Properties

Label 8712k
Number of curves 4
Conductor 8712
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("8712.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8712k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8712.f3 8712k1 [0, 0, 0, -13431, -577654] [2] 15360 \(\Gamma_0(N)\)-optimal
8712.f2 8712k2 [0, 0, 0, -35211, 1770230] [2, 2] 30720  
8712.f1 8712k3 [0, 0, 0, -514371, 141972446] [2] 61440  
8712.f4 8712k4 [0, 0, 0, 95469, 11832590] [2] 61440  

Rank

sage: E.rank()
 

The elliptic curves in class 8712k have rank \(1\).

Modular form 8712.2.a.f

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.