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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 650.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
650.f1 | 650b2 | \([1, 1, 0, -11330, -468940]\) | \(-6434774386429585/140608\) | \(-3515200\) | \([]\) | \(1080\) | \(0.78239\) | |
650.f2 | 650b1 | \([1, 1, 0, -130, -780]\) | \(-9836106385/3407872\) | \(-85196800\) | \([]\) | \(360\) | \(0.23308\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 650.f have rank \(1\).
Complex multiplication
The elliptic curves in class 650.f do not have complex multiplication.Modular form 650.2.a.f
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.