Properties

Label 3675.j
Number of curves 8
Conductor 3675
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("3675.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3675.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3675.j1 3675e7 [1, 1, 0, -2646025, 1655579500] [2] 36864  
3675.j2 3675e5 [1, 1, 0, -165400, 25808875] [2, 2] 18432  
3675.j3 3675e8 [1, 1, 0, -134775, 35700750] [2] 36864  
3675.j4 3675e3 [1, 1, 0, -98025, -11853750] [2] 9216  
3675.j5 3675e4 [1, 1, 0, -12275, 237000] [2, 2] 9216  
3675.j6 3675e2 [1, 1, 0, -6150, -185625] [2, 2] 4608  
3675.j7 3675e1 [1, 1, 0, -25, -8000] [2] 2304 \(\Gamma_0(N)\)-optimal
3675.j8 3675e6 [1, 1, 0, 42850, 1835625] [2] 18432  

Rank

sage: E.rank()
 

The elliptic curves in class 3675.j have rank \(0\).

Modular form 3675.2.a.j

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.