Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("10001.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 10001.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10001.a1 10001a2 [1, -1, 0, -53959, 4720704] [2] 46656  
10001.a2 10001a1 [1, -1, 0, -53594, 4788959] [2] 23328 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Modular form 10001.2.a.a

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

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.