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SageMath
E = EllipticCurve("kw1")
E.isogeny_class()
Elliptic curves in class 463680kw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.kw1 | 463680kw1 | \([0, 0, 0, -141132, -19481744]\) | \(1626794704081/83462400\) | \(15949913024102400\) | \([2]\) | \(4718592\) | \(1.8669\) | \(\Gamma_0(N)\)-optimal |
463680.kw2 | 463680kw2 | \([0, 0, 0, 89268, -76897424]\) | \(411664745519/13605414480\) | \(-2600035196841492480\) | \([2]\) | \(9437184\) | \(2.2135\) |
Rank
sage: E.rank()
The elliptic curves in class 463680kw have rank \(0\).
Complex multiplication
The elliptic curves in class 463680kw do not have complex multiplication.Modular form 463680.2.a.kw
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.