Abelian variety isogeny class 2.71.abc_mz over $\F_{71}$

• Label: 2.71.abc_mz
• Base field: $\F_{71}$
• 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:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 - 15 x + 71 x^{2} )( 1 - 13 x + 71 x^{2} )$
	$1 - 28 x + 337 x^{2} - 1988 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.150643965450$, $\pm0.219552767034$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3363$	$24869385$	$128241198288$	$646072426895625$	$3255499132227788883$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$4932$	$358304$	$25424228$	$1804371004$	$128101355982$	$9095125404724$	$645753535218628$	$45848500478862944$	$3255243547988068452$

Jacobians and polarizations

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

• $y^2=7x^6+31x^5+22x^4+42x^3+33x^2+52x+68$
• $y^2=56x^6+4x^5+50x^4+62x^3+54x^2+25x+23$
• $y^2=46x^6+49x^5+53x^4+25x^3+28x^2+9x+68$
• $y^2=69x^6+39x^5+61x^4+56x^3+65x^2+68x+70$
• $y^2=8x^6+40x^5+28x^4+19x^3+51x^2+32x+36$
• $y^2=35x^6+42x^5+69x^4+45x^3+42x^2+62x+51$
• $y^2=2x^6+25x^5+4x^4+18x^3+50x^2+19x+19$
• $y^2=11x^6+50x^5+30x^4+56x^3+37x^2+9x+52$
• $y^2=63x^6+23x^5+60x^4+50x^3+57x^2+22x+52$
• $y^2=35x^6+20x^5+x^4+50x^3+16x^2+8x+11$
• $y^2=44x^6+6x^5+41x^4+50x^3+28x^2+27x+14$
• $y^2=17x^6+29x^5+36x^4+34x^3+38x^2+15x+66$
• $y^2=46x^6+36x^5+28x^4+11x^3+63x^2+58x+7$
• $y^2=46x^6+17x^5+64x^4+57x^3+43x^2+59x+33$
• $y^2=39x^6+28x^5+32x^4+6x^3+19x^2+34x+53$
• $y^2=63x^6+16x^5+61x^4+55x^3+21x^2+45x+65$
• $y^2=70x^6+56x^5+55x^4+32x^3+35x^2+35x+23$
• $y^2=59x^6+68x^5+33x^4+37x^3+51x^2+21x+33$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71}$.

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ap $\times$ 1.71.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.ap : \(\Q(\sqrt{-59}) \).
• 1.71.an : \(\Q(\sqrt{-115}) \).

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
2.71.ac_acb	$2$	(not in LMFDB)
2.71.c_acb	$2$	(not in LMFDB)
2.71.bc_mz	$2$	(not in LMFDB)