Properties

Label 28224ck
Number of curves $4$
Conductor $28224$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ck1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224ck

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28224.bd4 28224ck1 [0, 0, 0, 1764, 148176] [2] 49152 \(\Gamma_0(N)\)-optimal
28224.bd3 28224ck2 [0, 0, 0, -33516, 2222640] [2, 2] 98304  
28224.bd2 28224ck3 [0, 0, 0, -104076, -10224144] [2] 196608  
28224.bd1 28224ck4 [0, 0, 0, -527436, 147435120] [2] 196608  

Rank

sage: E.rank()
 

The elliptic curves in class 28224ck have rank \(1\).

Complex multiplication

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

Modular form 28224.2.a.ck

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 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.