Properties

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

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

Elliptic curves in class 8470.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.j1 8470h4 \([1, -1, 0, -32390, 2251706]\) \(2121328796049/120050\) \(212675898050\) \([2]\) \(20480\) \(1.2379\)  
8470.j2 8470h3 \([1, -1, 0, -10610, -390450]\) \(74565301329/5468750\) \(9688224218750\) \([2]\) \(20480\) \(1.2379\)  
8470.j3 8470h2 \([1, -1, 0, -2140, 31356]\) \(611960049/122500\) \(217016222500\) \([2, 2]\) \(10240\) \(0.89136\)  
8470.j4 8470h1 \([1, -1, 0, 280, 2800]\) \(1367631/2800\) \(-4960370800\) \([2]\) \(5120\) \(0.54479\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 8470.j do not have complex multiplication.

Modular form 8470.2.a.j

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.