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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1600.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1600.e1 | 1600f2 | \([0, 1, 0, -633, 5863]\) | \(438976/5\) | \(320000000\) | \([2]\) | \(768\) | \(0.44525\) | |
1600.e2 | 1600f1 | \([0, 1, 0, -8, 238]\) | \(-64/25\) | \(-25000000\) | \([2]\) | \(384\) | \(0.098674\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1600.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1600.e do not have complex multiplication.Modular form 1600.2.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.