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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 4560.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4560.w1 | 4560bd4 | \([0, 1, 0, -96120, 11438100]\) | \(23977812996389881/146611125\) | \(600519168000\) | \([4]\) | \(18432\) | \(1.4473\) | |
4560.w2 | 4560bd3 | \([0, 1, 0, -19800, -875052]\) | \(209595169258201/41748046875\) | \(171000000000000\) | \([2]\) | \(18432\) | \(1.4473\) | |
4560.w3 | 4560bd2 | \([0, 1, 0, -6120, 170100]\) | \(6189976379881/456890625\) | \(1871424000000\) | \([2, 2]\) | \(9216\) | \(1.1007\) | |
4560.w4 | 4560bd1 | \([0, 1, 0, 360, 11988]\) | \(1256216039/15582375\) | \(-63825408000\) | \([2]\) | \(4608\) | \(0.75414\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4560.w have rank \(1\).
Complex multiplication
The elliptic curves in class 4560.w do not have complex multiplication.Modular form 4560.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.