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

• Label: 2.19.g_be
• 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:	$1 + 6 x + 30 x^{2} + 114 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.458878150664$, $\pm0.804406540814$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-174 +2 \sqrt{17}})\)
Galois group:	$D_{4}$
Jacobians:	$48$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$512$	$139264$	$47180288$	$16974610432$	$6118612470272$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$386$	$6878$	$130254$	$2471066$	$47067410$	$893894846$	$16983376990$	$322687483994$	$6131062716386$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-174 +2 \sqrt{17}})\).

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.ag_be	$2$	(not in LMFDB)