Properties

Label 242550cw
Number of curves $4$
Conductor $242550$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cw1")
 
E.isogeny_class()
 

Elliptic curves in class 242550cw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
242550.cw4 242550cw1 \([1, -1, 0, 38358, -1771484]\) \(4657463/3696\) \(-4952993487750000\) \([2]\) \(1572864\) \(1.6995\) \(\Gamma_0(N)\)-optimal
242550.cw3 242550cw2 \([1, -1, 0, -182142, -15221984]\) \(498677257/213444\) \(286035373917562500\) \([2, 2]\) \(3145728\) \(2.0461\)  
242550.cw2 242550cw3 \([1, -1, 0, -1394892, 623897266]\) \(223980311017/4278582\) \(5733709086256593750\) \([2]\) \(6291456\) \(2.3926\)  
242550.cw1 242550cw4 \([1, -1, 0, -2497392, -1517819234]\) \(1285429208617/614922\) \(824054291524406250\) \([2]\) \(6291456\) \(2.3926\)  

Rank

sage: E.rank()
 

The elliptic curves in class 242550cw have rank \(1\).

Complex multiplication

The elliptic curves in class 242550cw do not have complex multiplication.

Modular form 242550.2.a.cw

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - q^{11} + 2 q^{13} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.