Properties

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

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("47040.es1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040.es

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.es1 47040cg7 [0, 1, 0, -16725921, 26323353279] [2] 1327104  
47040.es2 47040cg8 [0, 1, 0, -1422241, 88380095] [2] 1327104  
47040.es3 47040cg6 [0, 1, 0, -1045921, 410585279] [2, 2] 663552  
47040.es4 47040cg5 [0, 1, 0, -904801, -331564801] [2] 442368  
47040.es5 47040cg4 [0, 1, 0, -214881, 32951295] [2] 442368  
47040.es6 47040cg2 [0, 1, 0, -58081, -4900225] [2, 2] 221184  
47040.es7 47040cg3 [0, 1, 0, -42401, 10983615] [2] 331776  
47040.es8 47040cg1 [0, 1, 0, 4639, -371841] [2] 110592 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Modular form 47040.2.a.es

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.