Properties

Label 5415k
Number of curves 8
Conductor 5415
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("5415.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5415k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5415.j7 5415k1 [1, 0, 1, -8, -1279] [2] 1728 \(\Gamma_0(N)\)-optimal
5415.j6 5415k2 [1, 0, 1, -1813, -29437] [2, 2] 3456  
5415.j4 5415k3 [1, 0, 1, -28888, -1892197] [2] 6912  
5415.j5 5415k4 [1, 0, 1, -3618, 38431] [2, 2] 6912  
5415.j2 5415k5 [1, 0, 1, -48743, 4135781] [2, 2] 13824  
5415.j8 5415k6 [1, 0, 1, 12627, 291853] [2] 13824  
5415.j1 5415k7 [1, 0, 1, -779768, 264965501] [2] 27648  
5415.j3 5415k8 [1, 0, 1, -39718, 5716961] [2] 27648  

Rank

sage: E.rank()
 

The elliptic curves in class 5415k have rank \(0\).

Modular form 5415.2.a.j

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

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.