Properties

Label 21675.s
Number of curves 8
Conductor 21675
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("21675.s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21675.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21675.s1 21675c8 [1, 1, 0, -15606150, -23736178125] [2] 491520  
21675.s2 21675c6 [1, 1, 0, -975525, -371070000] [2, 2] 245760  
21675.s3 21675c7 [1, 1, 0, -794900, -512499375] [2] 491520  
21675.s4 21675c4 [1, 1, 0, -578150, 168962625] [2] 122880  
21675.s5 21675c3 [1, 1, 0, -72400, -3498125] [2, 2] 122880  
21675.s6 21675c2 [1, 1, 0, -36275, 2607000] [2, 2] 61440  
21675.s7 21675c1 [1, 1, 0, -150, 114375] [2] 30720 \(\Gamma_0(N)\)-optimal
21675.s8 21675c5 [1, 1, 0, 252725, -25931750] [2] 245760  

Rank

sage: E.rank()
 

The elliptic curves in class 21675.s have rank \(1\).

Modular form 21675.2.a.s

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.