# Properties

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

## Invariants

 Base field: $\F_{199}$ Dimension: $2$ L-polynomial: $( 1 - 27 x + 199 x^{2} )( 1 - 23 x + 199 x^{2} )$ Frobenius angles: $\pm0.0936959350875$, $\pm0.196619630811$ Angle rank: $2$ (numerical) Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=56x^6+155x^5+138x^4+156x^3+38x^2+125x+119$
• $y^2=67x^6+183x^5+3x^4+23x^3+3x^2+183x+67$
• $y^2=78x^6+59x^5+187x^4+96x^3+28x^2+192x+71$
• $y^2=195x^6+194x^5+114x^4+183x^3+173x^2+x+85$
• $y^2=135x^6+87x^5+120x^4+165x^3+70x^2+13x+33$
• $y^2=19x^6+91x^5+134x^4+4x^3+134x^2+91x+19$
• $y^2=22x^6+7x^5+155x^4+78x^3+155x^2+7x+22$
• $y^2=20x^6+62x^5+x^4+60x^3+x^2+62x+20$
• $y^2=160x^6+118x^5+173x^4+122x^3+183x^2+22x+19$
• $y^2=141x^6+144x^5+26x^4+25x^3+26x^2+144x+141$
• $y^2=59x^6+30x^5+150x^4+53x^3+150x^2+30x+59$
• $y^2=4x^6+198x^5+64x^4+168x^3+154x^2+117x+1$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30621 1550065641 62088089585904 2459423877760491849 97393991078214868601901 3856888034022403438445097216 152736585098315399587521240270141 6048521417772032149087608069943787529 239527496521742950221530264785458129627504 9485528389643528757153063878549530712660313801

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 150 39140 7878600 1568270884 312080606250 62103856973006 12358664467959750 2459374193055320644 489415464124909579800 97393677359686357343300

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The isogeny class factors as 1.199.abb $\times$ 1.199.ax 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 is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.199.ae_aip $2$ (not in LMFDB) 2.199.e_aip $2$ (not in LMFDB) 2.199.by_bnf $2$ (not in LMFDB)