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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 481338.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
481338.e1 | 481338e1 | \([1, -1, 0, -3066696, 2067838416]\) | \(-36159554681206301257/125899594704\) | \(-11105477349245136\) | \([]\) | \(17252352\) | \(2.2983\) | \(\Gamma_0(N)\)-optimal |
481338.e2 | 481338e2 | \([1, -1, 0, -1966311, 3568473981]\) | \(-9531638527140434617/56831105229410304\) | \(-5013014961181053505536\) | \([]\) | \(51757056\) | \(2.8476\) |
Rank
sage: E.rank()
The elliptic curves in class 481338.e have rank \(1\).
Complex multiplication
The elliptic curves in class 481338.e do not have complex multiplication.Modular form 481338.2.a.e
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.