# Properties

 Label 2.19.ak_ck 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: $( 1 - 6 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$ Frobenius angles: $\pm0.258380448083$, $\pm0.348268167089$ Angle rank: $2$ (numerical) Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=4x^6+17x^5+11x^3+17x+4$
• $y^2=4x^6+14x^5+x^4+17x^2+18x+6$
• $y^2=9x^6+18x^5+5x^4+17x^3+5x^2+18x+9$
• $y^2=x^6+17x^5+7x^4+3x^3+17x^2+x+11$
• $y^2=14x^6+15x^5+10x^4+11x^3+10x^2+15x+14$
• $y^2=8x^6+8x^5+x^4+5x^3+4x^2+14x+18$
• $y^2=10x^6+18x^5+10x^4+14x^3+10x^2+18x+10$
• $y^2=13x^6+4x^5+17x^4+x^2+x+15$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 224 139776 49069664 17108582400 6130448043104 2212593545949696 798952242414789344 288440549357971046400 104127493568053717359584 37589983010397427949154816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 386 7150 131278 2475850 47030546 893810830 16983512158 322688141770 6131067815906

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ag $\times$ 1.19.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{19}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ac_o $2$ (not in LMFDB) 2.19.c_o $2$ (not in LMFDB) 2.19.k_ck $2$ (not in LMFDB)