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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 759.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
759.a1 | 759a2 | \([1, 1, 1, -1238, -17152]\) | \(209849322390625/1882056627\) | \(1882056627\) | \([2]\) | \(320\) | \(0.60249\) | |
759.a2 | 759a1 | \([1, 1, 1, -23, -628]\) | \(-1349232625/164333367\) | \(-164333367\) | \([2]\) | \(160\) | \(0.25591\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 759.a have rank \(1\).
Complex multiplication
The elliptic curves in class 759.a do not have complex multiplication.Modular form 759.2.a.a
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.