Properties

Label 4851.p
Number of curves 6
Conductor 4851
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4851.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4851.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4851.p1 4851o5 [1, -1, 0, -1992888, -1082361771] [2] 61440  
4851.p2 4851o3 [1, -1, 0, -125253, -16689240] [2, 2] 30720  
4851.p3 4851o2 [1, -1, 0, -17208, 489915] [2, 2] 15360  
4851.p4 4851o1 [1, -1, 0, -15003, 710856] [2] 7680 \(\Gamma_0(N)\)-optimal
4851.p5 4851o6 [1, -1, 0, 13662, -51779169] [2] 61440  
4851.p6 4851o4 [1, -1, 0, 55557, 3502386] [2] 30720  

Rank

sage: E.rank()
 

The elliptic curves in class 4851.p have rank \(0\).

Modular form 4851.2.a.p

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.