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

• Label: 2.19.a_abm
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$( 1 - 19 x^{2} )^{2}$
	$1 - 38 x^{2} + 361 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{19}) \)
Galois group:	$C_2$
Jacobians:	$2$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$104976$	$47032164$	$16796160000$	$6131061305604$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$286$	$6860$	$128878$	$2476100$	$47018446$	$893871740$	$16983041758$	$322687697780$	$6131056353406$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=x^6+10 x^3+12$
• $y^2=8 x^6+7 x^5+7 x^4+5 x^2+10 x+12$

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 the quaternion algebra over \(\Q(\sqrt{19}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{19^{2}}$ is 1.361.abm^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.a_t	$3$	(not in LMFDB)
2.19.a_bm	$4$	(not in LMFDB)
2.19.a_a	$8$	(not in LMFDB)