# Properties

 Label 2.41.at_gp 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: $1 - 19 x + 171 x^{2} - 779 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.188836750463$, $\pm0.272870090458$ Angle rank: $2$ (numerical) Number field: $$\Q(\zeta_{5})$$ Galois group: $C_4$ Jacobians: 7

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 7 curves, and hence is principally polarizable:

• $y^2=6x^5+40$
• $y^2=34x^6+40x^5+5x^4+23x^3+14x^2+23x+6$
• $y^2=4x^6+38x^5+32x^4+21x^3+11x^2+6$
• $y^2=35x^6+17x^5+15x^4+24x^3+39x^2+26x+12$
• $y^2=38x^6+6x^5+19x^4+6x^3+15x^2+15x+9$
• $y^2=29x^6+26x^5+x^4+5x^3+10x^2+22x+27$
• $y^2=30x^6+32x^5+5x^4+9x^3+23x^2+23x+22$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1055 2796805 4788266255 8000878796405 13426146538750000 22563815119488249805 37929153369600036433055 63758991523444769269272005 107178922379466635845639789655 180167782082809772797913620000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 23 1663 69473 2831403 115886298 4750172623 194753894813 7984920295923 327381908161403 13422659245067598

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{5})$$.
All geometric endomorphisms are defined over $\F_{41}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.t_gp $2$ (not in LMFDB) 2.41.aj_ct $5$ (not in LMFDB) 2.41.b_acr $5$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.t_gp $2$ (not in LMFDB) 2.41.aj_ct $5$ (not in LMFDB) 2.41.b_acr $5$ (not in LMFDB) 2.41.l_eh $5$ (not in LMFDB) 2.41.q_ew $5$ (not in LMFDB) 2.41.aq_ew $10$ (not in LMFDB) 2.41.al_eh $10$ (not in LMFDB) 2.41.ab_acr $10$ (not in LMFDB) 2.41.j_ct $10$ (not in LMFDB)