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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5850.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5850.e1 | 5850ba2 | \([1, -1, 0, -31707, -2158299]\) | \(38686490446661/141927552\) | \(12933148176000\) | \([2]\) | \(28672\) | \(1.3763\) | |
5850.e2 | 5850ba1 | \([1, -1, 0, -2907, 1701]\) | \(29819839301/17252352\) | \(1572120576000\) | \([2]\) | \(14336\) | \(1.0298\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5850.e have rank \(0\).
Complex multiplication
The elliptic curves in class 5850.e do not have complex multiplication.Modular form 5850.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.