Properties

Label 2.169.aca_bna
Base Field $\F_{13^{2}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 13 x )^{4}$
Frobenius angles:  $0$, $0$, $0$, $0$
Angle rank:  $0$ (numerical)
Jacobians:  3

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2, 1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 20736 796594176 23255696077056 665323421736960000 19004759032135892541696 542800320552882493433450496 15502931814404303545726027489536 442779261605637620799330792898560000 12646218547960214014894253456394002739456 361188648074051463355435127582466684175650816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 118 27886 4818022 815616478 137857006678 23298065815246 3937376134705222 665416605920256958 112455406909539395638 19004963774329365471406

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The isogeny class factors as 1.169.aba 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.
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_aba

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.a_ana$2$(not in LMFDB)
2.169.ca_bna$2$(not in LMFDB)
2.169.ba_tn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.169.a_ana$2$(not in LMFDB)
2.169.ca_bna$2$(not in LMFDB)
2.169.ba_tn$3$(not in LMFDB)
2.169.a_na$4$(not in LMFDB)
2.169.n_gn$5$(not in LMFDB)
2.169.aba_tn$6$(not in LMFDB)
2.169.a_a$8$(not in LMFDB)
2.169.an_gn$10$(not in LMFDB)