Properties

Label 28224.by
Number of curves $4$
Conductor $28224$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("by1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28224.by1 28224gd4 [0, 0, 0, -527436, -147435120] [2] 196608  
28224.by2 28224gd3 [0, 0, 0, -104076, 10224144] [2] 196608  
28224.by3 28224gd2 [0, 0, 0, -33516, -2222640] [2, 2] 98304  
28224.by4 28224gd1 [0, 0, 0, 1764, -148176] [2] 49152 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 28224.by have rank \(0\).

Complex multiplication

The elliptic curves in class 28224.by do not have complex multiplication.

Modular form 28224.2.a.by

sage: E.q_eigenform(10)
 
\( q - 2q^{5} + 4q^{11} + 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.