Properties

Label 4005.d
Number of curves 4
Conductor 4005
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4005.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4005.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4005.d1 4005c3 [1, -1, 0, -21375, 1208196] [2] 7168  
4005.d2 4005c2 [1, -1, 0, -1350, 18711] [2, 2] 3584  
4005.d3 4005c1 [1, -1, 0, -225, -864] [2] 1792 \(\Gamma_0(N)\)-optimal
4005.d4 4005c4 [1, -1, 0, 675, 68526] [2] 7168  

Rank

sage: E.rank()
 

The elliptic curves in class 4005.d have rank \(0\).

Modular form 4005.2.a.d

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.