Properties

Label 1287e
Number of curves $6$
Conductor $1287$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1287.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1287e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1287.e5 1287e1 [1, -1, 0, -216, -1701] [2] 512 \(\Gamma_0(N)\)-optimal
1287.e4 1287e2 [1, -1, 0, -3861, -91368] [2, 2] 1024  
1287.e1 1287e3 [1, -1, 0, -61776, -5894451] [2] 2048  
1287.e3 1287e4 [1, -1, 0, -4266, -70713] [2, 2] 2048  
1287.e2 1287e5 [1, -1, 0, -27081, 1667790] [4] 4096  
1287.e6 1287e6 [1, -1, 0, 12069, -492156] [2] 4096  

Rank

sage: E.rank()
 

The elliptic curves in class 1287e have rank \(0\).

Modular form 1287.2.a.e

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.