Properties

Label 47040hf
Number of curves $6$
Conductor $47040$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("47040.fw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040hf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.fw4 47040hf1 [0, 1, 0, -34365, 2440563] [2] 98304 \(\Gamma_0(N)\)-optimal
47040.fw3 47040hf2 [0, 1, 0, -35345, 2292975] [2, 2] 196608  
47040.fw5 47040hf3 [0, 1, 0, 46975, 11463423] [2] 393216  
47040.fw2 47040hf4 [0, 1, 0, -133345, -16307425] [2, 2] 393216  
47040.fw6 47040hf5 [0, 1, 0, 219455, -87643585] [2] 786432  
47040.fw1 47040hf6 [0, 1, 0, -2054145, -1133828865] [2] 786432  

Rank

sage: E.rank()
 

The elliptic curves in class 47040hf have rank \(1\).

Modular form 47040.2.a.fw

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.