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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 54450gk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54450.hi1 | 54450gk1 | \([1, -1, 1, -2409980, 1454371647]\) | \(-76711450249/851840\) | \(-17189438667390000000\) | \([]\) | \(2419200\) | \(2.5058\) | \(\Gamma_0(N)\)-optimal |
54450.hi2 | 54450gk2 | \([1, -1, 1, 8071645, 7533714147]\) | \(2882081488391/2883584000\) | \(-58188380811264000000000\) | \([]\) | \(7257600\) | \(3.0551\) |
Rank
sage: E.rank()
The elliptic curves in class 54450gk have rank \(0\).
Complex multiplication
The elliptic curves in class 54450gk do not have complex multiplication.Modular form 54450.2.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 Cremona 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 Cremona labels.