Properties

Label 48.a
Number of curves $6$
Conductor $48$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("48.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 48.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
48.a1 48a5 [0, 1, 0, -384, 2772] [4] 8  
48.a2 48a2 [0, 1, 0, -64, -220] [2] 4  
48.a3 48a3 [0, 1, 0, -24, 36] [2, 4] 4  
48.a4 48a1 [0, 1, 0, -4, -4] [2, 2] 2 \(\Gamma_0(N)\)-optimal
48.a5 48a4 [0, 1, 0, 1, 0] [2] 4  
48.a6 48a6 [0, 1, 0, 16, 180] [8] 8  

Rank

sage: E.rank()
 

The elliptic curves in class 48.a have rank \(0\).

Modular form 48.2.a.a

sage: E.q_eigenform(10)
 
\( q + q^{3} - 2q^{5} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.