Properties

Label 85176.n
Number of curves $6$
Conductor $85176$
CM no
Rank $1$
Graph

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

Elliptic curves in class 85176.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
85176.n1 85176bw6 \([0, 0, 0, -4305136251, 108724761767414]\) \(1224522642327678150914/66339\) \(478064521955284992\) \([2]\) \(24772608\) \(3.7831\)  
85176.n2 85176bw4 \([0, 0, 0, -269071491, 1698818100590]\) \(597914615076708388/4400862921\) \(15857161160995825542144\) \([2, 2]\) \(12386304\) \(3.4365\)  
85176.n3 85176bw5 \([0, 0, 0, -263535051, 1772072952806]\) \(-280880296871140514/25701087819771\) \(-185211990869467747735001088\) \([2]\) \(24772608\) \(3.7831\)  
85176.n4 85176bw3 \([0, 0, 0, -57409131, -137947911946]\) \(5807363790481348/1079211743883\) \(3888608860760305831898112\) \([2]\) \(12386304\) \(3.4365\)  
85176.n5 85176bw2 \([0, 0, 0, -17163471, 25393123730]\) \(620742479063632/49991146569\) \(45031944984295758338304\) \([2, 2]\) \(6193152\) \(3.0899\)  
85176.n6 85176bw1 \([0, 0, 0, 1096134, 1798062149]\) \(2587063175168/26304786963\) \(-1480956880167846205488\) \([2]\) \(3096576\) \(2.7433\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 85176.n have rank \(1\).

Complex multiplication

The elliptic curves in class 85176.n do not have complex multiplication.

Modular form 85176.2.a.n

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - q^{7} - 4q^{11} + 6q^{17} - 4q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.