Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 23 x + 169 x^{2} )^{2}$ |
$1 - 46 x + 867 x^{2} - 7774 x^{3} + 28561 x^{4}$ | |
Frobenius angles: | $\pm0.154420958311$, $\pm0.154420958311$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $12$ |
This isogeny class is not simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21609$ | $804913641$ | $23293210300416$ | $665450286236357769$ | $19005118247840541139449$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $124$ | $28180$ | $4825798$ | $815772004$ | $137859612364$ | $23298103917646$ | $3937376628621196$ | $665416611594000964$ | $112455406966352476822$ | $19004963774804457594100$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8ax^6+9ax^5+7ax^4+10ax^3+7ax^2+6ax+11a$
- $y^2=6ax^6+11ax^5+4ax^4+4ax^3+4ax^2+4ax+10a$
- $y^2=(4a+4)x^6+(4a+4)x^5+(4a+2)x^4+(12a+1)x^3+(5a+11)x^2+(6a+11)x+9a+8$
- $y^2=(9a+8)x^6+(9a+8)x^5+(9a+6)x^4+ax^3+(8a+3)x^2+(7a+4)x+4a+4$
- $y^2=x^6+6x^3+8$
- $y^2=(9a+12)x^6+(8a+3)x^5+(12a+10)x^4+(a+9)x^3+(7a+7)x^2+(10a+3)x+8a$
- $y^2=(12a+9)x^6+(a+7)x^5+(a+6)x^4+(11a+4)x^3+(8a+9)x^2+(a+10)x+10a+11$
- $y^2=(5a+9)x^6+(3a+12)x^5+(6a+9)x^4+(6a+7)x^3+(9a+3)x^2+(10a+8)x+10a+12$
- $y^2=(7a+3)x^6+(4a+3)x^5+(10a+2)x^4+(8a+12)x^3+(4a+5)x^2+(4a+12)x+12a$
- $y^2=4ax^6+(11a+2)x^5+(7a+10)x^4+(11a+5)x^3+(8a+2)x^2+(a+10)x+5a+9$
- $y^2=(6a+10)x^6+(9a+7)x^5+(3a+12)x^4+(5a+7)x^3+(9a+9)x^2+(9a+3)x+a+12$
- $y^2=4ax^6+(12a+11)x^5+(12a+3)x^4+(7a+8)x^3+(9a+11)x^2+(6a+3)x+3a+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The isogeny class factors as 1.169.ax 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
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.ao_cx |
$\F_{13}$ | 2.13.a_ax |
$\F_{13}$ | 2.13.o_cx |