Properties

Label 3600.f
Number of curves $8$
Conductor $3600$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("3600.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 3600.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3600.f1 3600bk7 [0, 0, 0, -19200675, -32383430750] [2] 110592  
3600.f2 3600bk8 [0, 0, 0, -1632675, -109286750] [2] 110592  
3600.f3 3600bk6 [0, 0, 0, -1200675, -505430750] [2, 2] 55296  
3600.f4 3600bk5 [0, 0, 0, -1038675, 407439250] [2] 36864  
3600.f5 3600bk4 [0, 0, 0, -246675, -40616750] [2] 36864  
3600.f6 3600bk2 [0, 0, 0, -66675, 6003250] [2, 2] 18432  
3600.f7 3600bk3 [0, 0, 0, -48675, -13526750] [2] 27648  
3600.f8 3600bk1 [0, 0, 0, 5325, 459250] [2] 9216 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3600.f have rank \(1\).

Modular form 3600.2.a.f

sage: E.q_eigenform(10)
 
\( q - 4q^{7} - 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.