LMFDB - Elliptic curve isogeny class with LMFDB label 1089.g (Cremona label 1089e)

Properties

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Show commands: SageMath

E = EllipticCurve("g1")

E.isogeny_class()

Elliptic curves in class 1089.g

sage: E.isogeny_class().curves

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
1089.g1	1089e2	$[0, 0, 1, -7986, 281839]$	$-32768$	$-1718943866739$	$[]$	$1320$	$1.1252$		$-11$
1089.g2	1089e1	$[0, 0, 1, -66, -212]$	$-32768$	$-970299$	$[]$	$120$	$-0.073767$	$\Gamma_0(N)$-optimal	$-11$

sage: E.rank()

The elliptic curves in class 1089.g have rank $0$.

Each elliptic curve in class 1089.g has complex multiplication by an order in the imaginary quadratic field $\Q(\sqrt{-11}) $.

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.

$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$

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

The vertices are labelled with LMFDB labels.

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LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
1089.g1	1089e2	\([0, 0, 1, -7986, 281839]\)	\(-32768\)	\(-1718943866739\)	\([]\)	\(1320\)	\(1.1252\)		\(-11\)
1089.g2	1089e1	\([0, 0, 1, -66, -212]\)	\(-32768\)	\(-970299\)	\([]\)	\(120\)	\(-0.073767\)	\(\Gamma_0(N)\)-optimal	\(-11\)