Properties

Label 2.169.abw_bje
Base Field $\F_{13^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 24 x + 169 x^{2} )^{2}$
Frobenius angles:  $\pm0.125665916378$, $\pm0.125665916378$
Angle rank:  $1$ (numerical)
Jacobians:  9

This isogeny class is not 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 9 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 21316 802248976 23282111123716 665417390653440000 19005044101524556310596 542801092413737557716007696 15502933721134081189770835107076 442779265947739709696484208803840000 12646218557102226267280733954285890174276 361188648091785272621232733845069559875820816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 122 28086 4823498 815731678 137859074522 23298098945046 3937376618969258 665416612445645758 112455406990833950522 19004963775262480076406

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.ay 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
All geometric endomorphisms are defined over $\F_{13^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.

SubfieldPrimitive Model
$\F_{13}$2.13.a_ay

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.a_aje$2$(not in LMFDB)
2.169.bw_bje$2$(not in LMFDB)
2.169.y_pr$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.a_aje$2$(not in LMFDB)
2.169.bw_bje$2$(not in LMFDB)
2.169.y_pr$3$(not in LMFDB)
2.169.abi_wg$4$(not in LMFDB)
2.169.au_qw$4$(not in LMFDB)
2.169.ao_du$4$(not in LMFDB)
2.169.a_je$4$(not in LMFDB)
2.169.o_du$4$(not in LMFDB)
2.169.u_qw$4$(not in LMFDB)
2.169.bi_wg$4$(not in LMFDB)
2.169.ay_pr$6$(not in LMFDB)
2.169.a_ajg$8$(not in LMFDB)
2.169.a_jg$8$(not in LMFDB)
2.169.ak_acr$12$(not in LMFDB)
2.169.k_acr$12$(not in LMFDB)