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SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 31680dw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31680.df4 | 31680dw1 | \([0, 0, 0, 468, 3024]\) | \(59319/55\) | \(-10510663680\) | \([2]\) | \(16384\) | \(0.61021\) | \(\Gamma_0(N)\)-optimal |
31680.df3 | 31680dw2 | \([0, 0, 0, -2412, 27216]\) | \(8120601/3025\) | \(578086502400\) | \([2, 2]\) | \(32768\) | \(0.95679\) | |
31680.df2 | 31680dw3 | \([0, 0, 0, -16812, -819504]\) | \(2749884201/73205\) | \(13989693358080\) | \([2]\) | \(65536\) | \(1.3034\) | |
31680.df1 | 31680dw4 | \([0, 0, 0, -34092, 2422224]\) | \(22930509321/6875\) | \(1313832960000\) | \([2]\) | \(65536\) | \(1.3034\) |
Rank
sage: E.rank()
The elliptic curves in class 31680dw have rank \(1\).
Complex multiplication
The elliptic curves in class 31680dw do not have complex multiplication.Modular form 31680.2.a.dw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.