Properties

Label 458640fr
Number of curves $4$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fr1")
 
E.isogeny_class()
 

Elliptic curves in class 458640fr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.fr3 458640fr1 \([0, 0, 0, -49098, -4177397]\) \(9538484224/26325\) \(36124690165200\) \([2]\) \(1769472\) \(1.4744\) \(\Gamma_0(N)\)-optimal*
458640.fr2 458640fr2 \([0, 0, 0, -68943, -482258]\) \(1650587344/950625\) \(20872043206560000\) \([2, 2]\) \(3538944\) \(1.8210\) \(\Gamma_0(N)\)-optimal*
458640.fr1 458640fr3 \([0, 0, 0, -730443, 239377642]\) \(490757540836/2142075\) \(188126682768460800\) \([2]\) \(7077888\) \(2.1676\) \(\Gamma_0(N)\)-optimal*
458640.fr4 458640fr4 \([0, 0, 0, 275037, -3853262]\) \(26198797244/15234375\) \(-1337951487600000000\) \([2]\) \(7077888\) \(2.1676\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 458640fr1.

Rank

sage: E.rank()
 

The elliptic curves in class 458640fr have rank \(1\).

Complex multiplication

The elliptic curves in class 458640fr do not have complex multiplication.

Modular form 458640.2.a.fr

sage: E.q_eigenform(10)
 
\(q - q^{5} + 4 q^{11} - q^{13} - 6 q^{17} + O(q^{20})\) 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 Cremona 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 Cremona labels.