Properties

Label 49725.f
Number of curves $6$
Conductor $49725$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("49725.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 49725.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
49725.f1 49725i6 [1, -1, 1, -4539155, 3722855222] [2] 1048576  
49725.f2 49725i4 [1, -1, 1, -312530, 45691472] [2, 2] 524288  
49725.f3 49725i2 [1, -1, 1, -122405, -15909028] [2, 2] 262144  
49725.f4 49725i1 [1, -1, 1, -121280, -16226278] [2] 131072 \(\Gamma_0(N)\)-optimal
49725.f5 49725i3 [1, -1, 1, 49720, -57219028] [2] 524288  
49725.f6 49725i5 [1, -1, 1, 872095, 308678222] [2] 1048576  

Rank

sage: E.rank()
 

The elliptic curves in class 49725.f have rank \(1\).

Modular form 49725.2.a.f

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.