Properties

Label 32487.d
Number of curves $4$
Conductor $32487$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32487.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32487.d1 32487n4 \([1, 0, 0, -1212849, 514011504]\) \(1677087406638588673/4641\) \(546009009\) \([2]\) \(245760\) \(1.7949\)  
32487.d2 32487n2 \([1, 0, 0, -75804, 8026479]\) \(409460675852593/21538881\) \(2534027810769\) \([2, 2]\) \(122880\) \(1.4483\)  
32487.d3 32487n3 \([1, 0, 0, -71639, 8948610]\) \(-345608484635233/94427721297\) \(-11109326982870753\) \([2]\) \(245760\) \(1.7949\)  
32487.d4 32487n1 \([1, 0, 0, -4999, 110480]\) \(117433042273/22801233\) \(2682542261217\) \([2]\) \(61440\) \(1.1018\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 32487.d have rank \(1\).

Complex multiplication

The elliptic curves in class 32487.d do not have complex multiplication.

Modular form 32487.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} - q^{13} - 2 q^{15} - q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.