Properties

Label 12789f
Number of curves $6$
Conductor $12789$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("12789.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12789f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12789.k5 12789f1 [1, -1, 0, -345753, 83869344] [2] 147456 \(\Gamma_0(N)\)-optimal
12789.k4 12789f2 [1, -1, 0, -5639958, 5156776575] [2, 2] 294912  
12789.k3 12789f3 [1, -1, 0, -5748003, 4949006040] [2, 2] 589824  
12789.k1 12789f4 [1, -1, 0, -90239193, 329967079434] [2] 589824  
12789.k2 12789f5 [1, -1, 0, -18728838, -25356051351] [2] 1179648  
12789.k6 12789f6 [1, -1, 0, 5504112, 21955452651] [2] 1179648  

Rank

sage: E.rank()
 

The elliptic curves in class 12789f have rank \(1\).

Modular form 12789.2.a.k

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.