Properties

Label 8470h
Number of curves $4$
Conductor $8470$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("8470.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8470.j4 8470h1 [1, -1, 0, 280, 2800] [2] 5120 \(\Gamma_0(N)\)-optimal
8470.j3 8470h2 [1, -1, 0, -2140, 31356] [2, 2] 10240  
8470.j2 8470h3 [1, -1, 0, -10610, -390450] [2] 20480  
8470.j1 8470h4 [1, -1, 0, -32390, 2251706] [2] 20480  

Rank

sage: E.rank()
 

The elliptic curves in class 8470h have rank \(1\).

Modular form 8470.2.a.j

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3q^{9} + q^{10} + 6q^{13} - q^{14} + q^{16} - 2q^{17} + 3q^{18} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.