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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 223146.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
223146.o1 | 223146cq6 | \([1, -1, 0, -30374798463, -2037589022314301]\) | \(36136672427711016379227705697/1011258101510224722\) | \(86731684696356216244243362\) | \([2]\) | \(314572800\) | \(4.4879\) | |
223146.o2 | 223146cq3 | \([1, -1, 0, -2173743693, 38932836408409]\) | \(13244420128496241770842177/29965867631164664892\) | \(2570056229124452020051723932\) | \([2]\) | \(157286400\) | \(4.1413\) | |
223146.o3 | 223146cq4 | \([1, -1, 0, -1900835253, -31752073571135]\) | \(8856076866003496152467137/46664863048067576004\) | \(4002264290628992539735760484\) | \([2, 2]\) | \(157286400\) | \(4.1413\) | |
223146.o4 | 223146cq5 | \([1, -1, 0, -872030763, -65989246672049]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-1838751280183569793207364370018\) | \([2]\) | \(314572800\) | \(4.4879\) | |
223146.o5 | 223146cq2 | \([1, -1, 0, -185512833, 123762959725]\) | \(8232463578739844255617/4687062591766850064\) | \(401991177380049266377881744\) | \([2, 2]\) | \(78643200\) | \(3.7947\) | |
223146.o6 | 223146cq1 | \([1, -1, 0, 45959247, 15387731869]\) | \(125177609053596564863/73635189229502208\) | \(-6315404549315383141095168\) | \([2]\) | \(39321600\) | \(3.4481\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 223146.o have rank \(1\).
Complex multiplication
The elliptic curves in class 223146.o do not have complex multiplication.Modular form 223146.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.