Properties

Label 457776e
Number of curves $4$
Conductor $457776$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 457776e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
457776.e3 457776e1 \([0, 0, 0, -1873587, 719508850]\) \(10091699281/2737152\) \(197278573965650952192\) \([2]\) \(19660800\) \(2.6022\) \(\Gamma_0(N)\)-optimal
457776.e4 457776e2 \([0, 0, 0, 4784973, 4681352050]\) \(168105213359/228637728\) \(-16478925881568281100288\) \([2]\) \(39321600\) \(2.9488\)  
457776.e1 457776e3 \([0, 0, 0, -418865907, -3299594922830]\) \(112763292123580561/1932612\) \(139291840346792804352\) \([2]\) \(98304000\) \(3.4069\)  
457776.e2 457776e4 \([0, 0, 0, -418449747, -3306478625390]\) \(-112427521449300721/466873642818\) \(-33649635269536991900540928\) \([2]\) \(196608000\) \(3.7535\)  

Rank

sage: E.rank()
 

The elliptic curves in class 457776e have rank \(1\).

Complex multiplication

The elliptic curves in class 457776e do not have complex multiplication.

Modular form 457776.2.a.e

sage: E.q_eigenform(10)
 
\(q - 4 q^{5} - 2 q^{7} - q^{11} + 4 q^{13} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.