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

• Label: 2.19.a_a
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 + 361 x^{4}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(i, \sqrt{38})\)
Galois group:	$C_2^2$
Jacobians:	$32$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$362$	$131044$	$47045882$	$17172529936$	$6131066257802$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$362$	$6860$	$131766$	$2476100$	$47045882$	$893871740$	$16983041758$	$322687697780$	$6131066257802$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{4}}$.

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{38})\).

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

The base change of $A$ to $\F_{19^{4}}$ is 1.130321.bbu^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{19^{2}}$

  The base change of $A$ to $\F_{19^{2}}$ is 1.361.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.a_abm	$8$	(not in LMFDB)
2.19.a_bm	$8$	(not in LMFDB)
2.19.a_at	$24$	(not in LMFDB)