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-a2
27.1-a
$2$
$3$
5.5.36497.1
$5$
$[5, 0]$
27.1
\( 3^{3} \)
\( 3^{3} \)
$23.73578$
$(a^2-1)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$9$
\( 1 \)
$1$
$26.28466615$
1.23827296
\( 1418630703 a^{4} - 1053833686 a^{3} - 5351126785 a^{2} - 8280788 a + 937167631 \)
\( \bigl[a^{4} - a^{3} - 3 a^{2} + 2 a\) , \( a^{2} - a - 2\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 1\) , \( 374 a^{4} - 252 a^{3} - 1451 a^{2} - 59 a + 277\) , \( 5982 a^{4} - 3984 a^{3} - 23258 a^{2} - 1118 a + 4479\bigr] \)
${y}^2+\left(a^{4}-a^{3}-3a^{2}+2a\right){x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(374a^{4}-252a^{3}-1451a^{2}-59a+277\right){x}+5982a^{4}-3984a^{3}-23258a^{2}-1118a+4479$
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
\( -180015 a^{4} + 286223 a^{3} + 657933 a^{2} - 631336 a - 439430 \)
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 6 a + 1\) , \( -a^{4} + a^{3} + 3 a^{2} - a + 1\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\) , \( -5 a^{4} + 6 a^{3} + 16 a^{2} - 9 a - 5\) , \( -3 a^{4} + 3 a^{3} + 10 a^{2} - 5 a - 3\bigr] \)
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+6a+1\right){x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+3a^{2}-a+1\right){x}^{2}+\left(-5a^{4}+6a^{3}+16a^{2}-9a-5\right){x}-3a^{4}+3a^{3}+10a^{2}-5a-3$
27.1-b2
27.1-b
$2$
$3$
5.5.36497.1
$5$
$[5, 0]$
27.1
\( 3^{3} \)
\( 3^{3} \)
$23.73578$
$(a^2-1)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.004258859$
$16840.33282$
1.87709116
\( 1418630703 a^{4} - 1053833686 a^{3} - 5351126785 a^{2} - 8280788 a + 937167631 \)
\( \bigl[2 a^{4} - 3 a^{3} - 7 a^{2} + 8 a + 3\) , \( a^{3} - 5 a - 2\) , \( a^{4} - a^{3} - 3 a^{2} + a\) , \( -5 a^{3} + 2 a^{2} + 16 a - 1\) , \( 4 a^{4} - 4 a^{3} - 16 a^{2} + 6 a + 10\bigr] \)
${y}^2+\left(2a^{4}-3a^{3}-7a^{2}+8a+3\right){x}{y}+\left(a^{4}-a^{3}-3a^{2}+a\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-5a^{3}+2a^{2}+16a-1\right){x}+4a^{4}-4a^{3}-16a^{2}+6a+10$
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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.