# Properties

 Label 78400gz Number of curves $4$ Conductor $78400$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gz1")

sage: E.isogeny_class()

## Elliptic curves in class 78400gz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
78400.ey4 78400gz1 [0, 0, 0, 4900, 686000] [2] 196608 $$\Gamma_0(N)$$-optimal
78400.ey3 78400gz2 [0, 0, 0, -93100, 10290000] [2, 2] 393216
78400.ey2 78400gz3 [0, 0, 0, -289100, -47334000] [2] 786432
78400.ey1 78400gz4 [0, 0, 0, -1465100, 682570000] [2] 786432

## Rank

sage: E.rank()

The elliptic curves in class 78400gz have rank $$1$$.

## Complex multiplication

The elliptic curves in class 78400gz do not have complex multiplication.

## Modular form 78400.2.a.gz

sage: E.q_eigenform(10)

$$q - 3q^{9} - 4q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.