Properties

Label 458640.gr
Number of curves $2$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("gr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 458640.gr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.gr1 458640gr2 \([0, 0, 0, -324723, 63802802]\) \(10779215329/1232010\) \(432802687931228160\) \([2]\) \(6635520\) \(2.1164\) \(\Gamma_0(N)\)-optimal*
458640.gr2 458640gr1 \([0, 0, 0, 28077, 5026322]\) \(6967871/35100\) \(-12330560909721600\) \([2]\) \(3317760\) \(1.7698\) \(\Gamma_0(N)\)-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 458640.gr1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 458640.2.a.gr

sage: E.q_eigenform(10)
 
\(q - q^{5} + 4q^{11} + q^{13} + 8q^{17} - 6q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.