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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 507.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
507.a1 | 507c3 | \([1, 1, 1, -11749, -495058]\) | \(37159393753/1053\) | \(5082629877\) | \([2]\) | \(672\) | \(0.96411\) | |
507.a2 | 507c4 | \([1, 1, 1, -3299, 64670]\) | \(822656953/85683\) | \(413575475547\) | \([2]\) | \(672\) | \(0.96411\) | |
507.a3 | 507c2 | \([1, 1, 1, -764, -7324]\) | \(10218313/1521\) | \(7341576489\) | \([2, 2]\) | \(336\) | \(0.61753\) | |
507.a4 | 507c1 | \([1, 1, 1, 81, -564]\) | \(12167/39\) | \(-188245551\) | \([4]\) | \(168\) | \(0.27096\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 507.a have rank \(1\).
Complex multiplication
The elliptic curves in class 507.a do not have complex multiplication.Modular form 507.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.