# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 121968.ez

sage: E.isogeny_class().curves

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

## Rank

sage: E.rank()

The elliptic curves in class 121968.ez 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 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.