Properties

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

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("2112.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2112u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2112.e4 2112u1 [0, -1, 0, -129, 33] [2] 768 \(\Gamma_0(N)\)-optimal
2112.e2 2112u2 [0, -1, 0, -1409, 20769] [2, 2] 1536  
2112.e1 2112u3 [0, -1, 0, -22529, 1309089] [4] 3072  
2112.e3 2112u4 [0, -1, 0, -769, 39073] [2] 3072  

Rank

sage: E.rank()
 

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

Modular form 2112.2.a.e

sage: E.q_eigenform(10)
 
\( q - q^{3} - 2q^{5} + 4q^{7} + q^{9} - q^{11} + 6q^{13} + 2q^{15} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.