Properties

Label 223146.o
Number of curves $6$
Conductor $223146$
CM no
Rank $1$
Graph

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Show commands: 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)
 
\(q - q^{2} + q^{4} - 2 q^{5} - q^{8} + 2 q^{10} + q^{11} + 2 q^{13} + q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.