Properties

Label 2112z
Number of curves 2
Conductor 2112
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2112.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2112z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.p1 2112z1 [0, 1, 0, -32065, 2199359] [2] 5376 \(\Gamma_0(N)\)-optimal
2112.p2 2112z2 [0, 1, 0, -31905, 2222559] [2] 10752  

Rank

sage: E.rank()
 

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

Modular form 2112.2.a.p

sage: E.q_eigenform(10)
 
\( q + q^{3} - 4q^{5} + 2q^{7} + q^{9} - q^{11} - 4q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.