# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 1638.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.d1 1638i3 $$[1, -1, 0, -3753, -87561]$$ $$8020417344913/187278$$ $$136525662$$ $$$$ $$1536$$ $$0.67328$$
1638.d2 1638i2 $$[1, -1, 0, -243, -1215]$$ $$2181825073/298116$$ $$217326564$$ $$[2, 2]$$ $$768$$ $$0.32670$$
1638.d3 1638i1 $$[1, -1, 0, -63, 189]$$ $$38272753/4368$$ $$3184272$$ $$$$ $$384$$ $$-0.019868$$ $$\Gamma_0(N)$$-optimal
1638.d4 1638i4 $$[1, -1, 0, 387, -6885]$$ $$8780064047/32388174$$ $$-23610978846$$ $$$$ $$1536$$ $$0.67328$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1638.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2 q^{5} + q^{7} - q^{8} + 2 q^{10} + 4 q^{11} - q^{13} - q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 