Properties

Label 8085.g
Number of curves $6$
Conductor $8085$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("8085.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8085.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8085.g1 8085p5 [1, 0, 0, -649251, 201293406] [2] 98304  
8085.g2 8085p3 [1, 0, 0, -42876, 2766231] [2, 2] 49152  
8085.g3 8085p2 [1, 0, 0, -13231, -548080] [2, 2] 24576  
8085.g4 8085p1 [1, 0, 0, -12986, -570669] [2] 12288 \(\Gamma_0(N)\)-optimal
8085.g5 8085p4 [1, 0, 0, 12494, -2415715] [2] 49152  
8085.g6 8085p6 [1, 0, 0, 89179, 16473540] [2] 98304  

Rank

sage: E.rank()
 

The elliptic curves in class 8085.g have rank \(0\).

Modular form 8085.2.a.g

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 3q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + 2q^{13} - q^{15} - q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.