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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 5200j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5200.o2 | 5200j1 | \([0, 0, 0, -35, 50]\) | \(37044/13\) | \(1664000\) | \([2]\) | \(640\) | \(-0.10492\) | \(\Gamma_0(N)\)-optimal |
5200.o1 | 5200j2 | \([0, 0, 0, -235, -1350]\) | \(5606442/169\) | \(43264000\) | \([2]\) | \(1280\) | \(0.24165\) |
Rank
sage: E.rank()
The elliptic curves in class 5200j have rank \(1\).
Complex multiplication
The elliptic curves in class 5200j do not have complex multiplication.Modular form 5200.2.a.j
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 Cremona 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 Cremona labels.