Abelian variety isogeny class 2.19.ak_cc over $\F_{19}$

• Label: 2.19.ak_cc
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 19 x^{2} )( 1 - 2 x + 19 x^{2} )$
	$1 - 10 x + 54 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.130073469147$, $\pm0.426318466621$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$216$	$133056$	$47396664$	$16933238784$	$6127476737976$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19}$.

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:

• 1.19.ai : \(\Q(\sqrt{-3}) \).
• 1.19.ac : \(\Q(\sqrt{-2}) \).

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)