# 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

## 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:

• $y^2=x^5+12x$
• $y^2=2ax^6+(3a+8)x^4+(3a+8)x^2+2a$
• $y^2=x^6+2x^3+8$

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

 $r$ 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 $\F_{13^{2}}$.

 Subfield Primitive Model $\F_{13}$ 2.13.a_aba

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)