Properties

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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)