Abelian variety isogeny class 3.19.a_cf_a over $\F_{19}$

• Label: 3.19.a_cf_a
• Base field: $\F_{19}$
• Dimension: $3$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$3$
L-polynomial:	$( 1 + 19 x^{2} )^{3}$
	$1 + 57 x^{2} + 1083 x^{4} + 6859 x^{6}$
Frobenius angles:	$\pm0.5$, $\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

This isogeny class is not 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:	$[1/2, 1/2, 1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8000$	$64000000$	$322828856000$	$2176782336000000$	$15181145423081000000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$476$	$6860$	$128156$	$2476100$	$47087036$	$893871740$	$16982781116$	$322687697780$	$6131081114396$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.a^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-19}) \)$)$

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

The base change of $A$ to $\F_{19^{2}}$ is 1.361.bm^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, 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
3.19.a_a_a	$3$	(not in LMFDB)
3.19.a_at_a	$4$	(not in LMFDB)
3.19.a_t_a	$8$	(not in LMFDB)