Properties

Label 1122e
Number of curves 6
Conductor 1122
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1122.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1122e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1122.e5 1122e1 [1, 1, 1, -284, 1277] [4] 512 \(\Gamma_0(N)\)-optimal
1122.e4 1122e2 [1, 1, 1, -1564, -23299] [2, 4] 1024  
1122.e2 1122e3 [1, 1, 1, -24684, -1502979] [2, 2] 2048  
1122.e6 1122e4 [1, 1, 1, 1076, -90883] [4] 2048  
1122.e1 1122e5 [1, 1, 1, -394944, -95697123] [2] 4096  
1122.e3 1122e6 [1, 1, 1, -24344, -1545955] [2] 4096  

Rank

sage: E.rank()
 

The elliptic curves in class 1122e have rank \(1\).

Modular form 1122.2.a.e

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{8} + q^{9} - 2q^{10} - q^{11} - q^{12} - 2q^{13} + 2q^{15} + q^{16} + q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.