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SageMath
E = EllipticCurve("pd1")
E.isogeny_class()
Elliptic curves in class 310464.pd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
310464.pd1 | 310464pd2 | \([0, 0, 0, -719308044, -7425410853392]\) | \(144106117295241933/247808\) | \(70778502348357500928\) | \([2]\) | \(52985856\) | \(3.4977\) | |
310464.pd2 | 310464pd1 | \([0, 0, 0, -44942604, -116098722320]\) | \(-35148950502093/46137344\) | \(-13177670255403287445504\) | \([2]\) | \(26492928\) | \(3.1512\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 310464.pd have rank \(1\).
Complex multiplication
The elliptic curves in class 310464.pd do not have complex multiplication.Modular form 310464.2.a.pd
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.