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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 27200ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27200.i1 | 27200ce1 | \([0, 1, 0, -13633, 604863]\) | \(68417929/425\) | \(1740800000000\) | \([2]\) | \(49152\) | \(1.1868\) | \(\Gamma_0(N)\)-optimal |
27200.i2 | 27200ce2 | \([0, 1, 0, -5633, 1316863]\) | \(-4826809/180625\) | \(-739840000000000\) | \([2]\) | \(98304\) | \(1.5333\) |
Rank
sage: E.rank()
The elliptic curves in class 27200ce have rank \(0\).
Complex multiplication
The elliptic curves in class 27200ce do not have complex multiplication.Modular form 27200.2.a.ce
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.