3 a^{3} + 10 a^{2} - 5 a - 3\bigr] \)
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+6a+1\right){x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+3a^{2}-a+1\right){x}^{2}+\left(-5a^{4}+6a^{3}+16a^{2}-9a-5\right){x}-3a^{4}+3a^{3}+10a^{2}-5a-3$
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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.