Properties

Label 424536.h
Number of curves $4$
Conductor $424536$
CM no
Rank $1$
Graph

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

Elliptic curves in class 424536.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424536.h1 424536h4 \([0, -1, 0, -57176744, -138074633556]\) \(1823652903746/328593657\) \(3724765824353030027741184\) \([2]\) \(88473600\) \(3.4338\)  
424536.h2 424536h2 \([0, -1, 0, -16845824, 24620297724]\) \(93280467172/7800849\) \(44213172009188403078144\) \([2, 2]\) \(44236800\) \(3.0872\)  
424536.h3 424536h1 \([0, -1, 0, -16492044, 25783950900]\) \(350104249168/2793\) \(3957498389651665152\) \([2]\) \(22118400\) \(2.7406\) \(\Gamma_0(N)\)-optimal*
424536.h4 424536h3 \([0, -1, 0, 17824616, 112835765260]\) \(55251546334/517244049\) \(-5863207994198086595954688\) \([2]\) \(88473600\) \(3.4338\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 424536.h1.

Rank

sage: E.rank()
 

The elliptic curves in class 424536.h have rank \(1\).

Complex multiplication

The elliptic curves in class 424536.h do not have complex multiplication.

Modular form 424536.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.