Properties

Label 47040ee
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.u1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040ee

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.u8 47040ee1 [0, -1, 0, 4639, 371841] [2] 110592 \(\Gamma_0(N)\)-optimal
47040.u6 47040ee2 [0, -1, 0, -58081, 4900225] [2, 2] 221184  
47040.u7 47040ee3 [0, -1, 0, -42401, -10983615] [2] 331776  
47040.u5 47040ee4 [0, -1, 0, -214881, -32951295] [2] 442368  
47040.u4 47040ee5 [0, -1, 0, -904801, 331564801] [2] 442368  
47040.u3 47040ee6 [0, -1, 0, -1045921, -410585279] [2, 2] 663552  
47040.u1 47040ee7 [0, -1, 0, -16725921, -26323353279] [2] 1327104  
47040.u2 47040ee8 [0, -1, 0, -1422241, -88380095] [2] 1327104  

Rank

sage: E.rank()
 

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

Modular form 47040.2.a.u

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.