Abelian variety isogeny class 2.49.o_fr over F72\F_{7^{2}}

• Label: 2.49.o_fr
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• pp-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 + 7 x + 49 x^{2} )^{2}$
	$1 + 14 x + 147 x^{2} + 686 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.666666666667$, $\pm0.666666666667$
Angle rank:	$0$ (numerical)
Number field:	$\Q(\sqrt{-3})$
Galois group:	$C_2$
Jacobians:	$17$

This isogeny class is simple but not geometrically simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3249$	$6007401$	$13680577296$	$33260630443209$	$79801762268091249$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$2500$	$116278$	$5769604$	$282508864$	$13840816606$	$678224719936$	$33232942099204$	$1628413436496022$	$79792266862562500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 17 curves (of which all are hyperelliptic):

• $y^2=(6a+4)x^6+(2a+4)x^5+(4a+5)x^4+5ax^3+(6a+6)x^2+(6a+1)x+a+2$
• $y^2=4x^6+(2a+5)x^5+(2a+6)x^4+(6a+3)x^3+6ax^2+(a+5)x+6$
• $y^2=(2a+1)x^6+(a+3)x^5+(6a+5)x^4+(3a+1)x^3+(4a+4)x^2+(6a+1)x+a+1$
• $y^2=(4a+3)x^6+4x^5+(4a+5)x^4+(a+4)x^3+(5a+4)x^2+(5a+4)x+2a+5$
• $y^2=(2a+6)x^6+(5a+1)x^5+(3a+6)x^4+(6a+6)x^3+(3a+4)x^2+(a+2)x+3a+5$
• $y^2=x^6+4ax^5+5ax^4+(6a+6)x^3+(6a+5)x^2+(6a+1)x+2a+2$
• $y^2=(6a+5)x^6+(2a+1)x^5+(a+6)x^4+(3a+6)x^3+(5a+3)x^2+3x+2$
• $y^2=(3a+6)x^6+4x^5+(4a+4)x^4+(a+5)x^3+4ax^2+5a+3$
• $y^2=(a+4)x^6+2x^5+x^4+(2a+6)x^3+(3a+4)x^2+(2a+1)x+5a+6$
• $y^2=ax^6+ax^5+(3a+1)x^4+5x^3+(2a+1)x^2+a$
• $y^2=(a+2)x^6+3ax^5+(5a+3)x^4+(a+6)x^3+(3a+6)x^2+2ax+2$
• $y^2=(4a+4)x^6+2ax^5+(3a+3)x^4+(6a+5)x^3+(a+1)x^2+(6a+4)x+4a$
• $y^2=(6a+4)x^6+(2a+5)x^5+(6a+2)x^4+(3a+2)x^3+(6a+5)x^2+(a+4)x+4a+3$
• $y^2=6x^6+(3a+4)x^5+(a+5)x^4+(4a+5)x^3+(3a+6)x^2+(3a+1)x+2a+2$
• $y^2=2x^6+(4a+6)x^5+x^4+(5a+2)x^3+(2a+3)x^2+(2a+2)x+2a+1$
• $y^2=3x^6+(3a+4)x^5+6x^4+2x^3+3x^2+4x+2a+6$
• $y^2=(a+2)x^6+(a+5)x^5+(3a+3)x^4+3ax^3+(5a+4)x^2+(6a+4)x+5a+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{6}}$.

Endomorphism algebra over $\F_{7^{2}}$

The endomorphism algebra of this simple isogeny class is the quaternion algebra over $\Q(\sqrt{-3})$ with the following ramification data at primes above $7$, and unramified at all archimedean places:

$v$	($7$,$\pi + 2$)	($7$,$\pi + 4$)
$\operatorname{inv}_v$	$1/2$	$1/2$

where $\pi$ is a root of $x^{2} - x + 1$.

Endomorphism algebra over $\overline{\F}_{7^{2}}$

The base change of $A$ to $\F_{7^{6}}$ is 1.117649.abak^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $\Q$ ramified at $7$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.

Subfield	Primitive Model
$\F_{7}$	2.7.a_h

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.49.ao_fr	$2$	(not in LMFDB)
2.49.abc_li	$3$	(not in LMFDB)
2.49.a_abx	$4$	(not in LMFDB)