Properties

Label 1001b
Number of curves 4
Conductor 1001
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("1001.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1001b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1001.b4 1001b1 [1, -1, 1, -16, -198] [4] 152 \(\Gamma_0(N)\)-optimal
1001.b3 1001b2 [1, -1, 1, -621, -5764] [2, 2] 304  
1001.b1 1001b3 [1, -1, 1, -9916, -377564] [2] 608  
1001.b2 1001b4 [1, -1, 1, -1006, 2552] [2] 608  

Rank

sage: E.rank()
 

The elliptic curves in class 1001b have rank \(0\).

Modular form 1001.2.a.b

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

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 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.