Properties

Label 4719.k
Number of curves $6$
Conductor $4719$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4719.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4719.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4719.k1 4719j3 [1, 0, 1, -830547, -291405149] [2] 30720  
4719.k2 4719j5 [1, 0, 1, -364092, 81851779] [2] 61440  
4719.k3 4719j4 [1, 0, 1, -57357, -3543245] [2, 2] 30720  
4719.k4 4719j2 [1, 0, 1, -51912, -4556015] [2, 2] 15360  
4719.k5 4719j1 [1, 0, 1, -2907, -86759] [2] 7680 \(\Gamma_0(N)\)-optimal
4719.k6 4719j6 [1, 0, 1, 162258, -24099209] [2] 61440  

Rank

sage: E.rank()
 

The elliptic curves in class 4719.k have rank \(1\).

Modular form 4719.2.a.k

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{8} + q^{9} - 2q^{10} - q^{12} - q^{13} - 2q^{15} - q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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.