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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 187200.fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.fk1 | 187200dn2 | \([0, 0, 0, -1188300, -498418000]\) | \(497169541448/190125\) | \(70963776000000000\) | \([2]\) | \(2949120\) | \(2.1993\) | |
187200.fk2 | 187200dn1 | \([0, 0, 0, -63300, -10168000]\) | \(-601211584/609375\) | \(-28431000000000000\) | \([2]\) | \(1474560\) | \(1.8528\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.fk have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.fk do not have complex multiplication.Modular form 187200.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.