Properties

Label 10780.k
Number of curves 4
Conductor 10780
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("10780.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 10780.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10780.k1 10780f4 [0, -1, 0, -347916, 79103816] [2] 62208  
10780.k2 10780f3 [0, -1, 0, -21821, 1232330] [2] 31104  
10780.k3 10780f2 [0, -1, 0, -4916, 76616] [2] 20736  
10780.k4 10780f1 [0, -1, 0, -2221, -38730] [2] 10368 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10780.k have rank \(1\).

Modular form 10780.2.a.k

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.