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Properties

Label 8085.r
Number of curves $4$
Conductor $8085$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 8085.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8085.r1 8085f3 \([1, 1, 0, -28175858, -57577447563]\) \(21026497979043461623321/161783881875\) \(19033711918711875\) \([2]\) \(368640\) \(2.7162\)  
8085.r2 8085f2 \([1, 1, 0, -1762163, -898940832]\) \(5143681768032498601/14238434358225\) \(1675137563810813025\) \([2, 2]\) \(184320\) \(2.3697\)  
8085.r3 8085f4 \([1, 1, 0, -1067588, -1613936337]\) \(-1143792273008057401/8897444448004035\) \(-1046775441863226713715\) \([2]\) \(368640\) \(2.7162\)  
8085.r4 8085f1 \([1, 1, 0, -154718, -1665033]\) \(3481467828171481/2005331497785\) \(235925245382907465\) \([2]\) \(92160\) \(2.0231\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 8085.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.