# Properties

 Label 2.41.av_hf Base Field $\F_{41}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{41}$ Dimension: $2$ L-polynomial: $1 - 21 x + 187 x^{2} - 861 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0153876165359$, $\pm0.278522863005$ Angle rank: $2$ (numerical) Number field: 4.0.16317.1 Galois group: $D_{4}$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=20x^6+11x^5+27x^4+29x^3+20x^2+18x+24$
• $y^2=20x^6+5x^5+33x^4+37x^3+21x^2+29x+7$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 987 2715237 4745662803 7984493803125 13421063357891952 22562527591813070925 37928895301445060277267 63758955100453314303853125 107178920473686561823608718083 180167782631303010314312004218112

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 21 1615 68859 2825611 115842426 4749901567 194752569711 7984915734451 327381902340129 13422659285930830

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.16317.1.
All geometric endomorphisms are defined over $\F_{41}$.

## 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 2.41.v_hf $2$ (not in LMFDB)