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

• Label: 2.19.d_bc
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 2 x + 19 x^{2} )( 1 + 5 x + 19 x^{2} )$
	$1 + 3 x + 28 x^{2} + 57 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.426318466621$, $\pm0.694430027533$
Angle rank:	$2$ (numerical)
Jacobians:	$24$
Isomorphism classes:	150
Cyclic group of points:	no
Non-cyclic primes:	$3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$450$	$148500$	$46672200$	$16999092000$	$6124615224750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$409$	$6806$	$130441$	$2473493$	$47036482$	$893985647$	$16983590161$	$322685903954$	$6131067780649$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ac $\times$ 1.19.f 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.ac : \(\Q(\sqrt{-2}) \).
• 1.19.f : \(\Q(\sqrt{-51}) \).

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.ah_bw	$2$	(not in LMFDB)
2.19.ad_bc	$2$	(not in LMFDB)
2.19.h_bw	$2$	(not in LMFDB)