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

• Label: 2.19.d_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:	$( 1 + 19 x^{2} )( 1 + 3 x + 19 x^{2} )$
	$1 + 3 x + 38 x^{2} + 57 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.611823923823$
Angle rank:	$1$ (numerical)
Jacobians:	$8$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$460$	$156400$	$46071760$	$16874308800$	$6138729787300$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$429$	$6716$	$129481$	$2479193$	$47052582$	$893834027$	$16983548881$	$322687674884$	$6131066595549$

Jacobians and polarizations

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

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

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.a $\times$ 1.19.d 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.a : \(\Q(\sqrt{-19}) \).
• 1.19.d : \(\Q(\sqrt{-67}) \).

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

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

• 1.361.bd : \(\Q(\sqrt{-67}) \).
• 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.ad_bm	$2$	(not in LMFDB)