Properties

Label 122199.f
Number of curves 6
Conductor 122199
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("122199.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 122199.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
122199.f1 122199f6 [1, 1, 1, -2390562, 1421653488] [2] 2027520  
122199.f2 122199f4 [1, 1, 1, -150247, 21904676] [2, 2] 1013760  
122199.f3 122199f2 [1, 1, 1, -20642, -646594] [2, 2] 506880  
122199.f4 122199f1 [1, 1, 1, -17997, -936486] [2] 253440 \(\Gamma_0(N)\)-optimal
122199.f5 122199f5 [1, 1, 1, 16388, 68029244] [2] 2027520  
122199.f6 122199f3 [1, 1, 1, 66643, -4591876] [2] 1013760  

Rank

sage: E.rank()
 

The elliptic curves in class 122199.f have rank \(1\).

Modular form 122199.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} - 2q^{10} + q^{11} + q^{12} + 6q^{13} + q^{14} - 2q^{15} - q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.