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

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

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 19 x^{2} )^{2}$
	$1 - 14 x + 87 x^{2} - 266 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.203259864187$, $\pm0.203259864187$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Cyclic group of points:	no
Non-cyclic primes:	$13$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$169$	$123201$	$47831056$	$17140831929$	$6146645853049$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$340$	$6972$	$131524$	$2482386$	$47067046$	$893900454$	$16983361924$	$322685744388$	$6131056405300$

Jacobians and polarizations

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

• $y^2=10 x^6+7 x^5+6 x^4+13 x^3+6 x^2+7 x+10$
• $y^2=2 x^6+2 x^3+14$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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_al	$2$	(not in LMFDB)
2.19.o_dj	$2$	(not in LMFDB)
2.19.ai_bt	$3$	(not in LMFDB)
2.19.ac_bn	$3$	(not in LMFDB)
2.19.b_as	$3$	(not in LMFDB)
2.19.h_be	$3$	(not in LMFDB)
2.19.q_dy	$3$	(not in LMFDB)