# Properties

 Label 16830.bk Number of curves $8$ Conductor $16830$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bk1")

sage: E.isogeny_class()

## Elliptic curves in class 16830.bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16830.bk1 16830cd8 $$[1, -1, 1, -1811128343, 29667338187311]$$ $$901247067798311192691198986281/552431869440$$ $$402722832821760$$ $$[6]$$ $$5308416$$ $$3.5101$$
16830.bk2 16830cd7 $$[1, -1, 1, -113955863, 457030494767]$$ $$224494757451893010998773801/6152490825146276160000$$ $$4485165811531635320640000$$ $$[6]$$ $$5308416$$ $$3.5101$$
16830.bk3 16830cd6 $$[1, -1, 1, -113195543, 463573200431]$$ $$220031146443748723000125481/172266701724057600$$ $$125582425556837990400$$ $$[2, 6]$$ $$2654208$$ $$3.1635$$
16830.bk4 16830cd5 $$[1, -1, 1, -22364168, 40683483731]$$ $$1696892787277117093383481/1440538624914939000$$ $$1050152657562990531000$$ $$[2]$$ $$1769472$$ $$2.9608$$
16830.bk5 16830cd4 $$[1, -1, 1, -14646488, -21340722733]$$ $$476646772170172569823801/5862293314453125000$$ $$4273611826236328125000$$ $$[2]$$ $$1769472$$ $$2.9608$$
16830.bk6 16830cd3 $$[1, -1, 1, -7027223, 7346695727]$$ $$-52643812360427830814761/1504091705903677440$$ $$-1096482853603780853760$$ $$[6]$$ $$1327104$$ $$2.8169$$
16830.bk7 16830cd2 $$[1, -1, 1, -1709168, 331875731]$$ $$757443433548897303481/373234243041000000$$ $$272087763176889000000$$ $$[2, 2]$$ $$884736$$ $$2.6142$$
16830.bk8 16830cd1 $$[1, -1, 1, 390352, 39622547]$$ $$9023321954633914439/6156756739584000$$ $$-4488275663156736000$$ $$[2]$$ $$442368$$ $$2.2676$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 16830.bk have rank $$0$$.

## Complex multiplication

The elliptic curves in class 16830.bk do not have complex multiplication.

## Modular form 16830.2.a.bk

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.