Properties

Label 210210be
Number of curves $4$
Conductor $210210$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("be1")
 
E.isogeny_class()
 

Elliptic curves in class 210210be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.eh4 210210be1 \([1, 0, 0, 3184, 90096]\) \(30342134159/47190000\) \(-5551856310000\) \([2]\) \(589824\) \(1.1307\) \(\Gamma_0(N)\)-optimal
210210.eh3 210210be2 \([1, 0, 0, -21316, 908396]\) \(9104453457841/2226896100\) \(261992099268900\) \([2, 2]\) \(1179648\) \(1.4773\)  
210210.eh1 210210be3 \([1, 0, 0, -317766, 68914026]\) \(30161840495801041/2799263610\) \(329330564452890\) \([2]\) \(2359296\) \(1.8238\)  
210210.eh2 210210be4 \([1, 0, 0, -116866, -14628034]\) \(1500376464746641/83599963590\) \(9835452116399910\) \([2]\) \(2359296\) \(1.8238\)  

Rank

sage: E.rank()
 

The elliptic curves in class 210210be have rank \(0\).

Complex multiplication

The elliptic curves in class 210210be do not have complex multiplication.

Modular form 210210.2.a.be

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.