Properties

Label 1005a
Number of curves 2
Conductor 1005
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1005.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1005a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1005.b1 1005a1 [1, 1, 0, -297, -1944] [2] 360 \(\Gamma_0(N)\)-optimal
1005.b2 1005a2 [1, 1, 0, 328, -8319] [2] 720  

Rank

sage: E.rank()
 

The elliptic curves in class 1005a have rank \(0\).

Modular form 1005.2.a.b

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

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 Cremona 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 Cremona labels.