Properties

Label 28224.eu
Number of curves $6$
Conductor $28224$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("28224.eu1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 28224.eu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28224.eu1 28224cb4 [0, 0, 0, -37933644, 89925934448] [2] 1179648  
28224.eu2 28224cb6 [0, 0, 0, -25797324, -49948258192] [2] 2359296  
28224.eu3 28224cb3 [0, 0, 0, -2935884, 685259120] [2, 2] 1179648  
28224.eu4 28224cb2 [0, 0, 0, -2371404, 1404406640] [2, 2] 589824  
28224.eu5 28224cb1 [0, 0, 0, -113484, 32494448] [2] 294912 \(\Gamma_0(N)\)-optimal
28224.eu6 28224cb5 [0, 0, 0, 10893876, 5293335152] [2] 2359296  

Rank

sage: E.rank()
 

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

Modular form 28224.2.a.eu

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.