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

• Label: 2.19.a_ak
• 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:	$1 - 10 x^{2} + 361 x^{4}$
Frobenius angles:	$\pm0.207617990860$, $\pm0.792382009140$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$34$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$352$	$123904$	$47055712$	$17146331136$	$6131061446752$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$342$	$6860$	$131566$	$2476100$	$47065542$	$893871740$	$16983310558$	$322687697780$	$6131056635702$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 34 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{3}, \sqrt{-7})\).

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

The base change of $A$ to $\F_{19^{2}}$ is 1.361.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.a_k	$4$	(not in LMFDB)
2.19.am_cp	$12$	(not in LMFDB)
2.19.m_cp	$12$	(not in LMFDB)