Properties

Label 5040bc
Number of curves 8
Conductor 5040
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("5040.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5040bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5040.g7 5040bc1 [0, 0, 0, -71643, 7378058] [2] 18432 \(\Gamma_0(N)\)-optimal
5040.g6 5040bc2 [0, 0, 0, -83163, 4845962] [2, 2] 36864  
5040.g5 5040bc3 [0, 0, 0, -212043, -28562182] [2] 55296  
5040.g4 5040bc4 [0, 0, 0, -627483, -187952182] [2] 73728  
5040.g8 5040bc5 [0, 0, 0, 276837, 35589962] [2] 73728  
5040.g2 5040bc6 [0, 0, 0, -3161163, -2163135238] [2, 2] 110592  
5040.g1 5040bc7 [0, 0, 0, -50577483, -138447122182] [2] 221184  
5040.g3 5040bc8 [0, 0, 0, -2930763, -2491823878] [2] 221184  

Rank

sage: E.rank()
 

The elliptic curves in class 5040bc have rank \(1\).

Modular form 5040.2.a.g

sage: E.q_eigenform(10)
 
\( q - q^{5} - q^{7} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20}) \)

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.