Properties

Label 450800.h
Number of curves $2$
Conductor $450800$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 450800.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450800.h1 450800h1 \([0, 1, 0, -294408, -100428812]\) \(-7649089/7360\) \(-2715451863040000000\) \([]\) \(8709120\) \(2.2333\) \(\Gamma_0(N)\)-optimal*
450800.h2 450800h2 \([0, 1, 0, 2449592, 1738051188]\) \(4405959551/6083500\) \(-2244490680544000000000\) \([]\) \(26127360\) \(2.7826\) \(\Gamma_0(N)\)-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 450800.h1.

Rank

sage: E.rank()
 

The elliptic curves in class 450800.h have rank \(2\).

Complex multiplication

The elliptic curves in class 450800.h do not have complex multiplication.

Modular form 450800.2.a.h

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} + q^{9} - 6 q^{11} + 4 q^{13} - 3 q^{17} - 2 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.