Elliptic curves over $\Q$ of conductor 19

Conductor	j-invariant	Rank	Torsion	Complex multiplication
Bad$\ p$	Discriminant	Regulator	Analytic order of Ш	Galois image
Isogeny class size	Isogeny class degree	Cyclic isogeny degree	$p\ $div$\ $|Ш|	Nonmax$\ \ell$
Curves per isogeny class	Reduction	Integral points	Faltings height

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	CM	Sato-Tate	Semistable	Potentially good	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Regulator	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	Weierstrass coefficients	Weierstrass equation
19.a1	19a2	19.a	19a	$3$	$9$	\( 19 \)	\( -19 \)	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	27.72.0.2	3B.1.2	$1$	$1$		$0$	$3$	$0.033439$	$-50357871050752/19$	$[0, 1, 1, -769, -8470]$	\(y^2+y=x^3+x^2-769x-8470\)
19.a2	19a1	19.a	19a	$3$	$9$	\( 19 \)	\( - 19^{3} \)	$0$	$\Z/3\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	9.72.0.3	3Cs.1.1	$1$	$1$		$2$	$1$	$-0.515867$	$-89915392/6859$	$[0, 1, 1, -9, -15]$	\(y^2+y=x^3+x^2-9x-15\)
19.a3	19a3	19.a	19a	$3$	$9$	\( 19 \)	\( -19 \)	$0$	$\Z/3\Z$	$\Q$		$\mathrm{SU}(2)$	✓		$3$	27.72.0.1	3B.1.1	$1$	$1$		$2$	$3$	$-1.065172$	$32768/19$	$[0, 1, 1, 1, 0]$	\(y^2+y=x^3+x^2+x\)