# Properties

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

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("ey1")

sage: E.isogeny_class()

## Elliptic curves in class 78400.ey

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78400.ey1 78400gz4 $$[0, 0, 0, -1465100, 682570000]$$ $$1443468546/7$$ $$1686616064000000$$ $$[2]$$ $$786432$$ $$2.1222$$
78400.ey2 78400gz3 $$[0, 0, 0, -289100, -47334000]$$ $$11090466/2401$$ $$578509309952000000$$ $$[2]$$ $$786432$$ $$2.1222$$
78400.ey3 78400gz2 $$[0, 0, 0, -93100, 10290000]$$ $$740772/49$$ $$5903156224000000$$ $$[2, 2]$$ $$393216$$ $$1.7756$$
78400.ey4 78400gz1 $$[0, 0, 0, 4900, 686000]$$ $$432/7$$ $$-210827008000000$$ $$[2]$$ $$196608$$ $$1.4291$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 78400.2.a.ey

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.