# Properties

 Label 1638.p Number of curves $4$ Conductor $1638$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 1638.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.p1 1638m3 $$[1, -1, 1, -8615, -305585]$$ $$3592121380875/246064$$ $$4843277712$$ $$$$ $$1728$$ $$0.91259$$
1638.p2 1638m4 $$[1, -1, 1, -8075, -345977]$$ $$-2958077788875/946054564$$ $$-18621191983212$$ $$$$ $$3456$$ $$1.2592$$
1638.p3 1638m1 $$[1, -1, 1, -215, 623]$$ $$40530337875/18264064$$ $$493129728$$ $$$$ $$576$$ $$0.36329$$ $$\Gamma_0(N)$$-optimal
1638.p4 1638m2 $$[1, -1, 1, 745, 4079]$$ $$1695802078125/1272491584$$ $$-34357272768$$ $$$$ $$1152$$ $$0.70986$$

## Rank

sage: E.rank()

The elliptic curves in class 1638.p have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1638.p do not have complex multiplication.

## Modular form1638.2.a.p

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{7} + q^{8} + q^{13} + q^{14} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 