Properties

Label 2112f
Number of curves 4
Conductor 2112
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2112.l1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2112f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.l3 2112f1 [0, -1, 0, -132, 630] [2] 256 \(\Gamma_0(N)\)-optimal
2112.l2 2112f2 [0, -1, 0, -137, 585] [2, 2] 512  
2112.l1 2112f3 [0, -1, 0, -577, -4607] [2] 1024  
2112.l4 2112f4 [0, -1, 0, 223, 2817] [4] 1024  

Rank

sage: E.rank()
 

The elliptic curves in class 2112f have rank \(0\).

Modular form 2112.2.a.l

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.