Properties

Label 82368eq
Number of curves $6$
Conductor $82368$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("82368.be1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 82368eq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
82368.be5 82368eq1 [0, 0, 0, -13836, 898576] [2] 262144 \(\Gamma_0(N)\)-optimal
82368.be4 82368eq2 [0, 0, 0, -247116, 47274640] [2, 2] 524288  
82368.be3 82368eq3 [0, 0, 0, -273036, 36751120] [2, 2] 1048576  
82368.be1 82368eq4 [0, 0, 0, -3953676, 3025866256] [2] 1048576  
82368.be6 82368eq5 [0, 0, 0, 772404, 250439056] [2] 2097152  
82368.be2 82368eq6 [0, 0, 0, -1733196, -850442096] [2] 2097152  

Rank

sage: E.rank()
 

The elliptic curves in class 82368eq have rank \(2\).

Modular form 82368.2.a.be

sage: E.q_eigenform(10)
 
\( q - 2q^{5} + q^{11} - q^{13} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.