Properties

Label 3630.f
Number of curves $6$
Conductor $3630$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3630.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3630.f1 3630d5 [1, 1, 0, -20701287, 36244390329] [4] 184320  
3630.f2 3630d3 [1, 1, 0, -1430827, -657916211] [2] 92160  
3630.f3 3630d4 [1, 1, 0, -1295307, 564555501] [2, 2] 92160  
3630.f4 3630d6 [1, 1, 0, -629807, 1144738401] [2] 184320  
3630.f5 3630d2 [1, 1, 0, -124027, -1641251] [2, 2] 46080  
3630.f6 3630d1 [1, 1, 0, 30853, -185379] [2] 23040 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 3630.f have rank \(1\).

Complex multiplication

The elliptic curves in class 3630.f do not have complex multiplication.

Modular form 3630.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 6q^{13} - q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.