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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 123840.fp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123840.fp1 | 123840x2 | \([0, 0, 0, -3372, -74736]\) | \(4792616856/46225\) | \(40896921600\) | \([2]\) | \(98304\) | \(0.85610\) | |
123840.fp2 | 123840x1 | \([0, 0, 0, -372, 864]\) | \(51478848/26875\) | \(2972160000\) | \([2]\) | \(49152\) | \(0.50953\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123840.fp have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.fp do not have complex multiplication.Modular form 123840.2.a.fp
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 LMFDB 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 LMFDB labels.