Elliptic curves over $\Q$ of conductor 27

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
27.a1	27a2	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{11} \)	$0$	$\mathsf{trivial}$	$\Q(\sqrt{-3})$	$-27$	$N(\mathrm{U}(1))$		✓	$3$	27.648.13.34	3B.1.2	$1$	$1$		$0$	$3$	$0.052148$	$-12288000$	$[0, 0, 1, -270, -1708]$	\(y^2+y=x^3-270x-1708\)
27.a2	27a4	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{5} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-27$	$N(\mathrm{U}(1))$		✓	$3$	27.648.13.25	3B.1.1	$1$	$1$		$2$	$9$	$-0.497158$	$-12288000$	$[0, 0, 1, -30, 63]$	\(y^2+y=x^3-30x+63\)
27.a3	27a1	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{9} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3$	27.1944.55.37	3Cs.1.1	$1$	$1$		$2$	$1$	$-0.497158$	$0$	$[0, 0, 1, 0, -7]$	\(y^2+y=x^3-7\)
27.a4	27a3	27.a	27a	$4$	$27$	\( 3^{3} \)	\( - 3^{3} \)	$0$	$\Z/3\Z$	$\Q(\sqrt{-3})$	$-3$	$N(\mathrm{U}(1))$		✓	$3$	27.1944.55.31	3Cs.1.1	$1$	$1$		$2$	$3$	$-1.046465$	$0$	$[0, 0, 1, 0, 0]$	\(y^2+y=x^3\)