Properties

Label 10005e
Number of curves 4
Conductor 10005
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("10005.d1")
sage: E.isogeny_class()

Elliptic curves in class 10005e

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
10005.d4 10005e1 [1, 1, 1, -5250, -144690] 4 12288 \(\Gamma_0(N)\)-optimal
10005.d2 10005e2 [1, 1, 1, -83375, -9300940] 4 24576  
10005.d1 10005e3 [1, 1, 1, -1334000, -593592940] 2 49152  
10005.d3 10005e4 [1, 1, 1, -82750, -9446440] 2 49152  

Rank

sage: E.rank()

The elliptic curves in class 10005e have rank \(1\).

Modular form 10005.2.a.d

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.