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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 627.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
627.a1 | 627b2 | \([0, 1, 1, -30063, -2016358]\) | \(-3004935183806464000/2037123\) | \(-2037123\) | \([]\) | \(540\) | \(0.96017\) | |
627.a2 | 627b1 | \([0, 1, 1, -363, -2995]\) | \(-5304438784000/497763387\) | \(-497763387\) | \([3]\) | \(180\) | \(0.41087\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 627.a have rank \(0\).
Complex multiplication
The elliptic curves in class 627.a do not have complex multiplication.Modular form 627.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.