Properties

Label 5040.k
Number of curves 8
Conductor 5040
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("5040.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5040.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.k1 5040bi7 [0, 0, 0, -276595203, -1770578063998] [2] 393216  
5040.k2 5040bi5 [0, 0, 0, -17287203, -27665272798] [2, 2] 196608  
5040.k3 5040bi8 [0, 0, 0, -17179203, -28028001598] [2] 393216  
5040.k4 5040bi4 [0, 0, 0, -2170083, 1230107618] [2] 98304  
5040.k5 5040bi3 [0, 0, 0, -1087203, -426592798] [2, 2] 98304  
5040.k6 5040bi2 [0, 0, 0, -154083, 13653218] [2, 2] 49152  
5040.k7 5040bi1 [0, 0, 0, 30237, 1524962] [2] 24576 \(\Gamma_0(N)\)-optimal
5040.k8 5040bi6 [0, 0, 0, 182877, -1363657822] [2] 196608  

Rank

sage: E.rank()
 

The elliptic curves in class 5040.k have rank \(0\).

Modular form 5040.2.a.k

sage: E.q_eigenform(10)
 
\( q - q^{5} + q^{7} - 4q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.