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Properties

Label 10001.a
Number of curves $2$
Conductor $10001$
CM no
Rank $1$
Graph

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

Elliptic curves in class 10001.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10001.a1 10001a2 \([1, -1, 0, -53959, 4720704]\) \(17374804109361438921/482665506294457\) \(482665506294457\) \([2]\) \(46656\) \(1.5969\)  
10001.a2 10001a1 \([1, -1, 0, -53594, 4788959]\) \(17024594875172176761/13702740137\) \(13702740137\) \([2]\) \(23328\) \(1.2503\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 10001.a have rank \(1\).

Complex multiplication

The elliptic curves in class 10001.a do not have complex multiplication.

Modular form 10001.2.a.a

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