Properties

Label 36822.e
Number of curves 6
Conductor 36822
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("36822.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 36822.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36822.e1 36822i6 [1, 1, 0, -10015591, 12195916411] [2] 884736  
36822.e2 36822i4 [1, 1, 0, -625981, 190361065] [2, 2] 442368  
36822.e3 36822i5 [1, 1, 0, -593491, 211044199] [2] 884736  
36822.e4 36822i2 [1, 1, 0, -41161, 2633845] [2, 2] 221184  
36822.e5 36822i1 [1, 1, 0, -12281, -490971] [2] 110592 \(\Gamma_0(N)\)-optimal
36822.e6 36822i3 [1, 1, 0, 81579, 15472449] [2] 442368  

Rank

sage: E.rank()
 

The elliptic curves in class 36822.e have rank \(1\).

Modular form 36822.2.a.e

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

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.