Properties

Label 2.173.abt_bgq
Base Field $\F_{173}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{173}$
Dimension:  $2$
L-polynomial:  $1 - 45 x + 848 x^{2} - 7785 x^{3} + 29929 x^{4}$
Frobenius angles:  $\pm0.116569177593$, $\pm0.216764155741$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:  $C_2^2$
Jacobians:  44

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 44 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 22948 885976384 26808759403456 802402518753628416 24013957048436101180468 718709580752390504984743936 21510249794143156738178611251892 643780251725664619672837476297237504 19267699140703639474908789608000549086144 576662967582142020174554028775116629789103424

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 129 29601 5177718 895793425 154964854869 26808765474822 4637914433102793 802359179025516289 138808137876363860814 24013807852611897137961

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{173}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).
Endomorphism algebra over $\overline{\F}_{173}$
The base change of $A$ to $\F_{173^{6}}$ is 1.26808753332089.nhlic 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
All geometric endomorphisms are defined over $\F_{173^{6}}$.
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
2.173.bt_bgq$2$(not in LMFDB)
2.173.a_mr$3$(not in LMFDB)
2.173.bt_bgq$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.173.bt_bgq$2$(not in LMFDB)
2.173.a_mr$3$(not in LMFDB)
2.173.bt_bgq$3$(not in LMFDB)
2.173.a_mr$6$(not in LMFDB)
2.173.a_amr$12$(not in LMFDB)