# Properties

 Label 1638.e Number of curves $4$ Conductor $1638$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 1638.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.e1 1638c3 $$[1, -1, 0, -1932, -14896]$$ $$40530337875/18264064$$ $$359491571712$$ $$$$ $$1728$$ $$0.91259$$
1638.e2 1638c1 $$[1, -1, 0, -957, 11637]$$ $$3592121380875/246064$$ $$6643728$$ $$$$ $$576$$ $$0.36329$$ $$\Gamma_0(N)$$-optimal
1638.e3 1638c2 $$[1, -1, 0, -897, 13113]$$ $$-2958077788875/946054564$$ $$-25543473228$$ $$$$ $$1152$$ $$0.70986$$
1638.e4 1638c4 $$[1, -1, 0, 6708, -116848]$$ $$1695802078125/1272491584$$ $$-25046451847872$$ $$$$ $$3456$$ $$1.2592$$

## Rank

sage: E.rank()

The elliptic curves in class 1638.e have rank $$1$$.

## Complex multiplication

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

## Modular form1638.2.a.e

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 