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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 51870f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51870.h2 | 51870f1 | \([1, 1, 0, -598, -6188]\) | \(-23711636464489/1513774080\) | \(-1513774080\) | \([2]\) | \(33280\) | \(0.51485\) | \(\Gamma_0(N)\)-optimal |
51870.h1 | 51870f2 | \([1, 1, 0, -9718, -372812]\) | \(101513598260088169/377613600\) | \(377613600\) | \([2]\) | \(66560\) | \(0.86143\) |
Rank
sage: E.rank()
The elliptic curves in class 51870f have rank \(0\).
Complex multiplication
The elliptic curves in class 51870f do not have complex multiplication.Modular form 51870.2.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 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.