Properties

Label 2142g
Number of curves $6$
Conductor $2142$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2142.h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2142g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2142.h5 2142g1 [1, -1, 0, -632196, -193075376] [2] 30720 \(\Gamma_0(N)\)-optimal
2142.h4 2142g2 [1, -1, 0, -816516, -71166128] [2, 2] 61440  
2142.h2 2142g3 [1, -1, 0, -7731396, 8219774992] [2, 2] 122880  
2142.h6 2142g4 [1, -1, 0, 3149244, -562127216] [2] 122880  
2142.h1 2142g5 [1, -1, 0, -123467436, 528082919464] [2] 245760  
2142.h3 2142g6 [1, -1, 0, -2633436, 18893883640] [2] 245760  

Rank

sage: E.rank()
 

The elliptic curves in class 2142g have rank \(1\).

Modular form 2142.2.a.h

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} + 2q^{5} + q^{7} - q^{8} - 2q^{10} - 4q^{11} - 2q^{13} - q^{14} + q^{16} - q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.