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

• Label: 2.19.a_aba
• 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:	$( 1 - 8 x + 19 x^{2} )( 1 + 8 x + 19 x^{2} )$
	$1 - 26 x^{2} + 361 x^{4}$
Frobenius angles:	$\pm0.130073469147$, $\pm0.869926530853$
Angle rank:	$1$ (numerical)
Jacobians:	$15$
Isomorphism classes:	242

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$336$	$112896$	$47056464$	$16995815424$	$6131069159376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$310$	$6860$	$130414$	$2476100$	$47067046$	$893871740$	$16984080094$	$322687697780$	$6131072060950$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

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

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.ai $\times$ 1.19.i 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.ai : \(\Q(\sqrt{-3}) \).
• 1.19.i : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{19^{2}}$ is 1.361.aba^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.aq_dy	$2$	(not in LMFDB)
2.19.q_dy	$2$	(not in LMFDB)
2.19.ap_dq	$3$	(not in LMFDB)
2.19.aj_bu	$3$	(not in LMFDB)
2.19.ag_bf	$3$	(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)