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

• Label: 2.19.i_bx
• 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:	$1 + 8 x + 49 x^{2} + 152 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.564853593150$, $\pm0.753721816602$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-55 +8 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$16$
Isomorphism classes:	16
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$571$	$143321$	$45643456$	$17012919305$	$6131877069571$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$396$	$6652$	$130548$	$2476428$	$47051526$	$893851252$	$16983318948$	$322689678148$	$6131063452156$

Jacobians and polarizations

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

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

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 \(\Q(\sqrt{-55 +8 \sqrt{5}})\).

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.ai_bx	$2$	(not in LMFDB)