# Properties

 Label 2.19.ak_cc 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: $( 1 - 8 x + 19 x^{2} )( 1 - 2 x + 19 x^{2} )$ Frobenius angles: $\pm0.130073469147$, $\pm0.426318466621$ Angle rank: $2$ (numerical) Jacobians: 16

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 16 curves, and hence is principally polarizable:

• $y^2=10x^6+5x^5+15x^4+9x^2+6x+15$
• $y^2=10x^5+8x^4+9x^3+8x^2+10x$
• $y^2=12x^6+4x^5+13x^3+4x+12$
• $y^2=3x^6+10x^5+12x^4+17x^3+18x^2+13x+3$
• $y^2=x^6+10x^5+14x^4+4x^3+15x^2+3x+12$
• $y^2=8x^6+x^5+11x^4+x^3+14x^2+13x+15$
• $y^2=12x^6+4x^5+7x^4+14x^3+3x^2+4x+8$
• $y^2=13x^6+12x^4+x^3+12x^2+13$
• $y^2=12x^6+4x^5+10x^4+12x^3+12x^2+5x+18$
• $y^2=2x^6+6x^5+3x^4+10x^3+7x^2+11x+12$
• $y^2=14x^6+10x^5+17x^3+x^2+1$
• $y^2=8x^6+15x^5+3x^4+6x^3+2x^2+6x+10$
• $y^2=4x^6+14x^5+8x^4+16x^3+2x^2+3x+13$
• $y^2=12x^6+4x^5+7x^4+9x^3+16x^2+13x+17$
• $y^2=3x^6+5x^5+4x^4+16x^3+x^2+8x+5$
• $y^2=10x^6+14x^5+7x^4+9x^3+7x^2+14x+10$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 216 133056 47396664 16933238784 6127476737976 2213929646824896 799111425953312856 288447031986733154304 104127345957440010149784 37589970685374618644016576

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 10 370 6910 129934 2474650 47058946 893988910 16983893854 322687684330 6131065805650

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The isogeny class factors as 1.19.ai $\times$ 1.19.ac 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.ag_w $2$ (not in LMFDB) 2.19.g_w $2$ (not in LMFDB) 2.19.k_cc $2$ (not in LMFDB) 2.19.ab_bk $3$ (not in LMFDB) 2.19.f_y $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.19.ag_w $2$ (not in LMFDB) 2.19.g_w $2$ (not in LMFDB) 2.19.k_cc $2$ (not in LMFDB) 2.19.ab_bk $3$ (not in LMFDB) 2.19.f_y $3$ (not in LMFDB) 2.19.aj_ca $6$ (not in LMFDB) 2.19.af_y $6$ (not in LMFDB) 2.19.ad_bo $6$ (not in LMFDB) 2.19.b_bk $6$ (not in LMFDB) 2.19.d_bo $6$ (not in LMFDB) 2.19.j_ca $6$ (not in LMFDB)