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Properties

Label 450450.ih
Number of curves $4$
Conductor $450450$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ih1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 450450.ih have rank \(0\).

Complex multiplication

The elliptic curves in class 450450.ih do not have complex multiplication.

Modular form 450450.2.a.ih

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{7} + q^{8} + q^{11} - q^{13} - q^{14} + q^{16} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 450450.ih

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450450.ih1 450450ih3 \([1, -1, 1, -27744980, 56255830647]\) \(207362104287019679089/5934929000000\) \(67602550640625000000\) \([2]\) \(38817792\) \(2.9062\) \(\Gamma_0(N)\)-optimal*
450450.ih2 450450ih4 \([1, -1, 1, -26619980, 61025830647]\) \(-183146792453150159089/35223382235041000\) \(-401216338271013890625000\) \([2]\) \(77635584\) \(3.2527\)  
450450.ih3 450450ih1 \([1, -1, 1, -600980, -53929353]\) \(2107441550633329/1108665958400\) \(12628398182400000000\) \([2]\) \(12939264\) \(2.3568\) \(\Gamma_0(N)\)-optimal*
450450.ih4 450450ih2 \([1, -1, 1, 2279020, -422569353]\) \(114926649504265871/73262465436160\) \(-834505270358760000000\) \([2]\) \(25878528\) \(2.7034\)  
*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 450450.ih1.