Abelian variety isogeny class 2.67.a_aeh over $\F_{67}$

• Label: 2.67.a_aeh
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 - 111 x^{2} + 4489 x^{4}$
Frobenius angles:	$\pm0.0946376280512$, $\pm0.905362371949$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{5}, \sqrt{-23})\)
Galois group:	$C_2^2$
Jacobians:	$35$
Isomorphism classes:	42

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4379$	$19175641$	$90458509376$	$405932998632841$	$1822837807213785539$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$4268$	$300764$	$20144436$	$1350125108$	$90458636582$	$6060711605324$	$406067735809828$	$27206534396294948$	$1822837809875809628$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 35 curves (of which all are hyperelliptic):

• $y^2=54 x^6+12 x^5+16 x^4+31 x^3+13 x^2+32 x+50$
• $y^2=65 x^6+28 x^5+30 x^4+4 x^3+20 x^2+32 x+41$
• $y^2=63 x^6+56 x^5+60 x^4+8 x^3+40 x^2+64 x+15$
• $y^2=34 x^6+56 x^5+20 x^4+41 x^3+21 x^2+56 x+55$
• $y^2=x^6+45 x^5+40 x^4+15 x^3+42 x^2+45 x+43$
• $y^2=24 x^6+47 x^5+49 x^4+46 x^3+58 x^2+62 x+3$
• $y^2=33 x^6+35 x^5+24 x^4+4 x^3+24 x^2+53 x+60$
• $y^2=66 x^6+3 x^5+48 x^4+8 x^3+48 x^2+39 x+53$
• $y^2=3 x^6+36 x^5+62 x^4+8 x^3+55 x^2+44 x+29$
• $y^2=6 x^6+5 x^5+57 x^4+16 x^3+43 x^2+21 x+58$
• $y^2=41 x^6+59 x^5+51 x^4+44 x^3+39 x^2+45 x+45$
• $y^2=15 x^6+51 x^5+35 x^4+21 x^3+11 x^2+23 x+23$
• $y^2=21 x^6+36 x^5+10 x^4+53 x^3+19 x^2+5 x+66$
• $y^2=42 x^6+5 x^5+20 x^4+39 x^3+38 x^2+10 x+65$
• $y^2=21 x^6+32 x^5+4 x^4+17 x^3+39 x^2+16 x+53$
• $y^2=42 x^6+64 x^5+8 x^4+34 x^3+11 x^2+32 x+39$
• $y^2=30 x^6+43 x^5+36 x^4+10 x^3+11 x^2+66 x+39$
• $y^2=43 x^6+17 x^5+49 x^4+4 x^3+49 x^2+x+21$
• $y^2=19 x^6+34 x^5+31 x^4+8 x^3+31 x^2+2 x+42$
• $y^2=53 x^6+54 x^5+23 x^4+41 x^3+16 x^2+52$
• and

• $y^2=39 x^6+41 x^5+46 x^4+15 x^3+32 x^2+37$
• $y^2=39 x^6+11 x^5+39 x^4+34 x^3+2 x^2+38 x+13$
• $y^2=20 x^6+24 x^5+4 x^4+19 x^3+16 x^2+29 x+45$
• $y^2=40 x^6+48 x^5+8 x^4+38 x^3+32 x^2+58 x+23$
• $y^2=47 x^6+34 x^5+46 x^4+27 x^3+18 x^2+37 x+30$
• $y^2=27 x^6+x^5+25 x^4+54 x^3+36 x^2+7 x+60$
• $y^2=6 x^6+23 x^5+51 x^4+63 x^3+16 x^2+23 x+61$
• $y^2=50 x^6+47 x^5+48 x^4+45 x^3+64 x^2+24 x+54$
• $y^2=28 x^6+62 x^5+64 x^4+63 x^3+45 x^2+47 x+58$
• $y^2=56 x^6+57 x^5+61 x^4+59 x^3+23 x^2+27 x+49$
• $y^2=51 x^6+33 x^5+28 x^4+36 x^3+62 x^2+41 x+54$
• $y^2=35 x^6+66 x^5+56 x^4+5 x^3+57 x^2+15 x+41$
• $y^2=64 x^6+49 x^5+35 x^4+19 x^3+58 x^2+6 x+2$
• $y^2=61 x^6+31 x^5+3 x^4+38 x^3+49 x^2+12 x+4$
• $y^2=24 x^6+34 x^5+40 x^4+32 x^3+53 x^2+38 x+45$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{2}}$.

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{5}, \sqrt{-23})\).

Endomorphism algebra over $\overline{\F}_{67}$

The base change of $A$ to $\F_{67^{2}}$ is 1.4489.aeh^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$

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
2.67.a_eh	$4$	(not in LMFDB)