Abelian variety isogeny class 2.199.aca_bpe over $\F_{199}$

• Label: 2.199.aca_bpe
• Base field: $\F_{199}$
• 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:	$\F_{199}$
Dimension:	$2$
L-polynomial:	$( 1 - 28 x + 199 x^{2} )( 1 - 24 x + 199 x^{2} )$
	$1 - 52 x + 1070 x^{2} - 10348 x^{3} + 39601 x^{4}$
Frobenius angles:	$\pm0.0391815390403$, $\pm0.176204172288$
Angle rank:	$2$ (numerical)
Jacobians:	$24$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$30272$	$1546051584$	$62066562721856$	$2459339257276661760$	$97393717264364738879552$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$148$	$39038$	$7875868$	$1568216926$	$312079728868$	$62103844451486$	$12358664299562092$	$2459374190696912446$	$489415464087522985012$	$97393677359029244399678$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=68x^6+197x^5+73x^4+30x^3+73x^2+197x+68$
• $y^2=123x^6+39x^5+114x^4+154x^3+9x^2+42x+116$
• $y^2=192x^6+31x^5+64x^4+163x^3+65x^2+2x+19$
• $y^2=39x^6+144x^5+197x^4+87x^3+197x^2+144x+39$
• $y^2=77x^6+38x^5+44x^4+88x^3+31x^2+195x+142$
• $y^2=44x^6+181x^5+101x^4+143x^3+45x^2+122x+122$
• $y^2=137x^6+189x^5+37x^4+143x^3+89x^2+127x+21$
• $y^2=87x^6+16x^5+99x^4+19x^3+189x^2+32x+97$
• $y^2=11x^6+100x^5+62x^4+152x^3+62x^2+100x+11$
• $y^2=198x^6+133x^5+88x^4+62x^3+11x^2+60x+38$
• $y^2=89x^6+126x^5+85x^4+174x^3+117x^2+39x+140$
• $y^2=100x^6+115x^5+12x^4+194x^3+109x^2+168x+43$
• $y^2=134x^6+104x^5+48x^4+26x^3+100x^2+171x+131$
• $y^2=55x^6+106x^5+42x^4+18x^3+42x^2+106x+55$
• $y^2=42x^6+156x^5+104x^4+40x^3+61x^2+179x+109$
• $y^2=153x^6+123x^5+57x^4+30x^3+57x^2+123x+153$
• $y^2=67x^6+189x^5+49x^4+64x^3+43x^2+185x+88$
• $y^2=115x^6+79x^5+82x^4+28x^3+168x^2+45x+16$
• $y^2=47x^6+46x^5+25x^4+48x^3+25x^2+46x+47$
• $y^2=126x^6+61x^5+26x^4+18x^3+26x^2+61x+126$
• $y^2=153x^6+167x^5+191x^4+146x^3+107x^2+146x+38$
• $y^2=11x^6+144x^5+29x^4+34x^3+29x^2+144x+11$
• $y^2=120x^6+127x^5+43x^4+22x^3+188x^2+138x+119$
• $y^2=68x^6+87x^5+95x^4+92x^3+187x^2+139x+166$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{199}$.

Endomorphism algebra over $\F_{199}$

The isogeny class factors as 1.199.abc $\times$ 1.199.ay and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.199.abc : \(\Q(\sqrt{-3}) \).
• 1.199.ay : \(\Q(\sqrt{-55}) \).

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
2.199.ae_ako	$2$	(not in LMFDB)
2.199.e_ako	$2$	(not in LMFDB)
2.199.ca_bpe	$2$	(not in LMFDB)
2.199.an_fe	$3$	(not in LMFDB)
2.199.ah_ak	$3$	(not in LMFDB)