# Properties

 Label 2.199.abz_bog 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: $( 1 - 27 x + 199 x^{2} )( 1 - 24 x + 199 x^{2} )$ Frobenius angles: $\pm0.0936959350875$, $\pm0.176204172288$ Angle rank: $2$ (numerical) Jacobians: 16

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 16 curves, and hence is principally polarizable:

• $y^2=64x^6+79x^5+18x^4+38x^3+105x^2+66x+198$
• $y^2=191x^6+16x^5+177x^4+29x^3+88x^2+188x+85$
• $y^2=186x^6+140x^5+191x^4+48x^3+59x^2+189x+101$
• $y^2=11x^6+117x^5+100x^4+142x^3+163x^2+171x+163$
• $y^2=110x^6+147x^5+17x^4+20x^3+164x^2+24x+99$
• $y^2=142x^6+16x^5+33x^4+25x^3+189x^2+31x+42$
• $y^2=26x^6+121x^5+19x^4+163x^3+32x^2+81x+40$
• $y^2=190x^6+89x^5+53x^4+32x^3+36x^2+104x+31$
• $y^2=107x^6+114x^5+9x^4+108x^3+52x^2+165x+125$
• $y^2=179x^6+111x^5+186x^4+119x^3+140x^2+191x+142$
• $y^2=2x^6+181x^5+128x^4+162x^3+141x^2+85x+37$
• $y^2=189x^6+80x^5+88x^4+138x^3+94x^2+106x+176$
• $y^2=39x^6+6x^5+37x^4+13x^3+139x^2+72x+114$
• $y^2=157x^6+176x^5+20x^4+128x^3+144x^2+36x+3$
• $y^2=198x^6+86x^5+18x^4+37x^3+16x^2+76x+60$
• $y^2=12x^6+62x^5+152x^4+189x^3+161x^2+197x+194$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30448 1548219904 62079739927744 2459401102663464960 97393967494055688776848 3856888170158997044063297536 152736586098917989578622601182288 6048521421685204791143189561577431040 239527496532198154018446454847219651164864 9485528389659396058259373625856413406510831104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 149 39093 7877540 1568256361 312080530679 62103859165086 12358664548923401 2459374194646446001 489415464146272214540 97393677359849276554653

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The isogeny class factors as 1.199.abb $\times$ 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:
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.ad_ajq $2$ (not in LMFDB) 2.199.d_ajq $2$ (not in LMFDB) 2.199.bz_bog $2$ (not in LMFDB)