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Properties

Label 422142o
Number of curves $4$
Conductor $422142$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("o1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 422142o have rank \(0\).

Complex multiplication

The elliptic curves in class 422142o do not have complex multiplication.

Modular form 422142.2.a.o

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - 2 q^{15} + q^{16} - 2 q^{17} - q^{18} - q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 422142o

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422142.o3 422142o1 \([1, 1, 0, -103959, -12913707]\) \(839362385737/2349312\) \(347782490458368\) \([2]\) \(2433024\) \(1.6625\) \(\Gamma_0(N)\)-optimal*
422142.o2 422142o2 \([1, 1, 0, -146279, -1461915]\) \(2338337977417/1347477264\) \(199474994683527696\) \([2, 2]\) \(4866048\) \(2.0091\) \(\Gamma_0(N)\)-optimal*
422142.o1 422142o3 \([1, 1, 0, -1553419, 741789433]\) \(2800418713303177/12058908372\) \(1785151221218562708\) \([2]\) \(9732096\) \(2.3557\) \(\Gamma_0(N)\)-optimal*
422142.o4 422142o4 \([1, 1, 0, 583741, -10952175]\) \(148599082115063/86360598996\) \(-12784468046945367444\) \([2]\) \(9732096\) \(2.3557\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 422142o1.