Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("47040.hb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040hc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.hb6 47040hc1 [0, 1, 0, 31295, -2971585] [2] 294912 \(\Gamma_0(N)\)-optimal
47040.hb5 47040hc2 [0, 1, 0, -219585, -30919617] [2, 2] 589824  
47040.hb4 47040hc3 [0, 1, 0, -1160385, 454345023] [2] 1179648  
47040.hb2 47040hc4 [0, 1, 0, -3292865, -2300844225] [2, 2] 1179648  
47040.hb3 47040hc5 [0, 1, 0, -3073345, -2620597057] [4] 2359296  
47040.hb1 47040hc6 [0, 1, 0, -52684865, -147207093825] [2] 2359296  

Rank

sage: E.rank()
 

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

Modular form 47040.2.a.hb

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.