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
53.2-a1
53.2-a
$2$
$3$
\(\Q(\zeta_{13})^+\)
$6$
$[6, 0]$
53.2
\( 53 \)
\( -53 \)
$75.80303$
$(2a^3-a^2-6a+1)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.003837573$
$60162.44104$
2.27340
\( -\frac{510635032}{53} a^{5} - \frac{391066624}{53} a^{4} + \frac{1601119885}{53} a^{3} + \frac{835805000}{53} a^{2} - \frac{816338829}{53} a - \frac{221982260}{53} \)
\( \bigl[a^{4} - 3 a^{2} + 1\) , \( a^{5} - 6 a^{3} + 8 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{4} - a^{3} + a^{2} + 2 a + 3\) , \( -a^{4} + 3 a^{2} - 1\bigr] \)
${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{5}-6a^{3}+8a-1\right){x}^{2}+\left(-a^{4}-a^{3}+a^{2}+2a+3\right){x}-a^{4}+3a^{2}-1$
53.2-a2
53.2-a
$2$
$3$
\(\Q(\zeta_{13})^+\)
$6$
$[6, 0]$
53.2
\( 53 \)
\( - 53^{3} \)
$75.80303$
$(2a^3-a^2-6a+1)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 3 \)
$0.001279191$
$60162.44104$
2.27340
\( -\frac{7340797392}{148877} a^{5} + \frac{20168559723}{148877} a^{4} + \frac{865808613}{148877} a^{3} - \frac{30080813866}{148877} a^{2} + \frac{9874050146}{148877} a + \frac{3817528150}{148877} \)
\( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( 2 a^{5} + a^{4} - 7 a^{3} - 4 a^{2} + 3 a + 1\) , \( 2 a^{5} + a^{4} - 7 a^{3} - 3 a^{2} + 4 a + 1\bigr] \)
${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+1\right){x}^{2}+\left(2a^{5}+a^{4}-7a^{3}-4a^{2}+3a+1\right){x}+2a^{5}+a^{4}-7a^{3}-3a^{2}+4a+1$
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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.