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Properties

Label 464607n
Number of curves $6$
Conductor $464607$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 464607n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
464607.n5 464607n1 \([1, -1, 1, -78044, 12057270]\) \(-1532808577/938223\) \(-32177715627298527\) \([2]\) \(3538944\) \(1.8695\) \(\Gamma_0(N)\)-optimal*
464607.n4 464607n2 \([1, -1, 1, -1393889, 633662448]\) \(8732907467857/1656369\) \(56807572033378881\) \([2, 2]\) \(7077888\) \(2.2160\) \(\Gamma_0(N)\)-optimal*
464607.n1 464607n3 \([1, -1, 1, -22301204, 40541545320]\) \(35765103905346817/1287\) \(44139527609463\) \([2]\) \(14155776\) \(2.5626\) \(\Gamma_0(N)\)-optimal*
464607.n3 464607n4 \([1, -1, 1, -1540094, 492720828]\) \(11779205551777/3763454409\) \(129073115612284970841\) \([2, 2]\) \(14155776\) \(2.5626\)  
464607.n6 464607n5 \([1, -1, 1, 4356841, 3353913690]\) \(266679605718863/296110251723\) \(-10155529628105980860027\) \([2]\) \(28311552\) \(2.9092\)  
464607.n2 464607n6 \([1, -1, 1, -9776309, -11390490174]\) \(3013001140430737/108679952667\) \(3727336263667582363083\) \([2]\) \(28311552\) \(2.9092\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 464607n1.

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 464607n do not have complex multiplication.

Modular form 464607.2.a.n

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.