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

• Label: 2.19.a_u
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 + 20 x^{2} + 361 x^{4}$
Frobenius angles:	$\pm0.338213510720$, $\pm0.661786489280$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-29})\)
Galois group:	$C_2^2$
Jacobians:	$9$
Cyclic group of points:	yes

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$382$	$145924$	$47032222$	$17067854736$	$6131068049902$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$402$	$6860$	$130966$	$2476100$	$47018562$	$893871740$	$16983876958$	$322687697780$	$6131069842002$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{2}}$.

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-29})\).

Endomorphism algebra over $\overline{\F}_{19}$

The base change of $A$ to $\F_{19^{2}}$ is 1.361.u^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-29}) \)$)$

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.a_au	$4$	(not in LMFDB)
2.19.ag_s	$8$	(not in LMFDB)
2.19.g_s	$8$	(not in LMFDB)