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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 132878.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
132878.j1 | 132878b3 | \([1, 1, 1, -4387094, -3538648557]\) | \(15698803397448457/20709376\) | \(12318419808157696\) | \([]\) | \(3024000\) | \(2.3635\) | |
132878.j2 | 132878b2 | \([1, 1, 1, -68559, -2100587]\) | \(59914169497/31554496\) | \(18769350103201216\) | \([]\) | \(1008000\) | \(1.8142\) | |
132878.j3 | 132878b1 | \([1, 1, 1, -39124, 2962233]\) | \(11134383337/316\) | \(187964169436\) | \([]\) | \(336000\) | \(1.2649\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 132878.j have rank \(0\).
Complex multiplication
The elliptic curves in class 132878.j do not have complex multiplication.Modular form 132878.2.a.j
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.