Properties

Label 21840bk
Number of curves 8
Conductor 21840
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("21840.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21840bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21840.w7 21840bk1 [0, -1, 0, -411600, -100957248] [2] 221184 \(\Gamma_0(N)\)-optimal
21840.w6 21840bk2 [0, -1, 0, -662480, 36926400] [2, 2] 442368  
21840.w5 21840bk3 [0, -1, 0, -2542800, 1493548992] [2] 663552  
21840.w8 21840bk4 [0, -1, 0, 2603440, 290361792] [4] 884736  
21840.w4 21840bk5 [0, -1, 0, -7942480, 8604030400] [2] 884736  
21840.w2 21840bk6 [0, -1, 0, -40190480, 98082436800] [2, 2] 1327104  
21840.w3 21840bk7 [0, -1, 0, -39696560, 100610121792] [4] 2654208  
21840.w1 21840bk8 [0, -1, 0, -643047280, 6276641208640] [2] 2654208  

Rank

sage: E.rank()
 

The elliptic curves in class 21840bk have rank \(0\).

Modular form 21840.2.a.w

sage: E.q_eigenform(10)
 
\( q - q^{3} + q^{5} - q^{7} + q^{9} + q^{13} - q^{15} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.