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

• Label: 2.19.ae_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:	$( 1 - 4 x + 19 x^{2} )( 1 + 19 x^{2} )$
	$1 - 4 x + 38 x^{2} - 76 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.348268167089$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$36$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$320$	$153600$	$48184640$	$16920576000$	$6125712929600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$422$	$7024$	$129838$	$2473936$	$47046422$	$893860144$	$16983506398$	$322688734096$	$6131071479302$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

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

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.ae $\times$ 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.ae : \(\Q(\sqrt{-15}) \).
• 1.19.a : \(\Q(\sqrt{-19}) \).

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

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

• 1.361.w : \(\Q(\sqrt{-15}) \).
• 1.361.bm : the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.

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.e_bm	$2$	(not in LMFDB)