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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 55440eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.dr2 | 55440eh1 | \([0, 0, 0, 2328, 108236]\) | \(7476617216/31444875\) | \(-5868368352000\) | \([]\) | \(82944\) | \(1.1326\) | \(\Gamma_0(N)\)-optimal |
55440.dr1 | 55440eh2 | \([0, 0, 0, -21432, -3317956]\) | \(-5833703071744/22107421875\) | \(-4125775500000000\) | \([]\) | \(248832\) | \(1.6819\) |
Rank
sage: E.rank()
The elliptic curves in class 55440eh have rank \(1\).
Complex multiplication
The elliptic curves in class 55440eh do not have complex multiplication.Modular form 55440.2.a.eh
sage: E.q_eigenform(10)
Isogeny matrix
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.