Properties

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

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

Elliptic curves in class 122199.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122199.f1 122199f6 \([1, 1, 1, -2390562, 1421653488]\) \(10206027697760497/5557167\) \(822660157166463\) \([2]\) \(2027520\) \(2.1905\)  
122199.f2 122199f4 \([1, 1, 1, -150247, 21904676]\) \(2533811507137/58110129\) \(8602384606419681\) \([2, 2]\) \(1013760\) \(1.8440\)  
122199.f3 122199f2 \([1, 1, 1, -20642, -646594]\) \(6570725617/2614689\) \(387067810573521\) \([2, 2]\) \(506880\) \(1.4974\)  
122199.f4 122199f1 \([1, 1, 1, -17997, -936486]\) \(4354703137/1617\) \(239374032513\) \([2]\) \(253440\) \(1.1508\) \(\Gamma_0(N)\)-optimal
122199.f5 122199f5 \([1, 1, 1, 16388, 68029244]\) \(3288008303/13504609503\) \(-1999166873374453167\) \([2]\) \(2027520\) \(2.1905\)  
122199.f6 122199f3 \([1, 1, 1, 66643, -4591876]\) \(221115865823/190238433\) \(-28162115551121937\) \([2]\) \(1013760\) \(1.8440\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 122199.f do not have complex multiplication.

Modular form 122199.2.a.f

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

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.