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

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

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 42 x^{2} - 171 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0659896528252$, $\pm0.482872009315$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-102 +10 \sqrt{65}})\)
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	16
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$224$	$130816$	$46315136$	$16828693504$	$6125033342624$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$365$	$6752$	$129129$	$2473661$	$47054486$	$893886599$	$16983349009$	$322687499168$	$6131072894525$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-102 +10 \sqrt{65}})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.j_bq	$2$	(not in LMFDB)