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
53.1-a1
53.1-a
$2$
$3$
5.5.70601.1
$5$
$[5, 0]$
53.1
\( 53 \)
\( 53 \)
$35.31600$
$(-a^4+2a^3+4a^2-6a)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.104754266$
$1594.251086$
3.14262831
\( -\frac{561873647340}{53} a^{4} + \frac{106058296968}{53} a^{3} + \frac{2895622251362}{53} a^{2} + \frac{1224893580269}{53} a - \frac{693126760539}{53} \)
\( \bigl[2 a^{4} - 2 a^{3} - 9 a^{2} + 4 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a - 1\) , \( 2 a^{4} - a^{3} - 10 a^{2} - a + 3\) , \( 30 a^{4} - 5 a^{3} - 157 a^{2} - 65 a + 36\) , \( 142 a^{4} - 64 a^{3} - 658 a^{2} - 250 a + 149\bigr] \)
${y}^2+\left(2a^{4}-2a^{3}-9a^{2}+4a+2\right){x}{y}+\left(2a^{4}-a^{3}-10a^{2}-a+3\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+5a-1\right){x}^{2}+\left(30a^{4}-5a^{3}-157a^{2}-65a+36\right){x}+142a^{4}-64a^{3}-658a^{2}-250a+149$
53.1-a2
53.1-a
$2$
$3$
5.5.70601.1
$5$
$[5, 0]$
53.1
\( 53 \)
\( 53^{3} \)
$35.31600$
$(-a^4+2a^3+4a^2-6a)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B
$1$
\( 1 \)
$0.034918088$
$4782.753259$
3.14262831
\( -\frac{253459126857}{148877} a^{4} + \frac{707710137815}{148877} a^{3} - \frac{1843009819}{148877} a^{2} - \frac{502269847354}{148877} a + \frac{141355191473}{148877} \)
\( \bigl[2 a^{4} - 2 a^{3} - 9 a^{2} + 4 a + 3\) , \( -a^{4} + a^{3} + 4 a^{2} - a - 1\) , \( a^{4} - 6 a^{2} - 2 a + 4\) , \( a^{4} + 6 a^{3} - 11 a^{2} - 29 a + 7\) , \( 7 a^{4} + 2 a^{3} - 39 a^{2} - 30 a + 12\bigr] \)
${y}^2+\left(2a^{4}-2a^{3}-9a^{2}+4a+3\right){x}{y}+\left(a^{4}-6a^{2}-2a+4\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-a-1\right){x}^{2}+\left(a^{4}+6a^{3}-11a^{2}-29a+7\right){x}+7a^{4}+2a^{3}-39a^{2}-30a+12$
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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.