Properties

Label 121968fv
Number of curves 6
Conductor 121968
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("121968.ez1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 121968fv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121968.ez4 121968fv1 [0, 0, 0, -592779, -175609478] [2] 1228800 \(\Gamma_0(N)\)-optimal
121968.ez3 121968fv2 [0, 0, 0, -679899, -120601910] [2, 2] 2457600  
121968.ez6 121968fv3 [0, 0, 0, 2195061, -873266438] [4] 4915200  
121968.ez2 121968fv4 [0, 0, 0, -4948779, 4152546970] [2, 2] 4915200  
121968.ez5 121968fv5 [0, 0, 0, 539781, 12858500842] [2] 9830400  
121968.ez1 121968fv6 [0, 0, 0, -78739419, 268928121418] [2] 9830400  

Rank

sage: E.rank()
 

The elliptic curves in class 121968fv have rank \(1\).

Modular form 121968.2.a.ez

sage: E.q_eigenform(10)
 
\( q + 2q^{5} + q^{7} - 6q^{13} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.