Properties

Label 121680ep
Number of curves $6$
Conductor $121680$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("121680.eg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 121680ep

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121680.eg6 121680ep1 [0, 0, 0, 364533, 33926074] [2] 2064384 \(\Gamma_0(N)\)-optimal
121680.eg5 121680ep2 [0, 0, 0, -1582347, 281958586] [2, 2] 4128768  
121680.eg3 121680ep3 [0, 0, 0, -13750347, -19427767814] [2, 2] 8257536  
121680.eg2 121680ep4 [0, 0, 0, -20564427, 35865765754] [4] 8257536  
121680.eg4 121680ep5 [0, 0, 0, -2799147, -49523855654] [2] 16515072  
121680.eg1 121680ep6 [0, 0, 0, -219389547, -1250754169574] [2] 16515072  

Rank

sage: E.rank()
 

The elliptic curves in class 121680ep have rank \(0\).

Modular form 121680.2.a.eg

sage: E.q_eigenform(10)
 
\( q + q^{5} - 4q^{11} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.