Properties

Label 15.a
Number of curves 8
Conductor \(15\)
CM False
Rank \(0\)
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("15.a1")
sage: E.isogeny_class()

Elliptic curves in class 15.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
15.a1 15a5 [1, 1, 1, -2160, -39540] 2 4  
15.a2 15a2 [1, 1, 1, -135, -660] 4 2  
15.a3 15a6 [1, 1, 1, -110, -880] 2 4  
15.a4 15a7 [1, 1, 1, -80, 242] 4 4  
15.a5 15a1 [1, 1, 1, -10, -10] 8 1 \(\Gamma_0(N)\)-optimal
15.a6 15a3 [1, 1, 1, -5, 2] 8 2  
15.a7 15a8 [1, 1, 1, 0, 0] 4 4  
15.a8 15a4 [1, 1, 1, 35, -28] 8 2  

Rank

sage: E.rank()

The elliptic curves in class 15.a have rank \(0\).

Modular form 15.2.1.a

sage: E.q_eigenform(10)
\( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + O(q^{10}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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