# Properties

 Label 2.163.abt_bft Base Field $\F_{163}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{163}$ Dimension: $2$ L-polynomial: $1 - 45 x + 825 x^{2} - 7335 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.0521334521506$, $\pm0.217387748284$ Angle rank: $2$ (numerical) Number field: 4.0.3785341.1 Galois group: $D_{4}$ Jacobians: 24

This isogeny class is simple and geometrically 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 24 curves, and hence is principally polarizable:

• $y^2=2x^6+18x^5+141x^4+87x^3+98x^2+121x+50$
• $y^2=89x^6+91x^5+144x^4+144x^3+10x^2+53x+117$
• $y^2=99x^6+43x^5+159x^4+69x^3+116x^2+2x+2$
• $y^2=18x^6+88x^5+126x^4+152x^3+66x^2+5x+139$
• $y^2=150x^6+7x^5+96x^4+146x^3+72x^2+87x+11$
• $y^2=154x^6+78x^5+155x^4+3x^3+132x^2+157x+72$
• $y^2=154x^6+46x^5+15x^4+134x^3+126x^2+141x+14$
• $y^2=103x^6+56x^5+62x^4+30x^3+148x^2+158x+60$
• $y^2=101x^6+61x^5+143x^4+119x^3+59x^2+11x+78$
• $y^2=134x^6+50x^5+121x^4+26x^3+44x^2+40x+150$
• $y^2=143x^6+159x^5+61x^4+53x^3+53x^2+64x+63$
• $y^2=109x^6+13x^5+16x^4+95x^3+161x^2+59x+120$
• $y^2=144x^6+116x^5+107x^4+42x^3+141x^2+59x$
• $y^2=69x^6+135x^5+124x^4+31x^3+68x^2+142x+79$
• $y^2=76x^6+111x^5+154x^4+111x^3+47x^2+99x+91$
• $y^2=26x^6+39x^5+123x^4+135x^3+145x^2+105x+42$
• $y^2=30x^6+139x^5+160x^4+110x^3+24x^2+51x+128$
• $y^2=31x^6+20x^5+156x^4+63x^3+46x^2+59x+130$
• $y^2=70x^6+125x^5+130x^4+20x^3+126x^2+115x+26$
• $y^2=31x^6+78x^5+95x^4+157x^3+118x^2+111x+111$
• $y^2=141x^6+123x^5+12x^4+159x^3+146x^2+9x+107$
• $y^2=61x^6+157x^5+4x^4+3x^3+69x^2+146x+161$
• $y^2=131x^6+66x^5+101x^4+87x^3+80x^2+13x+11$
• $y^2=112x^6+57x^5+61x^4+154x^3+111x^2+135x+56$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20015 696021625 18747770750345 498316073399828125 13239657817526454753200 351763891573583642346909625 9346014578250550963865693903405 248314264978346769555751525146703125 6597461722190028935173450538665221753635 175287960536761524851540487092839794114400000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 119 26195 4328993 705918363 115063806944 18755369768135 3057125188150523 498311412990880003 81224760514255585769 13239635966836934121350

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.3785341.1.
All geometric endomorphisms are defined over $\F_{163}$.

## 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 2.163.bt_bft $2$ (not in LMFDB)