# Properties

 Label 2.19.ak_ch 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: $( 1 - 7 x + 19 x^{2} )( 1 - 3 x + 19 x^{2} )$ Frobenius angles: $\pm0.203259864187$, $\pm0.388176076177$ 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:

• $y^2=2x^6+8x^5+15x^4+17x^3+15x^2+8x+2$
• $y^2=9x^6+14x^5+16x^4+x^3+6x^2+4x+3$
• $y^2=3x^6+17x^5+10x^4+10x^3+10x^2+17x+3$
• $y^2=2x^6+11x^5+17x^4+8x^3+17x^2+11x+2$
• $y^2=13x^6+x^5+4x^4+8x^3+10x^2+9x+3$
• $y^2=10x^6+4x^5+15x^4+14x^3+18x^2+12x+14$
• $y^2=18x^6+12x^5+13x^4+11x^2+4x+5$
• $y^2=2x^6+17x^5+2x^4+8x^3+2x^2+17x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 221 137241 48439664 17046567369 6131185293701 2213482610416896 799053232013574941 288443891852288625609 104127042519008111932784 37589914962904844325667401

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 380 7060 130804 2476150 47049446 893923810 16983708964 322686743980 6131056717100

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ah $\times$ 1.19.ad 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 are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ae_r $2$ (not in LMFDB) 2.19.e_r $2$ (not in LMFDB) 2.19.k_ch $2$ (not in LMFDB) 2.19.ae_bp $3$ (not in LMFDB) 2.19.f_o $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ae_r $2$ (not in LMFDB) 2.19.e_r $2$ (not in LMFDB) 2.19.k_ch $2$ (not in LMFDB) 2.19.ae_bp $3$ (not in LMFDB) 2.19.f_o $3$ (not in LMFDB) 2.19.al_ck $6$ (not in LMFDB) 2.19.af_o $6$ (not in LMFDB) 2.19.ac_bj $6$ (not in LMFDB) 2.19.c_bj $6$ (not in LMFDB) 2.19.e_bp $6$ (not in LMFDB) 2.19.l_ck $6$ (not in LMFDB)