# Properties

 Label 1638.s Number of curves $4$ Conductor $1638$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("s1")

E.isogeny_class()

## Elliptic curves in class 1638.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.s1 1638q3 $$[1, -1, 1, -17609, -894787]$$ $$828279937799497/193444524$$ $$141021057996$$ $$[2]$$ $$3072$$ $$1.1299$$
1638.s2 1638q2 $$[1, -1, 1, -1229, -10267]$$ $$281397674377/96589584$$ $$70413806736$$ $$[2, 2]$$ $$1536$$ $$0.78338$$
1638.s3 1638q1 $$[1, -1, 1, -509, 4421]$$ $$19968681097/628992$$ $$458535168$$ $$[4]$$ $$768$$ $$0.43680$$ $$\Gamma_0(N)$$-optimal
1638.s4 1638q4 $$[1, -1, 1, 3631, -74419]$$ $$7264187703863/7406095788$$ $$-5399043829452$$ $$[2]$$ $$3072$$ $$1.1299$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1638.2.a.s

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2 q^{5} - q^{7} + q^{8} + 2 q^{10} + 4 q^{11} + q^{13} - q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.