# Properties

 Label 2.41.as_ga 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: $1 - 18 x + 156 x^{2} - 738 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.136556295479$, $\pm0.334734336659$ Angle rank: $2$ (numerical) Number field: 4.0.2750272.1 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=9x^6+7x^5+12x^4+5x^3+20x^2+17x+24$
• $y^2=38x^6+28x^5+16x^3+13x^2+4x+17$
• $y^2=34x^6+22x^5+26x^4+37x^3+3x^2+31x+30$
• $y^2=19x^6+2x^5+15x^4+3x^3+18x^2+40x+15$
• $y^2=6x^6+27x^5+18x^4+18x^3+x^2+40x+6$
• $y^2=26x^6+7x^5+12x^4+17x^3+30x^2+17x+11$
• $y^2=38x^6+20x^5+20x^4+25x^3+26x^2+35x+15$
• $y^2=29x^6+38x^5+8x^4+28x^3+30x^2+2x+14$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1082 2806708 4776217418 7990910985808 13422719888173082 22563387862602739348 37929316120910038511978 63759098076945360956347392 107178951716386911480770805098 180167785666375184195028472013908

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1670 69300 2827878 115856724 4750082678 194754730488 7984933640254 327381997772088 13422659512046390

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is 4.0.2750272.1.
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.s_ga $2$ (not in LMFDB)