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

• Label: 2.19.ao_di
• 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:	$( 1 - 8 x + 19 x^{2} )( 1 - 6 x + 19 x^{2} )$
	$1 - 14 x + 86 x^{2} - 266 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.130073469147$, $\pm0.258380448083$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$168$	$122304$	$47532744$	$17083422720$	$6139358981448$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$338$	$6930$	$131086$	$2479446$	$47054306$	$893879874$	$16983566686$	$322688082150$	$6131070448178$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.ag 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.ag : \(\Q(\sqrt{-10}) \).

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.ac_ak	$2$	(not in LMFDB)
2.19.c_ak	$2$	(not in LMFDB)
2.19.o_di	$2$	(not in LMFDB)
2.19.af_bg	$3$	(not in LMFDB)
2.19.b_ae	$3$	(not in LMFDB)