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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 217800.gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
217800.gk1 | 217800db2 | \([0, 0, 0, -343035, 77264550]\) | \(1000188\) | \(4463313300864000\) | \([2]\) | \(1966080\) | \(1.9223\) | |
217800.gk2 | 217800db1 | \([0, 0, 0, -16335, 1796850]\) | \(-432\) | \(-1115828325216000\) | \([2]\) | \(983040\) | \(1.5757\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 217800.gk have rank \(0\).
Complex multiplication
The elliptic curves in class 217800.gk do not have complex multiplication.Modular form 217800.2.a.gk
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.