# Properties

 Label 2.19.ak_cl Base Field $\F_{19}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $2$ L-polynomial: $( 1 - 5 x + 19 x^{2} )^{2}$ Frobenius angles: $\pm0.305569972467$, $\pm0.305569972467$ Angle rank: $1$ (numerical) Jacobians: 6

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

• $y^2=6x^6+13x^5+7x^4+4x^3+9x^2+4x+15$
• $y^2=13x^6+9x^5+6x^4+2x^3+6x^2+9x+13$
• $y^2=14x^6+11x^5+2x^4+2x^2+11x+14$
• $y^2=12x^6+8x^5+4x^4+6x^3+4x^2+8x+12$
• $y^2=x^6+17x^3+11$
• $y^2=10x^6+11x^5+12x^4+5x^3+12x^2+11x+10$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 225 140625 49280400 17128265625 6129709430625 2212197156000000 798909821626178025 288439879410444515625 104127868974500422131600 37590033254766349306640625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 388 7180 131428 2475550 47022118 893763370 16983472708 322689305140 6131076010948

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$
All geometric endomorphisms are defined over $\F_{19}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.a_n $2$ (not in LMFDB) 2.19.k_cl $2$ (not in LMFDB) 2.19.f_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.a_n $2$ (not in LMFDB) 2.19.k_cl $2$ (not in LMFDB) 2.19.f_g $3$ (not in LMFDB) 2.19.a_an $4$ (not in LMFDB) 2.19.af_g $6$ (not in LMFDB)