Abelian variety isogeny class 2.121.abk_vl over $\F_{11^{2}}$

• Label: 2.121.abk_vl
• Base field: $\F_{11^{2}}$
• 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:	$\F_{11^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 21 x + 121 x^{2} )( 1 - 15 x + 121 x^{2} )$
	$1 - 36 x + 557 x^{2} - 4356 x^{3} + 14641 x^{4}$
Frobenius angles:	$\pm0.0963413489042$, $\pm0.261189521777$
Angle rank:	$2$ (numerical)
Jacobians:	$44$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10807$	$211719937$	$3139193843968$	$45953732930873625$	$672754298622262473727$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$14460$	$1771994$	$214377556$	$25937590526$	$3138428495022$	$379749820441406$	$45949729773931876$	$5559917315671205354$	$672749995001954839980$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 44 curves (of which all are hyperelliptic):

• $y^2=(4 a+2) x^6+(9 a+5) x^5+(a+9) x^4+(10 a+7) x^3+(a+9) x^2+(9 a+5) x+4 a+2$
• $y^2=(7 a+6) x^6+(4 a+4) x^5+(4 a+5) x^4+(5 a+4) x^3+(7 a+1) x^2+(6 a+6) x+8 a+5$
• $y^2=(8 a+4) x^6+(8 a+7) x^5+a x^4+(9 a+5) x^3+(3 a+7) x^2+5 a x+4$
• $y^2=(3 a+7) x^6+(4 a+1) x^5+(3 a+4) x^4+(3 a+1) x^3+(5 a+3) x^2+3 x+6 a+9$
• $y^2=(9 a+9) x^6+(3 a+9) x^5+(8 a+5) x^4+(a+3) x^3+(10 a+2) x^2+6 a x+4 a+6$
• $y^2=(5 a+8) x^6+(3 a+4) x^5+(7 a+10) x^4+(7 a+1) x^3+(8 a+5) x^2+(6 a+1) x+a+7$
• $y^2=x^6+(5 a+1) x^5+(8 a+8) x^4+x^3+(6 a+7) x^2+(7 a+8) x+2 a+5$
• $y^2=6 a x^6+(9 a+9) x^5+(3 a+6) x^4+8 a x^2+(3 a+10) x+2 a+3$
• $y^2=(3 a+5) x^6+(a+9) x^5+(5 a+10) x^4+(8 a+6) x^3+(4 a+7) x^2+3 x+3 a+7$
• $y^2=(6 a+4) x^6+(5 a+6) x^5+(5 a+5) x^4+10 a x^3+(4 a+9) x^2+(10 a+8) x+10 a+5$
• $y^2=(3 a+7) x^6+(7 a+10) x^5+(10 a+2) x^4+2 a x^3+(2 a+10) x^2+(a+5) x+6 a+5$
• $y^2=(9 a+10) x^6+(8 a+1) x^5+(10 a+9) x^4+(7 a+9) x^3+(10 a+9) x^2+(8 a+1) x+9 a+10$
• $y^2=(8 a+4) x^6+(2 a+7) x^5+(2 a+6) x^4+3 x^3+4 x^2+(7 a+6) x+5 a+6$
• $y^2=(2 a+10) x^6+a x^5+(6 a+3) x^4+(9 a+4) x^3+(6 a+6) x^2+(4 a+8) x+2 a+10$
• $y^2=(a+7) x^6+10 x^5+(10 a+2) x^4+(6 a+7) x^3+(4 a+4) x^2+(7 a+3) x+a+5$
• $y^2=(a+6) x^6+(8 a+8) x^5+(10 a+5) x^4+2 x^3+(9 a+2) x^2+(9 a+4) x+2 a+10$
• $y^2=(7 a+8) x^6+(2 a+10) x^5+(6 a+5) x^4+(7 a+9) x^3+(6 a+5) x^2+(2 a+10) x+7 a+8$
• $y^2=(2 a+9) x^6+(4 a+2) x^5+(4 a+7) x^4+(2 a+4) x^3+(4 a+7) x^2+(4 a+2) x+2 a+9$
• $y^2=(2 a+3) x^6+(6 a+10) x^5+(7 a+6) x^4+4 a x^3+3 x^2+(8 a+5) x+3 a+7$
• $y^2=9 a x^6+(2 a+1) x^5+(7 a+9) x^4+4 x^3+(2 a+3) x^2+7 x+5 a+2$
• and

• $y^2=(5 a+10) x^6+(5 a+10) x^5+(8 a+3) x^4+(a+5) x^3+(6 a+6) x^2+(a+5) x+3 a+1$
• $y^2=(10 a+10) x^6+(5 a+7) x^5+(6 a+8) x^4+(6 a+5) x^3+(2 a+9) x^2+7 x+9 a+5$
• $y^2=(4 a+2) x^6+(7 a+2) x^5+(7 a+4) x^4+(a+2) x^3+(7 a+4) x^2+(7 a+2) x+4 a+2$
• $y^2=(a+6) x^6+(8 a+8) x^5+7 x^4+(5 a+2) x^3+(9 a+7) x^2+(8 a+3) x+4 a+4$
• $y^2=(9 a+3) x^6+(4 a+5) x^5+2 a x^4+2 x^3+10 a x^2+(7 a+3) x+5 a+8$
• $y^2=(9 a+9) x^6+(10 a+8) x^5+(10 a+8) x^4+(10 a+2) x^3+(8 a+6) x^2+(2 a+7) x+10 a$
• $y^2=8 x^6+(a+5) x^5+(4 a+1) x^4+9 a x^3+(4 a+1) x^2+(a+5) x+8$
• $y^2=(6 a+6) x^6+(a+6) x^5+(2 a+10) x^4+(4 a+4) x^3+(2 a+10) x^2+(a+6) x+6 a+6$
• $y^2=(a+2) x^6+(7 a+2) x^5+7 a x^4+(3 a+1) x^3+7 a x^2+(7 a+2) x+a+2$
• $y^2=(10 a+6) x^6+(4 a+8) x^5+(4 a+6) x^4+(8 a+2) x^3+(4 a+6) x^2+(4 a+8) x+10 a+6$
• $y^2=(9 a+8) x^6+6 a x^5+(8 a+5) x^4+(4 a+1) x^3+(8 a+5) x^2+6 a x+9 a+8$
• $y^2=(2 a+10) x^6+(8 a+3) x^5+(9 a+10) x^4+(a+3) x^3+(9 a+3) x^2+4 a x+8 a+7$
• $y^2=(10 a+4) x^6+(7 a+8) x^5+(7 a+1) x^4+(4 a+6) x^3+(8 a+1) x^2+(9 a+3) x+4 a+2$
• $y^2=(4 a+9) x^6+(8 a+10) x^5+(2 a+7) x^4+(5 a+10) x^3+(2 a+7) x^2+(8 a+10) x+4 a+9$
• $y^2=10 x^6+(10 a+9) x^5+2 a x^4+(6 a+3) x^3+3 x^2+(8 a+5) x+10 a+10$
• $y^2=2 x^6+(4 a+4) x^5+(6 a+2) x^4+(3 a+9) x^3+(4 a+5) x^2+(10 a+10) x+2 a+1$
• $y^2=(9 a+4) x^6+8 a x^5+(a+6) x^4+(a+8) x^3+(2 a+9) x^2+(6 a+10) x+2 a+3$
• $y^2=(6 a+4) x^6+(6 a+2) x^5+(10 a+2) x^4+(a+10) x^3+(9 a+3) x^2+2 a+6$
• $y^2=(5 a+7) x^6+(2 a+6) x^5+(9 a+7) x^4+(10 a+6) x^3+(9 a+8) x^2+8 a+4$
• $y^2=(6 a+3) x^6+(3 a+5) x^5+(8 a+6) x^4+3 a x^3+(6 a+10) x^2+(6 a+1) x+5 a+7$
• $y^2=(5 a+8) x^6+(a+2) x^5+(5 a+2) x^4+(3 a+3) x^3+(5 a+2) x^2+(a+2) x+5 a+8$
• $y^2=(7 a+8) x^6+(6 a+1) x^5+(4 a+3) x^4+a x^3+(7 a+5) x^2+(9 a+7) x+7 a+5$
• $y^2=(a+1) x^6+(a+3) x^5+(a+4) x^4+(2 a+8) x^3+(10 a+6) x^2+(6 a+3) x+10 a+3$
• $y^2=(8 a+7) x^6+3 x^5+(7 a+2) x^4+3 x^3+(4 a+8) x^2+6 x+8 a+6$

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11^{2}}$

The isogeny class factors as 1.121.av $\times$ 1.121.ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.121.av : \(\Q(\sqrt{-43}) \).
• 1.121.ap : \(\Q(\sqrt{-259}) \).

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
2.121.ag_acv	$2$	(not in LMFDB)
2.121.g_acv	$2$	(not in LMFDB)
2.121.bk_vl	$2$	(not in LMFDB)