Properties

Label 2112.v
Number of curves 4
Conductor 2112
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2112.v1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2112.v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.v1 2112x3 [0, 1, 0, -5153, 140127] [2] 2304  
2112.v2 2112x4 [0, 1, 0, -2593, 281951] [2] 4608  
2112.v3 2112x1 [0, 1, 0, -353, -2529] [2] 768 \(\Gamma_0(N)\)-optimal
2112.v4 2112x2 [0, 1, 0, 287, -10081] [2] 1536  

Rank

sage: E.rank()
 

The elliptic curves in class 2112.v have rank \(1\).

Modular form 2112.2.a.v

sage: E.q_eigenform(10)
 
\( q + q^{3} - 2q^{7} + q^{9} - q^{11} + 4q^{13} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.