Elliptic curves over \(\Q(\zeta_{13})^+\) of conductor 53.1

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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	$2$	$3$	\(\Q(\zeta_{13})^+\)	$6$	$[6, 0]$	53.1	\( 53 \)	\( -53 \)	$75.80303$	$(-a^4+a^3+3a^2-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$3$	3B	$1$	\( 1 \)	$0.003837573$	$60162.44104$	2.27340	\( \frac{1343121899}{53} a^{5} - \frac{1680516771}{53} a^{4} - \frac{6274189252}{53} a^{3} + \frac{6889341303}{53} a^{2} + \frac{6343404041}{53} a - \frac{5544409259}{53} \)	\( \bigl[a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( -a^{3} - a^{2} + 3 a + 3\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 3 a^{5} - 3 a^{4} - 17 a^{3} + 7 a^{2} + 23 a + 5\) , \( a^{5} - a^{4} - 7 a^{3} + 2 a^{2} + 11 a + 2\bigr] \)	${y}^2+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+3\right){x}^{2}+\left(3a^{5}-3a^{4}-17a^{3}+7a^{2}+23a+5\right){x}+a^{5}-a^{4}-7a^{3}+2a^{2}+11a+2$
53.1-a2	53.1-a	$2$	$3$	\(\Q(\zeta_{13})^+\)	$6$	$[6, 0]$	53.1	\( 53 \)	\( - 53^{3} \)	$75.80303$	$(-a^4+a^3+3a^2-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$3$	3B	$1$	\( 3 \)	$0.001279191$	$60162.44104$	2.27340	\( \frac{15669618624}{148877} a^{5} + \frac{14755246679}{148877} a^{4} - \frac{49850712165}{148877} a^{3} - \frac{33922667128}{148877} a^{2} + \frac{28694128602}{148877} a + \frac{8142934591}{148877} \)	\( \bigl[a^{2} + a - 1\) , \( a^{5} - a^{4} - 6 a^{3} + 3 a^{2} + 9 a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 3\) , \( a^{5} - 2 a^{4} - 8 a^{3} + 6 a^{2} + 17 a + 4\) , \( 2 a^{5} - a^{4} - 11 a^{3} + 2 a^{2} + 15 a + 4\bigr] \)	${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+3a^{2}+9a+1\right){x}^{2}+\left(a^{5}-2a^{4}-8a^{3}+6a^{2}+17a+4\right){x}+2a^{5}-a^{4}-11a^{3}+2a^{2}+15a+4$

 

  *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.