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Properties

Label 122199a
Number of curves $2$
Conductor $122199$
CM no
Rank $0$
Graph

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

Elliptic curves in class 122199a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122199.i2 122199a1 \([1, 1, 0, -38363, 4544880]\) \(-42180533641/36293103\) \(-5372681767173567\) \([2]\) \(1148928\) \(1.7158\) \(\Gamma_0(N)\)-optimal
122199.i1 122199a2 \([1, 1, 0, -707548, 228721855]\) \(264621653112601/81336717\) \(12040753209436413\) \([2]\) \(2297856\) \(2.0624\)  

Rank

sage: E.rank()
 

The elliptic curves in class 122199a have rank \(0\).

Complex multiplication

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

Modular form 122199.2.a.a

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.