Properties

Label 22080d
Number of curves $6$
Conductor $22080$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("22080.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 22080d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22080.k5 22080d1 [0, -1, 0, -26881, 1870081] [2] 73728 \(\Gamma_0(N)\)-optimal
22080.k4 22080d2 [0, -1, 0, -441601, 113097985] [2, 2] 147456  
22080.k3 22080d3 [0, -1, 0, -453121, 106897921] [2, 2] 294912  
22080.k1 22080d4 [0, -1, 0, -7065601, 7231248385] [2] 294912  
22080.k6 22080d5 [0, -1, 0, 562559, 517029505] [2] 589824  
22080.k2 22080d6 [0, -1, 0, -1653121, -700222079] [2] 589824  

Rank

sage: E.rank()
 

The elliptic curves in class 22080d have rank \(1\).

Modular form 22080.2.a.k

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

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.