Properties

Label 86394b
Number of curves $6$
Conductor $86394$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("86394.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 86394b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86394.d5 86394b1 [1, 1, 0, -8499526, -9529233644] [2] 4915200 \(\Gamma_0(N)\)-optimal
86394.d4 86394b2 [1, 1, 0, -10977606, -3522863340] [2, 2] 9830400  
86394.d6 86394b3 [1, 1, 0, 42339834, -27654336684] [2] 19660800  
86394.d2 86394b4 [1, 1, 0, -103944326, 405065871060] [2, 2] 19660800  
86394.d3 86394b5 [1, 1, 0, -35405086, 931351279324] [2] 39321600  
86394.d1 86394b6 [1, 1, 0, -1659951086, 26030318798796] [2] 39321600  

Rank

sage: E.rank()
 

The elliptic curves in class 86394b have rank \(0\).

Modular form 86394.2.a.d

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + 2q^{10} - q^{12} + 2q^{13} + q^{14} + 2q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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 Cremona 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 Cremona labels.