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

• Label: 2.19.b_aw
• 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:	$1 + x - 22 x^{2} + 19 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.186610687577$, $\pm0.896624766491$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-33 +2 \sqrt{241}})\)
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	18
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$360$	$115200$	$47913120$	$17024256000$	$6138278623800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$317$	$6984$	$130633$	$2479011$	$47063702$	$893867289$	$16983791953$	$322685981496$	$6131066661677$

Jacobians and polarizations

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

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

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 \(\Q(\sqrt{-33 +2 \sqrt{241}})\).

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.ab_aw	$2$	(not in LMFDB)