# Properties

 Label 38640.x Number of curves $2$ Conductor $38640$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 38640.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.x1 38640i1 $$[0, -1, 0, -96, 336]$$ $$96550276/16905$$ $$17310720$$ $$$$ $$8704$$ $$0.10873$$ $$\Gamma_0(N)$$-optimal
38640.x2 38640i2 $$[0, -1, 0, 184, 1680]$$ $$334568302/833175$$ $$-1706342400$$ $$$$ $$17408$$ $$0.45530$$

## Rank

sage: E.rank()

The elliptic curves in class 38640.x have rank $$1$$.

## Complex multiplication

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

## Modular form 38640.2.a.x

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{7} + q^{9} - 2q^{13} + q^{15} + 8q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 