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

• Label: 2.19.c_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:	$( 1 + 19 x^{2} )( 1 + 2 x + 19 x^{2} )$
	$1 + 2 x + 38 x^{2} + 38 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.573681533379$
Angle rank:	$1$ (numerical)
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$440$	$158400$	$46332440$	$16833484800$	$6138207330200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$434$	$6754$	$129166$	$2478982$	$47062082$	$893812018$	$16983374686$	$322688687926$	$6131067856274$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

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.c 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.c : \(\Q(\sqrt{-2}) \).

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

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

• 1.361.bi : \(\Q(\sqrt{-2}) \).
• 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.ac_bm	$2$	(not in LMFDB)