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

• Label: 2.19.ak_by
• 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:	$1 - 10 x + 50 x^{2} - 190 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0511346671405$, $\pm0.448865332859$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{13})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	11
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$212$	$129744$	$46568132$	$16833505536$	$6120057514052$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$362$	$6790$	$129166$	$2471650$	$47045882$	$893899870$	$16983416158$	$322686428890$	$6131066257802$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

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

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 \(\Q(i, \sqrt{13})\).

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

The base change of $A$ to $\F_{19^{4}}$ is 1.130321.awg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 2.361.a_awg and its endomorphism algebra is \(\Q(i, \sqrt{13})\).

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.k_by	$2$	(not in LMFDB)
2.19.a_am	$8$	(not in LMFDB)
2.19.a_m	$8$	(not in LMFDB)