Properties

Label 166410cm
Number of curves $2$
Conductor $166410$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cm1")
 
E.isogeny_class()
 

Elliptic curves in class 166410cm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
166410.x2 166410cm1 \([1, -1, 0, -636060969, -6174099116467]\) \(2097201152384001/64000000\) \(868480033426864704000000\) \([2]\) \(61028352\) \(3.6911\) \(\Gamma_0(N)\)-optimal
166410.x1 166410cm2 \([1, -1, 0, -10176900969, -395156054084467]\) \(8589947174918144001/8000\) \(108560004178358088000\) \([2]\) \(122056704\) \(4.0376\)  

Rank

sage: E.rank()
 

The elliptic curves in class 166410cm have rank \(0\).

Complex multiplication

The elliptic curves in class 166410cm do not have complex multiplication.

Modular form 166410.2.a.cm

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 2 q^{11} - 6 q^{13} + q^{16} - 2 q^{17} + 6 q^{19} + O(q^{20})\) 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.