Properties

Label 21.a
Number of curves 6
Conductor \(21\)
CM False
Rank \(0\)
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("21.a1")
sage: E.isogeny_class()

Elliptic curves in class 21.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
21.a1 21a5 [1, 0, 0, -784, -8515] 2 4  
21.a2 21a2 [1, 0, 0, -49, -136] 4 2  
21.a3 21a3 [1, 0, 0, -39, 90] 8 2  
21.a4 21a6 [1, 0, 0, -34, -217] 2 4  
21.a5 21a1 [1, 0, 0, -4, -1] 8 1 \(\Gamma_0(N)\)-optimal
21.a6 21a4 [1, 0, 0, 1, 0] 4 2  

Rank

sage: E.rank()

The elliptic curves in class 21.a have rank \(0\).

Modular form 21.2.1.a

sage: E.q_eigenform(10)
\( q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + O(q^{10}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()

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

Isogeny graph

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