Properties

Label 2.83.abe_ot
Base Field $\F_{83}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{83}$
Dimension:  $2$
L-polynomial:  $1 - 30 x + 383 x^{2} - 2490 x^{3} + 6889 x^{4}$
Frobenius angles:  $\pm0.0661835162234$, $\pm0.267149817110$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{2}, \sqrt{-3})\)
Galois group:  $C_2^2$
Jacobians:  21

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 21 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4753 46546129 326940010096 2252490483005721 15516034691728738993 106889770201572581929216 736364971293330790602485497 5072820124489153502937556016169 34946659039493167699189665926817904 240747534245290983104270476618202577649

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6756 571788 47462500 3939038994 326939646822 27136040228358 2252292154678084 186940255267540404 15516041195083027236

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\).
Endomorphism algebra over $\overline{\F}_{83}$
The base change of $A$ to $\F_{83^{6}}$ is 1.326940373369.aurkc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
All geometric endomorphisms are defined over $\F_{83^{6}}$.
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
2.83.be_ot$2$(not in LMFDB)
2.83.a_fe$3$(not in LMFDB)
2.83.be_ot$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.83.be_ot$2$(not in LMFDB)
2.83.a_fe$3$(not in LMFDB)
2.83.be_ot$3$(not in LMFDB)
2.83.a_afe$12$(not in LMFDB)
2.83.ai_bg$24$(not in LMFDB)
2.83.i_bg$24$(not in LMFDB)