Properties

Label 100002.a
Number of curves 2
Conductor 100002
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("100002.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 100002.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100002.a1 100002a2 [1, 1, 0, -448, -560] [2] 150272  
100002.a2 100002a1 [1, 1, 0, 112, 0] [2] 75136 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 100002.a have rank \(1\).

Modular form 100002.2.a.a

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

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.