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

• Label: 2.19.ai_bm
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 19 x^{2} )( 1 + 19 x^{2} )$
	$1 - 8 x + 38 x^{2} - 152 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.130073469147$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$30$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$240$	$134400$	$46675440$	$16895692800$	$6134616985200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$374$	$6804$	$129646$	$2477532$	$47070182$	$893929188$	$16983560926$	$322688674476$	$6131074111574$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ai $\times$ 1.19.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.19.ai : \(\Q(\sqrt{-3}) \).
• 1.19.a : \(\Q(\sqrt{-19}) \).

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

The base change of $A$ to $\F_{19^{2}}$ is 1.361.aba $\times$ 1.361.bm. The endomorphism algebra for each factor is:

• 1.361.aba : \(\Q(\sqrt{-3}) \).
• 1.361.bm : the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.

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.i_bm	$2$	(not in LMFDB)
2.19.b_bm	$3$	(not in LMFDB)
2.19.h_bm	$3$	(not in LMFDB)