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Properties

Label 417910i
Number of curves $2$
Conductor $417910$
CM no
Rank $0$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 417910i have rank \(0\).

Complex multiplication

The elliptic curves in class 417910i do not have complex multiplication.

Modular form 417910.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} + 2 q^{7} + q^{8} + q^{9} - q^{10} + 4 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + q^{16} + 4 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 417910i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
417910.i2 417910i1 \([1, 0, 0, -13236, -719984]\) \(-1732323601/505600\) \(-74846945478400\) \([2]\) \(1622016\) \(1.3769\) \(\Gamma_0(N)\)-optimal*
417910.i1 417910i2 \([1, 0, 0, -224836, -41050944]\) \(8490912541201/499280\) \(73911358659920\) \([2]\) \(3244032\) \(1.7235\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 417910i1.