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

• Label: 2.19.c_bn
• 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:	$( 1 + x + 19 x^{2} )^{2}$
	$1 + 2 x + 39 x^{2} + 38 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.536593197520$, $\pm0.536593197520$
Angle rank:	$1$ (numerical)
Jacobians:	$8$
Cyclic group of points:	no
Non-cyclic primes:	$3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$441$	$159201$	$46294416$	$16815605625$	$6139547351721$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$436$	$6748$	$129028$	$2479522$	$47067046$	$893785558$	$16983247108$	$322689651172$	$6131070307156$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.b^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.ac_bn	$2$	(not in LMFDB)
2.19.a_bl	$2$	(not in LMFDB)
2.19.aq_dy	$3$	(not in LMFDB)
2.19.ah_be	$3$	(not in LMFDB)
2.19.ab_as	$3$	(not in LMFDB)
2.19.i_bt	$3$	(not in LMFDB)
2.19.o_dj	$3$	(not in LMFDB)