# Properties

 Label 38640.z Number of curves $2$ Conductor $38640$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 38640.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.z1 38640h1 $$[0, -1, 0, -476, -3840]$$ $$46689225424/36225$$ $$9273600$$ $$[2]$$ $$16384$$ $$0.26837$$ $$\Gamma_0(N)$$-optimal
38640.z2 38640h2 $$[0, -1, 0, -376, -5600]$$ $$-5756278756/10498005$$ $$-10749957120$$ $$[2]$$ $$32768$$ $$0.61494$$

## Rank

sage: E.rank()

The elliptic curves in class 38640.z have rank $$0$$.

## Complex multiplication

The elliptic curves in class 38640.z do not have complex multiplication.

## Modular form 38640.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{7} + q^{9} + 6q^{11} - 4q^{13} + q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.