# Properties

 Label 6699b Number of curves $4$ Conductor $6699$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 6699b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6699.d3 6699b1 $$[1, 1, 0, -49, 112]$$ $$13430356633/180873$$ $$180873$$ $$$$ $$832$$ $$-0.18454$$ $$\Gamma_0(N)$$-optimal
6699.d2 6699b2 $$[1, 1, 0, -94, -185]$$ $$93391282153/44876601$$ $$44876601$$ $$[2, 2]$$ $$1664$$ $$0.16203$$
6699.d1 6699b3 $$[1, 1, 0, -1249, -17510]$$ $$215751695207833/163381911$$ $$163381911$$ $$$$ $$3328$$ $$0.50861$$
6699.d4 6699b4 $$[1, 1, 0, 341, -968]$$ $$4365111505607/3058314567$$ $$-3058314567$$ $$$$ $$3328$$ $$0.50861$$

## Rank

sage: E.rank()

The elliptic curves in class 6699b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 6699b do not have complex multiplication.

## Modular form6699.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - q^{7} - 3q^{8} + q^{9} + 2q^{10} + q^{11} + q^{12} - 2q^{13} - q^{14} - 2q^{15} - q^{16} - 2q^{17} + q^{18} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 