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

• Label: 2.71.abb_mc
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 - 27 x + 314 x^{2} - 1917 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0426059583755$, $\pm0.290727374958$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	10

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3412$	$24907600$	$128099786800$	$645756529838400$	$3255145589555706412$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$45$	$4941$	$357912$	$25411801$	$1804175055$	$128099289678$	$9095110591185$	$645753480436561$	$45848500718449032$	$3255243553635618501$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=25x^6+13x^5+25x^4+10x^3+55x^2+54x+61$
• $y^2=58x^6+21x^5+58x^4+53x^3+39x^2+65x+56$
• $y^2=10x^6+9x^5+31x^4+46x^3+41x^2+50x+65$
• $y^2=17x^6+43x^5+62x^4+27x^3+32x^2+64x+70$
• $y^2=66x^6+46x^5+19x^4+56x^3+15x^2+12x+53$
• $y^2=25x^6+44x^5+8x^4+9x^3+52x^2+20x+54$
• $y^2=69x^6+40x^5+20x^4+3x^3+6x^2+7x+25$
• $y^2=38x^6+26x^5+62x^4+4x^3+65x^2+10x+62$
• $y^2=46x^6+35x^5+23x^4+47x^3+41x^2+28x+46$
• $y^2=61x^6+26x^5+15x^4+17x^3+67x^2+44x+9$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71^{6}}$.

Endomorphism algebra over $\F_{71}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{41})\).

Endomorphism algebra over $\overline{\F}_{71}$

The base change of $A$ to $\F_{71^{6}}$ is 1.128100283921.abchkc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{71^{2}}$

  The base change of $A$ to $\F_{71^{2}}$ is the simple isogeny class 2.5041.adx_hqm and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).

• Endomorphism algebra over $\F_{71^{3}}$

  The base change of $A$ to $\F_{71^{3}}$ is the simple isogeny class 2.357911.a_abchkc and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).

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
2.71.bb_mc	$2$	(not in LMFDB)
2.71.a_dx	$3$	(not in LMFDB)
2.71.bb_mc	$3$	(not in LMFDB)