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

• Label: 2.19.b_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:	$1 + x + 34 x^{2} + 19 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.442674189896$, $\pm0.594930043603$
Angle rank:	$2$ (numerical)
Number field:	4.0.369053.1
Galois group:	$D_{4}$
Jacobians:	$22$
Isomorphism classes:	22
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$416$	$156416$	$46746752$	$16878537728$	$6132724936416$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$429$	$6816$	$129513$	$2476771$	$47049366$	$893876697$	$16983718929$	$322687064928$	$6131060238589$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 22 curves (of which all are hyperelliptic):

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

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 4.0.369053.1.

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.ab_bi	$2$	(not in LMFDB)