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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 129360.gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.gk1 | 129360hm2 | \([0, 1, 0, -501648520, 4324447450868]\) | \(9937296563535244838593567/1587762000000\) | \(2230691291136000000\) | \([2]\) | \(20643840\) | \(3.3656\) | |
129360.gk2 | 129360hm1 | \([0, 1, 0, -31356040, 67548038900]\) | \(2426796094451411844127/969756530688000\) | \(1362438103146430464000\) | \([2]\) | \(10321920\) | \(3.0190\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.gk have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.gk do not have complex multiplication.Modular form 129360.2.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.