# Properties

 Label 1638.f Number of curves $3$ Conductor $1638$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 1638.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.f1 1638j3 $$[1, -1, 0, -140967, 20406843]$$ $$-424962187484640625/182$$ $$-132678$$ $$$$ $$3240$$ $$1.2298$$
1638.f2 1638j2 $$[1, -1, 0, -1737, 28485]$$ $$-795309684625/6028568$$ $$-4394826072$$ $$$$ $$1080$$ $$0.68046$$
1638.f3 1638j1 $$[1, -1, 0, 63, 189]$$ $$37595375/46592$$ $$-33965568$$ $$[]$$ $$360$$ $$0.13115$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1638.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{7} - q^{8} + 3 q^{11} + q^{13} - q^{14} + q^{16} + 2 q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 