Abelian variety isogeny class 2.73.h_ga over $\F_{73}$

• Label: 2.73.h_ga
• Base field: $\F_{73}$
• 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:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$( 1 + 2 x + 73 x^{2} )( 1 + 5 x + 73 x^{2} )$
	$1 + 7 x + 156 x^{2} + 511 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.537340940774$, $\pm0.594521390912$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	90
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6004$	$29827872$	$150790796224$	$806077122666624$	$4297918404797268724$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$5593$	$387618$	$28384753$	$2073212721$	$151334656558$	$11047387844889$	$806460099258721$	$58871587349959314$	$4297625827224456793$

Jacobians and polarizations

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

• $y^2=22 x^6+9 x^5+26 x^4+59 x^3+30 x^2+31 x+67$
• $y^2=16 x^6+16 x^5+42 x^4+67 x^3+55 x^2+69 x+21$
• $y^2=65 x^6+64 x^5+53 x^4+59 x^3+53 x^2+71 x+23$
• $y^2=34 x^6+16 x^5+9 x^4+70 x^3+26 x^2+70 x+68$
• $y^2=48 x^6+28 x^5+34 x^4+11 x^3+6 x^2+41 x+48$
• $y^2=53 x^6+3 x^5+x^4+44 x^3+39 x^2+52 x+46$
• $y^2=65 x^6+68 x^5+36 x^4+5 x^3+27 x^2+42 x+19$
• $y^2=14 x^6+3 x^5+26 x^4+34 x^3+41 x+6$
• $y^2=18 x^6+65 x^5+8 x^4+48 x^3+55 x^2+66 x+3$
• $y^2=20 x^6+34 x^5+11 x^4+4 x^3+32 x^2+59 x+71$
• $y^2=69 x^6+63 x^5+69 x^4+42 x^3+25 x^2+68 x+10$
• $y^2=64 x^6+21 x^5+32 x^4+40 x^3+41 x^2+64 x+47$
• $y^2=25 x^6+51 x^5+39 x^4+7 x^3+72 x^2+19 x+71$
• $y^2=2 x^6+4 x^5+30 x^4+23 x^3+43 x^2+40 x+41$
• $y^2=65 x^6+7 x^5+33 x^4+71 x^3+45 x^2+10 x+26$
• $y^2=64 x^6+57 x^5+7 x^4+9 x^3+46 x^2+68 x+45$
• $y^2=64 x^6+18 x^5+64 x^4+28 x^3+8 x^2+22 x$
• $y^2=57 x^6+49 x^5+27 x^4+33 x^3+28 x^2+21 x+14$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73}$.

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.c $\times$ 1.73.f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.73.c : \(\Q(\sqrt{-2}) \).
• 1.73.f : \(\Q(\sqrt{-267}) \).

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
2.73.ah_ga	$2$	(not in LMFDB)
2.73.ad_fg	$2$	(not in LMFDB)
2.73.d_fg	$2$	(not in LMFDB)