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Properties

Label 10010.m
Number of curves $2$
Conductor $10010$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 10010.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10010.m1 10010p2 \([1, 0, 0, -1001, 10105]\) \(110931033861649/19352933600\) \(19352933600\) \([2]\) \(10240\) \(0.69377\)  
10010.m2 10010p1 \([1, 0, 0, 119, 921]\) \(186267240431/466385920\) \(-466385920\) \([2]\) \(5120\) \(0.34720\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10010.m have rank \(1\).

Complex multiplication

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

Modular form 10010.2.a.m

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