Properties

Label 10010.a
Number of curves $4$
Conductor $10010$
CM no
Rank $0$
Graph

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

Elliptic curves in class 10010.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10010.a1 10010f4 \([1, 0, 1, -78985959, -270129426454]\) \(54497099771831721530744218729/16209843781074944000000\) \(16209843781074944000000\) \([2]\) \(1451520\) \(3.2401\)  
10010.a2 10010f3 \([1, 0, 1, -5585639, -3040342038]\) \(19272683606216463573689449/7161126378530668544000\) \(7161126378530668544000\) \([2]\) \(725760\) \(2.8935\)  
10010.a3 10010f2 \([1, 0, 1, -2633144, 1164731142]\) \(2019051077229077416165369/582160888682835862400\) \(582160888682835862400\) \([6]\) \(483840\) \(2.6908\)  
10010.a4 10010f1 \([1, 0, 1, -2413624, 1442906886]\) \(1555006827939811751684089/221961497899581440\) \(221961497899581440\) \([6]\) \(241920\) \(2.3442\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10010.a have rank \(0\).

Complex multiplication

The elliptic curves in class 10010.a do not have complex multiplication.

Modular form 10010.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.