# Properties

 Label 2.199.abw_bkw Base Field $\F_{199}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

# Learn more about

## Invariants

 Base field: $\F_{199}$ Dimension: $2$ L-polynomial: $( 1 - 28 x + 199 x^{2} )( 1 - 20 x + 199 x^{2} )$ Frobenius angles: $\pm0.0391815390403$, $\pm0.249200223051$ Angle rank: $2$ (numerical) Jacobians: 90

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 90 curves, and hence is principally polarizable:

• $y^2=54x^6+113x^5+2x^4+119x^3+131x^2+104x+116$
• $y^2=155x^6+183x^5+75x^4+11x^3+91x^2+165x+151$
• $y^2=120x^6+113x^5+152x^4+91x^3+122x^2+53x+55$
• $y^2=28x^6+81x^5+145x^4+43x^3+159x^2+60x+35$
• $y^2=140x^6+95x^5+196x^4+34x^3+140x^2+169x+32$
• $y^2=69x^6+119x^5+42x^4+181x^3+134x^2+11x+44$
• $y^2=141x^6+145x^5+20x^4+x^3+10x^2+86x+142$
• $y^2=10x^6+64x^5+44x^4+16x^3+75x^2+175x+137$
• $y^2=55x^6+142x^5+113x^4+49x^3+113x^2+142x+55$
• $y^2=157x^6+13x^5+62x^4+30x^3+14x^2+168x+37$
• $y^2=176x^6+43x^5+175x^4+13x^3+160x^2+11x+172$
• $y^2=144x^6+71x^5+49x^4+150x^3+31x^2+197x+106$
• $y^2=29x^6+114x^5+174x^4+127x^3+134x^2+102x+96$
• $y^2=150x^6+186x^5+141x^4+122x^3+191x^2+91x+84$
• $y^2=33x^6+129x^5+15x^4+112x^3+112x^2+114x+39$
• $y^2=135x^6+16x^5+10x^4+183x^3+72x^2+83x+174$
• $y^2=173x^6+5x^5+29x^4+18x^3+29x^2+5x+173$
• $y^2=161x^6+37x^5+106x^4+137x^3+98x^2+115x+3$
• $y^2=153x^6+23x^5+159x^4+175x^3+143x^2+61x+68$
• $y^2=65x^6+17x^5+187x^4+113x^3+83x^2+24x+8$
• and 70 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30960 1552953600 62093622472560 2459388936883507200 97393642291828872658800 3856886308088067056375865600 152736579067150039017126835568880 6048521402099003438196377948155084800 239527496493560034455534281942260928654320 9485528389622582293594773478217100542574240000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 152 39214 7879304 1568248606 312079488632 62103829181902 12358663979948648 2459374186682549566 489415464067324730456 97393677359471287288174

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The isogeny class factors as 1.199.abc $\times$ 1.199.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{199}$.

## 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.ai_agg $2$ (not in LMFDB) 2.199.i_agg $2$ (not in LMFDB) 2.199.bw_bkw $2$ (not in LMFDB) 2.199.aj_gw $3$ (not in LMFDB) 2.199.ad_cg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.199.ai_agg $2$ (not in LMFDB) 2.199.i_agg $2$ (not in LMFDB) 2.199.bw_bkw $2$ (not in LMFDB) 2.199.aj_gw $3$ (not in LMFDB) 2.199.ad_cg $3$ (not in LMFDB) 2.199.abl_bck $6$ (not in LMFDB) 2.199.abf_xu $6$ (not in LMFDB) 2.199.d_cg $6$ (not in LMFDB) 2.199.j_gw $6$ (not in LMFDB) 2.199.bf_xu $6$ (not in LMFDB) 2.199.bl_bck $6$ (not in LMFDB)