Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8085.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8085.f1 8085g5 [1, 1, 1, -149410801, 702882354698] [2] 368640  
8085.f2 8085g4 [1, 1, 1, -9338176, 10979616248] [2, 2] 184320  
8085.f3 8085g6 [1, 1, 1, -9291871, 11093952554] [2] 368640  
8085.f4 8085g3 [1, 1, 1, -1246806, -283100196] [2] 184320  
8085.f5 8085g2 [1, 1, 1, -586531, 169584344] [2, 2] 92160  
8085.f6 8085g1 [1, 1, 1, 1714, 7934618] [2] 46080 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 8085.f do not have complex multiplication.

Modular form 8085.2.a.f

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.