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

• Label: 2.19.ad_ac
• 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:	$( 1 - 8 x + 19 x^{2} )( 1 + 5 x + 19 x^{2} )$
	$1 - 3 x - 2 x^{2} - 57 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.130073469147$, $\pm0.694430027533$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$300$	$126000$	$45586800$	$17061912000$	$6135298306500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$17$	$349$	$6644$	$130921$	$2477807$	$47044582$	$893983373$	$16983776401$	$322687870796$	$6131074035949$

Jacobians and polarizations

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

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

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.f 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.f : \(\Q(\sqrt{-51}) \).

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.an_da	$2$	(not in LMFDB)
2.19.d_ac	$2$	(not in LMFDB)
2.19.n_da	$2$	(not in LMFDB)
2.19.g_br	$3$	(not in LMFDB)
2.19.m_cv	$3$	(not in LMFDB)