Properties

Label 10647f
Number of curves 6
Conductor 10647
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("10647.d1")
sage: E.isogeny_class()

Elliptic curves in class 10647f

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
10647.d6 10647f1 [1, -1, 1, 1489, 4502] 2 9216 \(\Gamma_0(N)\)-optimal
10647.d5 10647f2 [1, -1, 1, -6116, 41006] 4 18432  
10647.d3 10647f3 [1, -1, 1, -59351, -5516728] 2 36864  
10647.d2 10647f4 [1, -1, 1, -74561, 7843736] 4 36864  
10647.d1 10647f5 [1, -1, 1, -1192496, 501523832] 2 73728  
10647.d4 10647f6 [1, -1, 1, -51746, 12717020] 2 73728  

Rank

sage: E.rank()

The elliptic curves in class 10647f have rank \(1\).

Modular form 10647.2.a.d

sage: E.q_eigenform(10)
\( q - q^{2} - q^{4} - 2q^{5} + q^{7} + 3q^{8} + 2q^{10} + 4q^{11} - q^{14} - q^{16} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.