Properties

Label 464607.n
Number of curves $6$
Conductor $464607$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("464607.n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 464607.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
464607.n1 464607n3 [1, -1, 1, -22301204, 40541545320] [2] 14155776 \(\Gamma_0(N)\)-optimal*
464607.n2 464607n6 [1, -1, 1, -9776309, -11390490174] [2] 28311552  
464607.n3 464607n4 [1, -1, 1, -1540094, 492720828] [2, 2] 14155776  
464607.n4 464607n2 [1, -1, 1, -1393889, 633662448] [2, 2] 7077888 \(\Gamma_0(N)\)-optimal*
464607.n5 464607n1 [1, -1, 1, -78044, 12057270] [2] 3538944 \(\Gamma_0(N)\)-optimal*
464607.n6 464607n5 [1, -1, 1, 4356841, 3353913690] [2] 28311552  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 464607.n5.

Rank

sage: E.rank()
 

The elliptic curves in class 464607.n have rank \(0\).

Modular form 464607.2.a.n

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 2q^{5} + 3q^{8} - 2q^{10} + q^{11} - q^{13} - q^{16} + 6q^{17} + O(q^{20}) \)

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.