Abelian variety isogeny class 2.79.ba_mp over $\F_{79}$

• Label: 2.79.ba_mp
• Base field: $\F_{79}$
• 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:	$\F_{79}$
Dimension:	$2$
L-polynomial:	$( 1 + 13 x + 79 x^{2} )^{2}$
	$1 + 26 x + 327 x^{2} + 2054 x^{3} + 6241 x^{4}$
Frobenius angles:	$\pm0.761089378095$, $\pm0.761089378095$
Angle rank:	$1$ (numerical)
Jacobians:	$12$
Cyclic group of points:	no
Non-cyclic primes:	$3, 31$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8649$	$38825361$	$242217528336$	$1518071964528249$	$9467716952809284849$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$106$	$6220$	$491272$	$38974804$	$3076874686$	$243087864766$	$19203918021394$	$1517108660118244$	$119851597216082968$	$9468276078425083900$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=11 x^6+18 x^5+6 x^4+19 x^3+6 x^2+18 x+11$
• $y^2=61 x^6+48 x^5+42 x^4+32 x^3+42 x^2+48 x+61$
• $y^2=16 x^6+55 x^5+6 x^4+64 x^3+54 x^2+31 x+62$
• $y^2=5 x^6+45 x^5+58 x^3+41 x^2+25 x+55$
• $y^2=26 x^6+34 x^5+70 x^4+43 x^3+70 x^2+34 x+26$
• $y^2=49 x^6+42 x^5+8 x^4+30 x^3+23 x^2+19 x+24$
• $y^2=19 x^6+73 x^5+23 x^4+25 x^3+23 x^2+73 x+19$
• $y^2=x^6+26 x^3+38$
• $y^2=15 x^6+60 x^5+36 x^4+3 x^3+36 x^2+60 x+15$
• $y^2=22 x^6+63 x^5+22 x^4+24 x^3+22 x^2+63 x+22$
• $y^2=73 x^6+71 x^5+76 x^4+34 x^3+6 x^2+55 x+23$
• $y^2=5 x^6+29 x^5+2 x^4+62 x^3+69 x^2+22 x+24$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79}$.

Endomorphism algebra over $\F_{79}$

The isogeny class factors as 1.79.n^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.79.aba_mp	$2$	(not in LMFDB)
2.79.a_al	$2$	(not in LMFDB)
2.79.abi_rf	$3$	(not in LMFDB)
2.79.an_dm	$3$	(not in LMFDB)
2.79.ae_acl	$3$	(not in LMFDB)
2.79.i_gs	$3$	(not in LMFDB)
2.79.r_ic	$3$	(not in LMFDB)