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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10766.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10766.e1 | 10766h1 | \([1, -1, 1, -450273, 116420913]\) | \(-10096027515422913423969/1328135939293184\) | \(-1328135939293184\) | \([7]\) | \(204624\) | \(1.9219\) | \(\Gamma_0(N)\)-optimal |
10766.e2 | 10766h2 | \([1, -1, 1, 2885367, -4130671167]\) | \(2656605299081089600290591/8905789086154159331384\) | \(-8905789086154159331384\) | \([]\) | \(1432368\) | \(2.8948\) |
Rank
sage: E.rank()
The elliptic curves in class 10766.e have rank \(1\).
Complex multiplication
The elliptic curves in class 10766.e do not have complex multiplication.Modular form 10766.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.