Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 25 x + 169 x^{2} )( 1 - 22 x + 169 x^{2} )$ |
$1 - 47 x + 888 x^{2} - 7943 x^{3} + 28561 x^{4}$ | |
Frobenius angles: | $\pm0.0885687144757$, $\pm0.178912375022$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $22$ |
This isogeny class is not simple, 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21460$ | $803462400$ | $23286296559760$ | $665425223211340800$ | $19005042298295333077300$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $123$ | $28129$ | $4824366$ | $815741281$ | $137859061443$ | $23298095471182$ | $3937376519962827$ | $665416610526622081$ | $112455406961804735934$ | $19004963774921533896049$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 22 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+(7a+1)x^5+(6a+3)x^4+(11a+8)x^3+9ax^2+(10a+3)x+5a+12$
- $y^2=(12a+9)x^6+(7a+6)x^5+(4a+6)x^4+7x^3+2ax^2+(10a+12)x+9a+3$
- $y^2=(11a+3)x^6+(3a+8)x^5+(10a+5)x^4+(9a+12)x^3+(9a+4)x^2+(10a+5)x+8a+5$
- $y^2=6ax^6+(7a+8)x^5+(5a+12)x^4+(a+5)x^3+(6a+2)x^2+(12a+10)x+8a+3$
- $y^2=(5a+9)x^6+(9a+4)x^5+(4a+3)x^4+(a+3)x^3+(a+5)x^2+(10a+3)x+9a+12$
- $y^2=(6a+7)x^6+(a+9)x^5+(a+6)x^4+(11a+1)x^3+(8a+7)x^2+(7a+2)x+5a+2$
- $y^2=(8a+7)x^6+(5a+11)x^5+(8a+5)x^4+(11a+2)x^3+(10a+8)x^2+(3a+7)x$
- $y^2=(10a+3)x^6+(4a+12)x^5+(2a+1)x^4+(5a+4)x^3+(8a+9)x^2+(a+4)x+11$
- $y^2=(7a+7)x^6+(8a+3)x^5+(12a+2)x^4+(6a+10)x^3+(8a+11)x^2+(2a+12)x+11a+4$
- $y^2=11ax^6+6x^5+(a+2)x^4+(9a+2)x^3+(4a+7)x^2+(10a+11)x+5a+8$
- $y^2=(2a+7)x^6+(3a+1)x^5+(11a+10)x^4+(11a+8)x^3+4ax^2+(2a+10)x+a+7$
- $y^2=(2a+12)x^6+(8a+2)x^5+(10a+1)x^4+(9a+6)x^3+10ax^2+(11a+7)x+a+7$
- $y^2=(9a+5)x^6+6ax^5+(4a+8)x^4+(6a+7)x^3+(8a+3)x^2+6ax+3a+12$
- $y^2=(3a+8)x^6+(12a+6)x^5+(2a+3)x^4+(10a+8)x^3+(8a+9)x^2+(7a+1)x+4a$
- $y^2=(a+10)x^6+(a+10)x^5+(8a+6)x^4+(10a+11)x^3+(4a+10)x^2+(4a+6)x+a+10$
- $y^2=(6a+6)x^6+(10a+9)x^5+(9a+8)x^4+(2a+1)x^3+(11a+5)x^2+(5a+8)x+2a+4$
- $y^2=(11a+9)x^6+(12a+7)x^5+8ax^4+2x^3+(3a+1)x^2+(10a+7)x+2a+10$
- $y^2=9ax^6+(2a+9)x^5+(10a+6)x^4+(6a+9)x^3+(6a+9)x^2+(3a+1)x+5a+3$
- $y^2=(4a+12)x^5+(5a+10)x^4+(7a+2)x^3+(9a+8)x^2+(9a+2)x+4a+8$
- $y^2=(9a+2)x^6+(9a+7)x^5+7ax^4+(6a+11)x^3+(a+1)x^2+(6a+3)x+5a+2$
- $y^2=(8a+8)x^6+(9a+9)x^5+(4a+2)x^4+(3a+9)x^3+(5a+12)x^2+(3a+6)x+9$
- $y^2=(5a+2)x^6+(8a+7)x^5+10x^3+(7a+8)x^2+(9a+4)x+11a+9$
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.az $\times$ 1.169.aw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.