Properties

Label 47040ec
Number of curves $8$
Conductor $47040$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("47040.z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040ec

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.z7 47040ec1 [0, -1, 0, -128641, -6205919] [2] 442368 \(\Gamma_0(N)\)-optimal
47040.z5 47040ec2 [0, -1, 0, -1132161, 459628065] [2, 2] 884736  
47040.z4 47040ec3 [0, -1, 0, -8407681, -9380638175] [2] 1327104  
47040.z6 47040ec4 [0, -1, 0, -254081, 1153486881] [2] 1769472  
47040.z2 47040ec5 [0, -1, 0, -18066561, 29563087905] [2] 1769472  
47040.z3 47040ec6 [0, -1, 0, -8470401, -9233509599] [2, 2] 2654208  
47040.z8 47040ec7 [0, -1, 0, 2286079, -31092828255] [2] 5308416  
47040.z1 47040ec8 [0, -1, 0, -20230401, 22041034401] [2] 5308416  

Rank

sage: E.rank()
 

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

Modular form 47040.2.a.z

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.