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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 480960ec
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
480960.ec2 | 480960ec1 | \([0, 0, 0, -48492, 2584656]\) | \(2444008923/855040\) | \(4411818592174080\) | \([2]\) | \(1843200\) | \(1.7034\) | \(\Gamma_0(N)\)-optimal |
480960.ec1 | 480960ec2 | \([0, 0, 0, -324972, -69410736]\) | \(735580702683/22311200\) | \(115120891389542400\) | \([2]\) | \(3686400\) | \(2.0499\) |
Rank
sage: E.rank()
The elliptic curves in class 480960ec have rank \(1\).
Complex multiplication
The elliptic curves in class 480960ec do not have complex multiplication.Modular form 480960.2.a.ec
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.