Properties

Label 3675.n
Number of curves 6
Conductor 3675
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("3675.n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3675.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3675.n1 3675j5 [1, 0, 1, -960426, 362199373] [2] 24576  
3675.n2 3675j4 [1, 0, 1, -60051, 5650873] [2, 2] 12288  
3675.n3 3675j3 [1, 0, 1, -47801, -4002127] [2] 12288  
3675.n4 3675j6 [1, 0, 1, -41676, 9178873] [2] 24576  
3675.n5 3675j2 [1, 0, 1, -4926, 28123] [2, 2] 6144  
3675.n6 3675j1 [1, 0, 1, 1199, 3623] [2] 3072 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3675.n have rank \(1\).

Modular form 3675.2.a.n

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{8} + q^{9} + 4q^{11} - q^{12} - 2q^{13} - q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.