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
89.4-a1
89.4-a
$2$
$3$
\(\Q(\zeta_{15})^+\)
$4$
$[4, 0]$
89.4
\( 89 \)
\( - 89^{3} \)
$5.25277$
$(-2a^3-a^2+8a-1)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$9$
\( 1 \)
$1$
$4.662275233$
1.251019722
\( \frac{113501404732035}{704969} a^{3} + \frac{29870758114798}{704969} a^{2} - \frac{399875876584351}{704969} a - \frac{61417651565281}{704969} \)
\( \bigl[a\) , \( a^{3} - a^{2} - 2 a + 3\) , \( a^{3} - 3 a + 1\) , \( 46 a^{3} + 15 a^{2} - 164 a - 37\) , \( 220 a^{3} + 76 a^{2} - 784 a - 178\bigr] \)
${y}^2+a{x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+3\right){x}^{2}+\left(46a^{3}+15a^{2}-164a-37\right){x}+220a^{3}+76a^{2}-784a-178$
89.4-a2
89.4-a
$2$
$3$
\(\Q(\zeta_{15})^+\)
$4$
$[4, 0]$
89.4
\( 89 \)
\( -89 \)
$5.25277$
$(-2a^3-a^2+8a-1)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 1 \)
$1$
$377.6442939$
1.251019722
\( \frac{71041}{89} a^{3} - \frac{8400}{89} a^{2} - \frac{224109}{89} a + \frac{125520}{89} \)
\( \bigl[a\) , \( a^{3} - a^{2} - 2 a + 3\) , \( a^{3} - 3 a + 1\) , \( a^{3} - 4 a + 3\) , \( a^{3} - 3 a\bigr] \)
${y}^2+a{x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+3\right){x}^{2}+\left(a^{3}-4a+3\right){x}+a^{3}-3a$
89.4-b1
89.4-b
$2$
$3$
\(\Q(\zeta_{15})^+\)
$4$
$[4, 0]$
89.4
\( 89 \)
\( - 89^{3} \)
$5.25277$
$(-2a^3-a^2+8a-1)$
$1$
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 3 \)
$0.019242294$
$1474.084412$
1.127565233
\( \frac{113501404732035}{704969} a^{3} + \frac{29870758114798}{704969} a^{2} - \frac{399875876584351}{704969} a - \frac{61417651565281}{704969} \)
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a\) , \( -32 a^{3} + 49 a^{2} + 129 a - 191\) , \( 232 a^{3} - 278 a^{2} - 873 a + 1119\bigr] \)
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(-32a^{3}+49a^{2}+129a-191\right){x}+232a^{3}-278a^{2}-873a+1119$
89.4-b2
89.4-b
$2$
$3$
\(\Q(\zeta_{15})^+\)
$4$
$[4, 0]$
89.4
\( 89 \)
\( -89 \)
$5.25277$
$(-2a^3-a^2+8a-1)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$1$
\( 1 \)
$0.006414098$
$1474.084412$
1.127565233
\( \frac{71041}{89} a^{3} - \frac{8400}{89} a^{2} - \frac{224109}{89} a + \frac{125520}{89} \)
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a\) , \( 3 a^{3} + 4 a^{2} - 6 a - 6\) , \( 4 a^{3} + 3 a^{2} - 10 a - 2\bigr] \)
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-2a-2\right){x}^{2}+\left(3a^{3}+4a^{2}-6a-6\right){x}+4a^{3}+3a^{2}-10a-2$
Download
displayed columns for
results
to
Text
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.