Properties

Label 5040.bg
Number of curves $4$
Conductor $5040$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bg1")
 
E.isogeny_class()
 

Elliptic curves in class 5040.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.bg1 5040bo3 \([0, 0, 0, -53787, 4801354]\) \(5763259856089/5670\) \(16930529280\) \([2]\) \(12288\) \(1.2560\)  
5040.bg2 5040bo2 \([0, 0, 0, -3387, 73834]\) \(1439069689/44100\) \(131681894400\) \([2, 2]\) \(6144\) \(0.90946\)  
5040.bg3 5040bo1 \([0, 0, 0, -507, -2774]\) \(4826809/1680\) \(5016453120\) \([2]\) \(3072\) \(0.56289\) \(\Gamma_0(N)\)-optimal
5040.bg4 5040bo4 \([0, 0, 0, 933, 249226]\) \(30080231/9003750\) \(-26885053440000\) \([4]\) \(12288\) \(1.2560\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5040.bg have rank \(1\).

Complex multiplication

The elliptic curves in class 5040.bg do not have complex multiplication.

Modular form 5040.2.a.bg

sage: E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - 4 q^{11} - 2 q^{13} + 6 q^{17} + O(q^{20})\) 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 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.