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

• Label: 2.73.as_is
• 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:	$( 1 - 10 x + 73 x^{2} )( 1 - 8 x + 73 x^{2} )$
	$1 - 18 x + 226 x^{2} - 1314 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.301013746420$, $\pm0.344915434243$
Angle rank:	$2$ (numerical)
Jacobians:	$32$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4224$	$29094912$	$152281793664$	$806814478761984$	$4297500417250026624$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$5458$	$391448$	$28410718$	$2073011096$	$151332828658$	$11047390478264$	$806460117044926$	$58871587464116984$	$4297625834507075218$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=39 x^6+59 x^5+45 x^4+46 x^3+45 x^2+59 x+39$
• $y^2=46 x^6+63 x^5+54 x^4+43 x^3+54 x^2+63 x+46$
• $y^2=29 x^6+6 x^5+6 x^4+65 x^3+6 x^2+6 x+29$
• $y^2=11 x^6+21 x^5+68 x^4+3 x^3+68 x^2+21 x+11$
• $y^2=29 x^6+47 x^5+37 x^4+31 x^3+18 x^2+30 x+42$
• $y^2=60 x^6+69 x^5+56 x^4+53 x^3+56 x^2+69 x+60$
• $y^2=35 x^6+41 x^5+35 x^4+60 x^3+9 x^2+16 x+32$
• $y^2=43 x^6+10 x^5+44 x^4+20 x^3+44 x^2+10 x+43$
• $y^2=41 x^6+48 x^5+70 x^4+45 x^3+71 x^2+70 x+50$
• $y^2=56 x^6+6 x^5+37 x^4+72 x^3+35 x^2+54 x+21$
• $y^2=36 x^6+27 x^5+5 x^4+62 x^3+42 x^2+48 x+12$
• $y^2=14 x^6+22 x^5+42 x^4+26 x^3+42 x^2+22 x+14$
• $y^2=24 x^6+14 x^5+63 x^4+33 x^3+15 x^2+68 x+65$
• $y^2=40 x^6+7 x^5+62 x^4+24 x^3+10 x^2+45 x+26$
• $y^2=23 x^6+66 x^5+29 x^4+57 x^3+56 x^2+14 x+32$
• $y^2=15 x^6+47 x^5+63 x^4+41 x^3+63 x^2+47 x+15$
• $y^2=60 x^6+36 x^5+47 x^4+22 x^3+47 x^2+36 x+60$
• $y^2=28 x^6+22 x^5+14 x^4+10 x^3+14 x^2+22 x+28$
• $y^2=38 x^6+62 x^5+2 x^4+34 x^3+2 x^2+62 x+38$
• $y^2=40 x^6+22 x^5+47 x^4+63 x^3+42 x^2+60 x+5$
• and

• $y^2=36 x^6+x^5+15 x^4+44 x^3+68 x^2+65 x+23$
• $y^2=65 x^6+29 x^5+19 x^4+67 x^3+19 x^2+29 x+65$
• $y^2=42 x^6+8 x^5+18 x^4+62 x^3+18 x^2+8 x+42$
• $y^2=26 x^6+14 x^5+23 x^4+10 x^3+61 x^2+39 x+47$
• $y^2=41 x^6+10 x^5+64 x^4+30 x^3+18 x^2+40 x+37$
• $y^2=51 x^6+65 x^5+32 x^4+24 x^3+32 x^2+65 x+51$
• $y^2=67 x^6+72 x^5+13 x^4+71 x^3+13 x^2+72 x+67$
• $y^2=20 x^6+69 x^5+9 x^4+57 x^3+9 x^2+69 x+20$
• $y^2=60 x^6+59 x^5+22 x^4+56 x^3+21 x^2+34 x+60$
• $y^2=54 x^6+35 x^5+59 x^4+63 x^3+40 x^2+25 x+6$
• $y^2=45 x^6+25 x^5+29 x^4+12 x^3+43 x^2+35 x+33$
• $y^2=8 x^6+54 x^5+55 x^4+36 x^3+55 x^2+54 x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ak $\times$ 1.73.ai 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.ak : \(\Q(\sqrt{-3}) \).
• 1.73.ai : \(\Q(\sqrt{-57}) \).

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
2.73.ac_co	$2$	(not in LMFDB)
2.73.c_co	$2$	(not in LMFDB)
2.73.s_is	$2$	(not in LMFDB)
2.73.ap_hu	$3$	(not in LMFDB)
2.73.j_k	$3$	(not in LMFDB)