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SageMath
E = EllipticCurve("kg1")
E.isogeny_class()
Elliptic curves in class 178752kg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
178752.ep3 | 178752kg1 | \([0, -1, 0, -182737, -30005807]\) | \(350104249168/2793\) | \(5383678476288\) | \([2]\) | \(983040\) | \(1.6150\) | \(\Gamma_0(N)\)-optimal |
178752.ep2 | 178752kg2 | \([0, -1, 0, -186657, -28647135]\) | \(93280467172/7800849\) | \(60146455937089536\) | \([2, 2]\) | \(1966080\) | \(1.9616\) | |
178752.ep1 | 178752kg3 | \([0, -1, 0, -633537, 161276865]\) | \(1823652903746/328593657\) | \(5067075112454455296\) | \([2]\) | \(3932160\) | \(2.3081\) | |
178752.ep4 | 178752kg4 | \([0, -1, 0, 197503, -131678847]\) | \(55251546334/517244049\) | \(-7976156544473628672\) | \([2]\) | \(3932160\) | \(2.3081\) |
Rank
sage: E.rank()
The elliptic curves in class 178752kg have rank \(2\).
Complex multiplication
The elliptic curves in class 178752kg do not have complex multiplication.Modular form 178752.2.a.kg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.