Properties

Label 84966da
Number of curves $6$
Conductor $84966$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("84966.cr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 84966da

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
84966.cr5 84966da1 [1, 1, 1, -994725579, -12060749479815] [2] 53084160 \(\Gamma_0(N)\)-optimal
84966.cr4 84966da2 [1, 1, 1, -1284742859, -4454988308359] [2, 2] 106168320  
84966.cr6 84966da3 [1, 1, 1, 4955160181, -35033009165575] [2] 212336640  
84966.cr2 84966da4 [1, 1, 1, -12164922379, 512897547867641] [2, 2] 212336640  
84966.cr3 84966da5 [1, 1, 1, -4143565539, 1179186740968137] [2] 424673280  
84966.cr1 84966da6 [1, 1, 1, -194269151539, 32957388273621545] [2] 424673280  

Rank

sage: E.rank()
 

The elliptic curves in class 84966da have rank \(1\).

Modular form 84966.2.a.cr

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

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.