Properties

Label 10005i
Number of curves 2
Conductor 10005
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("10005.h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 10005i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10005.h2 10005i1 [0, 1, 1, -124021, 14938111] [3] 64800 \(\Gamma_0(N)\)-optimal
10005.h1 10005i2 [0, 1, 1, -2379061, -1411043480] [] 194400  

Rank

sage: E.rank()
 

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

Modular form 10005.2.a.h

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

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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.