Properties

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

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

Elliptic curves in class 122199.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122199.e1 122199b2 \([1, 1, 1, -713103, -232076688]\) \(270902819202625/1004157\) \(148651274190573\) \([2]\) \(1148928\) \(1.9352\)  
122199.e2 122199b1 \([1, 1, 1, -43918, -3750766]\) \(-63282696625/4032567\) \(-596964640797063\) \([2]\) \(574464\) \(1.5887\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 122199.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.