Properties

Label 85782.a
Number of curves 6
Conductor 85782
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("85782.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 85782.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
85782.a1 85782b6 [1, 1, 0, -23332721, -43390387461] [2] 2867200  
85782.a2 85782b4 [1, 1, 0, -1458311, -678414495] [2, 2] 1433600  
85782.a3 85782b5 [1, 1, 0, -1382621, -751879209] [2] 2867200  
85782.a4 85782b2 [1, 1, 0, -95891, -9466275] [2, 2] 716800  
85782.a5 85782b1 [1, 1, 0, -28611, 1715661] [2] 358400 \(\Gamma_0(N)\)-optimal
85782.a6 85782b3 [1, 1, 0, 190049, -54816359] [2] 1433600  

Rank

sage: E.rank()
 

The elliptic curves in class 85782.a have rank \(1\).

Modular form 85782.2.a.a

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.