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

• Label: 2.19.ag_bf
• 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:	$( 1 - 7 x + 19 x^{2} )( 1 + x + 19 x^{2} )$
	$1 - 6 x + 31 x^{2} - 114 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.203259864187$, $\pm0.536593197520$
Angle rank:	$1$ (numerical)
Jacobians:	$40$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$273$	$140049$	$47056464$	$16977440025$	$6143095577073$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$388$	$6860$	$130276$	$2480954$	$47067046$	$893843006$	$16983304516$	$322687697780$	$6131063356228$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{6}}$.

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ah $\times$ 1.19.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.19.ah : \(\Q(\sqrt{-3}) \).
• 1.19.b : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.pra^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{19^{2}}$
  The base change of $A$ to $\F_{19^{2}}$ is 1.361.al $\times$ 1.361.bl. The endomorphism algebra for each factor is:

  • 1.361.al : \(\Q(\sqrt{-3}) \).
  • 1.361.bl : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{19^{3}}$
  The base change of $A$ to $\F_{19^{3}}$ is 1.6859.ace $\times$ 1.6859.ce. The endomorphism algebra for each factor is:

  • 1.6859.ace : \(\Q(\sqrt{-3}) \).
  • 1.6859.ce : \(\Q(\sqrt{-3}) \).

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.ai_bt	$2$	(not in LMFDB)
2.19.i_bt	$2$	(not in LMFDB)
2.19.ap_dq	$3$	(not in LMFDB)
2.19.aj_bu	$3$	(not in LMFDB)
2.19.a_aba	$3$	(not in LMFDB)
2.19.a_al	$3$	(not in LMFDB)
2.19.a_bl	$3$	(not in LMFDB)
2.19.g_bf	$3$	(not in LMFDB)
2.19.j_bu	$3$	(not in LMFDB)
2.19.p_dq	$3$	(not in LMFDB)