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

• Label: 2.19.f_bi
• 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:	$1 + 5 x + 34 x^{2} + 95 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.474356379535$, $\pm0.726916281199$
Angle rank:	$2$ (numerical)
Number field:	4.0.1380101.1
Galois group:	$D_{4}$
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$496$	$146816$	$46368064$	$16984262144$	$6124820650576$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$405$	$6760$	$130329$	$2473575$	$47052246$	$893961205$	$16983136689$	$322687094680$	$6131072972725$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

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 4.0.1380101.1.

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.af_bi	$2$	(not in LMFDB)