Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.e1 8470l4 \([1, 0, 1, -425923, 106370856]\) \(4823468134087681/30382271150\) \(53824046660765150\) \([2]\) \(138240\) \(2.0478\)  
8470.e2 8470l2 \([1, 0, 1, -32673, -2178244]\) \(2177286259681/105875000\) \(187564020875000\) \([2]\) \(46080\) \(1.4985\)  
8470.e3 8470l3 \([1, 0, 1, -10893, 3609428]\) \(-80677568161/3131816380\) \(-5548203757969180\) \([2]\) \(69120\) \(1.7012\)  
8470.e4 8470l1 \([1, 0, 1, 1207, -131892]\) \(109902239/4312000\) \(-7638971032000\) \([2]\) \(23040\) \(1.1519\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 8470.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - 2q^{12} - 2q^{13} + q^{14} - 2q^{15} + q^{16} - 6q^{17} - q^{18} - 2q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.