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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 258570cw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
258570.cw3 | 258570cw1 | \([1, -1, 0, -893619, -324472667]\) | \(22428153804601/35802000\) | \(125978064131322000\) | \([2]\) | \(7225344\) | \(2.1789\) | \(\Gamma_0(N)\)-optimal |
258570.cw2 | 258570cw2 | \([1, -1, 0, -1167399, -108898295]\) | \(50002789171321/27473062500\) | \(96670667267438062500\) | \([2, 2]\) | \(14450688\) | \(2.5255\) | |
258570.cw1 | 258570cw3 | \([1, -1, 0, -11251629, 14438611903]\) | \(44769506062996441/323730468750\) | \(1139124567159667968750\) | \([2]\) | \(28901376\) | \(2.8720\) | |
258570.cw4 | 258570cw4 | \([1, -1, 0, 4536351, -862934045]\) | \(2933972022568679/1789082460750\) | \(-6295322746678589880750\) | \([2]\) | \(28901376\) | \(2.8720\) |
Rank
sage: E.rank()
The elliptic curves in class 258570cw have rank \(0\).
Complex multiplication
The elliptic curves in class 258570cw do not have complex multiplication.Modular form 258570.2.a.cw
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.