Properties

Label 8470.i
Number of curves $2$
Conductor $8470$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.i1 8470a2 \([1, -1, 0, -116352110, -483040019500]\) \(73877525106256274859/48189030400\) \(113627212963208806400\) \([2]\) \(887040\) \(3.1661\)  
8470.i2 8470a1 \([1, -1, 0, -7316590, -7448888364]\) \(18370278334948779/460366807040\) \(1085520849673010544640\) \([2]\) \(443520\) \(2.8195\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 8470.2.a.i

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.