Properties

Label 1050a
Number of curves $8$
Conductor $1050$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1050.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1050a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1050.a7 1050a1 [1, 1, 0, -1025, -4875] [2] 1152 \(\Gamma_0(N)\)-optimal
1050.a5 1050a2 [1, 1, 0, -9025, 323125] [2, 2] 2304  
1050.a4 1050a3 [1, 1, 0, -67025, -6706875] [2] 3456  
1050.a2 1050a4 [1, 1, 0, -144025, 20978125] [2] 4608  
1050.a6 1050a5 [1, 1, 0, -2025, 820125] [2] 4608  
1050.a3 1050a6 [1, 1, 0, -67525, -6602375] [2, 2] 6912  
1050.a1 1050a7 [1, 1, 0, -161275, 15616375] [2] 13824  
1050.a8 1050a8 [1, 1, 0, 18225, -22123125] [2] 13824  

Rank

sage: E.rank()
 

The elliptic curves in class 1050a have rank \(1\).

Modular form 1050.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - q^{12} - 2q^{13} + q^{14} + q^{16} + 6q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.