Properties

Label 21021b
Number of curves $6$
Conductor $21021$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("21021.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21021b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21021.e5 21021b1 [1, 1, 1, -1177, -22786] [2] 24576 \(\Gamma_0(N)\)-optimal
21021.e4 21021b2 [1, 1, 1, -21022, -1181734] [2, 2] 49152  
21021.e3 21021b3 [1, 1, 1, -23227, -921544] [2, 2] 98304  
21021.e1 21021b4 [1, 1, 1, -336337, -75217696] [2] 98304  
21021.e6 21021b5 [1, 1, 1, 65708, -6186496] [2] 196608  
21021.e2 21021b6 [1, 1, 1, -147442, 21039668] [2] 196608  

Rank

sage: E.rank()
 

The elliptic curves in class 21021b have rank \(0\).

Modular form 21021.2.a.e

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + 3q^{8} + q^{9} - 2q^{10} - q^{11} + q^{12} - q^{13} - 2q^{15} - q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.