Properties

Label 121968.ez
Number of curves $6$
Conductor $121968$
CM no
Rank $1$
Graph

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

Elliptic curves in class 121968.ez

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121968.ez1 121968fv6 \([0, 0, 0, -78739419, 268928121418]\) \(10206027697760497/5557167\) \(29396595420708139008\) \([2]\) \(9830400\) \(3.0642\)  
121968.ez2 121968fv4 \([0, 0, 0, -4948779, 4152546970]\) \(2533811507137/58110129\) \(307394028658515972096\) \([2, 2]\) \(4915200\) \(2.7176\)  
121968.ez3 121968fv2 \([0, 0, 0, -679899, -120601910]\) \(6570725617/2614689\) \(13831319930456641536\) \([2, 2]\) \(2457600\) \(2.3710\)  
121968.ez4 121968fv1 \([0, 0, 0, -592779, -175609478]\) \(4354703137/1617\) \(8553691979255808\) \([2]\) \(1228800\) \(2.0245\) \(\Gamma_0(N)\)-optimal
121968.ez5 121968fv5 \([0, 0, 0, 539781, 12858500842]\) \(3288008303/13504609503\) \(-71437396406179878531072\) \([2]\) \(9830400\) \(3.0642\)  
121968.ez6 121968fv3 \([0, 0, 0, 2195061, -873266438]\) \(221115865823/190238433\) \(-1006333307667466555392\) \([4]\) \(4915200\) \(2.7176\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 121968.ez do not have complex multiplication.

Modular form 121968.2.a.ez

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.