Properties

Label 2.41.av_hg
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:  $1 - 21 x + 188 x^{2} - 861 x^{3} + 1681 x^{4}$
Frobenius angles:  $\pm0.0623248522748$, $\pm0.271008481058$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:  $C_2^2$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 988 2718976 4750051072 7987361780736 13422365037551548 22562985186608349184 37929030028367375781052 63758991734613322876145664 107178930967532391731290738432 180167786037728988982209161733376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 21 1617 68922 2826625 115853661 4749997902 194753261493 7984920322369 327381934393962 13422659539712577

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).
Endomorphism algebra over $\overline{\F}_{41}$
The base change of $A$ to $\F_{41^{6}}$ is 1.4750104241.adara 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
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.v_hg$2$(not in LMFDB)
2.41.a_cn$3$(not in LMFDB)
2.41.v_hg$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.41.v_hg$2$(not in LMFDB)
2.41.a_cn$3$(not in LMFDB)
2.41.v_hg$3$(not in LMFDB)
2.41.a_cn$6$(not in LMFDB)
2.41.a_acn$12$(not in LMFDB)