Properties

Label 4410bk
Number of curves 8
Conductor 4410
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4410.bi1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4410bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4410.bi7 4410bk1 [1, -1, 1, -219407, 39596631] [4] 36864 \(\Gamma_0(N)\)-optimal
4410.bi6 4410bk2 [1, -1, 1, -254687, 26034999] [2, 2] 73728  
4410.bi5 4410bk3 [1, -1, 1, -649382, -152913099] [4] 110592  
4410.bi4 4410bk4 [1, -1, 1, -1921667, -1006825809] [2] 147456  
4410.bi8 4410bk5 [1, -1, 1, 847813, 190527999] [2] 147456  
4410.bi2 4410bk6 [1, -1, 1, -9681062, -11590632651] [2, 2] 221184  
4410.bi1 4410bk7 [1, -1, 1, -154893542, -741951322059] [2] 442368  
4410.bi3 4410bk8 [1, -1, 1, -8975462, -13352374731] [2] 442368  

Rank

sage: E.rank()
 

The elliptic curves in class 4410bk have rank \(1\).

Modular form 4410.2.a.bi

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 2q^{13} + q^{16} - 6q^{17} - 8q^{19} + O(q^{20}) \)

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.