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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 87120.gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87120.gk1 | 87120gm1 | \([0, 0, 0, -1542387, -744329806]\) | \(-76711450249/851840\) | \(-4506108210024284160\) | \([]\) | \(2419200\) | \(2.3942\) | \(\Gamma_0(N)\)-optimal |
| 87120.gk2 | 87120gm2 | \([0, 0, 0, 5165853, -3858294814]\) | \(2882081488391/2883584000\) | \(-15253734899387990016000\) | \([]\) | \(7257600\) | \(2.9435\) |
Rank
sage: E.rank()
The elliptic curves in class 87120.gk have rank \(1\).
Complex multiplication
The elliptic curves in class 87120.gk do not have complex multiplication.Modular form 87120.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
