Properties

Label 32448.be
Number of curves $4$
Conductor $32448$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("be1")
 
E.isogeny_class()
 

Elliptic curves in class 32448.be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32448.be1 32448ch4 \([0, -1, 0, -751937, 251213793]\) \(37159393753/1053\) \(1332380926476288\) \([2]\) \(344064\) \(2.0038\)  
32448.be2 32448ch3 \([0, -1, 0, -211137, -33744543]\) \(822656953/85683\) \(108416329461792768\) \([2]\) \(344064\) \(2.0038\)  
32448.be3 32448ch2 \([0, -1, 0, -48897, 3603105]\) \(10218313/1521\) \(1924550227132416\) \([2, 2]\) \(172032\) \(1.6573\)  
32448.be4 32448ch1 \([0, -1, 0, 5183, 304225]\) \(12167/39\) \(-49347441721344\) \([2]\) \(86016\) \(1.3107\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 32448.be have rank \(0\).

Complex multiplication

The elliptic curves in class 32448.be do not have complex multiplication.

Modular form 32448.2.a.be

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} - 4 q^{7} + q^{9} - 4 q^{11} - 2 q^{15} + 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.