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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 40749.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40749.e1 | 40749m2 | \([1, 0, 0, -41333, -3237330]\) | \(323535264625/59643\) | \(1439637027867\) | \([2]\) | \(110592\) | \(1.3356\) | |
40749.e2 | 40749m1 | \([1, 0, 0, -2318, -61509]\) | \(-57066625/34263\) | \(-827025526647\) | \([2]\) | \(55296\) | \(0.98901\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40749.e have rank \(1\).
Complex multiplication
The elliptic curves in class 40749.e do not have complex multiplication.Modular form 40749.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.