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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 116688x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
116688.w1 | 116688x1 | \([0, 1, 0, -48328, 3703412]\) | \(3047678972871625/304559880768\) | \(1247477271625728\) | \([2]\) | \(516096\) | \(1.6330\) | \(\Gamma_0(N)\)-optimal |
116688.w2 | 116688x2 | \([0, 1, 0, 59832, 18023796]\) | \(5783051584712375/37533175779528\) | \(-153735887992946688\) | \([2]\) | \(1032192\) | \(1.9796\) |
Rank
sage: E.rank()
The elliptic curves in class 116688x have rank \(2\).
Complex multiplication
The elliptic curves in class 116688x do not have complex multiplication.Modular form 116688.2.a.x
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.