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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 25200.fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25200.fk1 | 25200em2 | \([0, 0, 0, -181200, -29688500]\) | \(-225637236736/1715\) | \(-5000940000000\) | \([]\) | \(103680\) | \(1.6116\) | |
25200.fk2 | 25200em1 | \([0, 0, 0, -1200, -78500]\) | \(-65536/875\) | \(-2551500000000\) | \([]\) | \(34560\) | \(1.0623\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25200.fk have rank \(0\).
Complex multiplication
The elliptic curves in class 25200.fk do not have complex multiplication.Modular form 25200.2.a.fk
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.