# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("78400.gq1")

sage: E.isogeny_class()

## Elliptic curves in class 78400gy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
78400.gq4 78400gy1 [0, 0, 0, 181300, -47334000] [2] 884736 $$\Gamma_0(N)$$-optimal
78400.gq3 78400gy2 [0, 0, 0, -1386700, -508326000] [2, 2] 1769472
78400.gq2 78400gy3 [0, 0, 0, -6874700, 6483386000] [2] 3538944
78400.gq1 78400gy4 [0, 0, 0, -20986700, -37003526000] [2] 3538944

## Rank

sage: E.rank()

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

## Modular form 78400.2.a.gq

sage: E.q_eigenform(10)

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

## 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.