Properties

Label 8085j
Number of curves $2$
Conductor $8085$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8085j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8085.b2 8085j1 [0, -1, 1, -438076, -130549434] [] 288000 \(\Gamma_0(N)\)-optimal
8085.b1 8085j2 [0, -1, 1, -1312726, 10945050906] [] 1440000  

Rank

sage: E.rank()
 

The elliptic curves in class 8085j have rank \(1\).

Complex multiplication

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

Modular form 8085.2.a.j

sage: E.q_eigenform(10)
 
\( q - 2q^{2} - q^{3} + 2q^{4} - q^{5} + 2q^{6} + q^{9} + 2q^{10} + q^{11} - 2q^{12} + 6q^{13} + q^{15} - 4q^{16} + 7q^{17} - 2q^{18} + 5q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.