LMFDB - Elliptic curve isogeny class with LMFDB label 486720.pf (Cremona label 486720pf)

Properties

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Show commands: SageMath

sage:E = EllipticCurve("pf1") E.isogeny_class()

Rank

sage:E.rank()

The elliptic curves in class 486720.pf have rank $1$.

The elliptic curves in class 486720.pf do not have complex multiplication.

sage:E.q_eigenform(10)

sage:E.isogeny_class().matrix()

The $i,j$ entry is the smallest degree of a cyclic isogeny between the $i$-th and $j$-th curve in the isogeny class, in the LMFDB numbering.

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$

sage:E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.

sage:E.isogeny_class().curves

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
486720.pf1	486720pf1	$[0, 0, 0, -494832, -12848056]$	$3718856704/2132325$	$7683179817138508800$	$[2]$	$8257536$	$2.3135$	$\Gamma_0(N)$-optimal
486720.pf2	486720pf2	$[0, 0, 0, 1969188, -102538384]$	$14647977776/8555625$	$-493241173445928960000$	$[2]$	$16515072$	$2.6600$

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LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
486720.pf1	486720pf1	\([0, 0, 0, -494832, -12848056]\)	\(3718856704/2132325\)	\(7683179817138508800\)	\([2]\)	\(8257536\)	\(2.3135\)	\(\Gamma_0(N)\)-optimal
486720.pf2	486720pf2	\([0, 0, 0, 1969188, -102538384]\)	\(14647977776/8555625\)	\(-493241173445928960000\)	\([2]\)	\(16515072\)	\(2.6600\)