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

• Label: 2.19.c_ap
• 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:	$1 + 2 x - 15 x^{2} + 38 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.240348200046$, $\pm0.907014866713$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$5$
Isomorphism classes:	15
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$387$	$118809$	$48525156$	$17040180825$	$6138208207827$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$328$	$7072$	$130756$	$2478982$	$47050846$	$893812018$	$16983490756$	$322685717488$	$6131069611528$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{3}}$.

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{19^{3}}$ is 1.6859.ec^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.ac_ap	$2$	(not in LMFDB)
2.19.ae_bq	$3$	(not in LMFDB)
2.19.a_bi	$6$	(not in LMFDB)