Properties

Label 2.41.as_ft
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:  $1 - 18 x + 149 x^{2} - 738 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0319832925581$, $\pm0.365316625891$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{14})\)
Galois group:  $C_2^2$
Jacobians:  6

This isogeny class is simple but not 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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1075 2781025 4749990700 7977339621225 13418341001726875 22562411650086490000 37929126827025765085675 63759043395676530206174025 107178930967531938606221248300 180167779849021299747820853550625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 24 1656 68922 2823076 115818924 4749877158 194753758524 7984926792196 327381934393962 13422659078648376

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{14})\).
Endomorphism algebra over $\overline{\F}_{41}$
The base change of $A$ to $\F_{41^{6}}$ is 1.4750104241.aglza 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$
All geometric endomorphisms are defined over $\F_{41^{6}}$.
Remainder of endomorphism lattice by field

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.a_ba$3$(not in LMFDB)
2.41.s_ft$3$(not in LMFDB)
2.41.a_ba$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.a_ba$3$(not in LMFDB)
2.41.s_ft$3$(not in LMFDB)
2.41.a_ba$6$(not in LMFDB)
2.41.a_aba$12$(not in LMFDB)