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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 407330k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
407330.k1 | 407330k1 | \([1, 0, 1, -531004128, -23443582709202]\) | \(-399704504508069289/5499970653900800\) | \(-227844595968561914344052019200\) | \([]\) | \(463574016\) | \(4.3148\) | \(\Gamma_0(N)\)-optimal* |
407330.k2 | 407330k2 | \([1, 0, 1, 4748196337, 610436352163906]\) | \(285779688176240568551/4051227901952000000\) | \(-167828238109262239496142848000000\) | \([]\) | \(1390722048\) | \(4.8641\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 407330k have rank \(1\).
Complex multiplication
The elliptic curves in class 407330k do not have complex multiplication.Modular form 407330.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.