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

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

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - 34 x^{2} + 361 x^{4}$
Frobenius angles:	$\pm0.0736815333795$, $\pm0.926318466621$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$5$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$328$	$107584$	$47043400$	$16870892544$	$6131069611528$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$294$	$6860$	$129454$	$2476100$	$47040918$	$893871740$	$16983707614$	$322687697780$	$6131072965254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{19^{2}}$ is 1.361.abi^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

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.ae_bq	$4$	(not in LMFDB)
2.19.a_bi	$4$	(not in LMFDB)
2.19.e_bq	$4$	(not in LMFDB)