Properties

Label 784j
Number of curves 6
Conductor 784
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath

sage: E = EllipticCurve("784.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 784j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
784.b5 784j1 [0, 1, 0, -408, 6292] [2] 384 \(\Gamma_0(N)\)-optimal
784.b4 784j2 [0, 1, 0, -8248, 285396] [2] 768  
784.b6 784j3 [0, 1, 0, 3512, -133260] [2] 1152  
784.b3 784j4 [0, 1, 0, -27848, -1475468] [2] 2304  
784.b2 784j5 [0, 1, 0, -133688, -18913196] [2] 3456  
784.b1 784j6 [0, 1, 0, -2140728, -1206278060] [2] 6912  

Rank

sage: E.rank()
 

The elliptic curves in class 784j have rank \(1\).

Modular form 784.2.a.b

sage: E.q_eigenform(10)
 
\( q - 2q^{3} + q^{9} + 4q^{13} - 6q^{17} + 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.