Properties

Label 228888d
Number of curves 4
Conductor 228888
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("228888.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 228888d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
228888.k4 228888d1 [0, 0, 0, 1734, -761515] [2] 884736 \(\Gamma_0(N)\)-optimal
228888.k3 228888d2 [0, 0, 0, -115311, -14689870] [2, 2] 1769472  
228888.k2 228888d3 [0, 0, 0, -271371, 33595094] [2] 3538944  
228888.k1 228888d4 [0, 0, 0, -1831971, -954389554] [2] 3538944  

Rank

sage: E.rank()
 

The elliptic curves in class 228888d have rank \(1\).

Modular form 228888.2.a.k

sage: E.q_eigenform(10)
 
\( q - 2q^{5} - 4q^{7} - q^{11} + 6q^{13} - 8q^{19} + O(q^{20}) \)

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.