Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 25 x + 169 x^{2} )( 1 - 23 x + 169 x^{2} )$ |
$1 - 48 x + 913 x^{2} - 8112 x^{3} + 28561 x^{4}$ | |
Frobenius angles: | $\pm0.0885687144757$, $\pm0.154420958311$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $14$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21315$ | $802190025$ | $23281414813440$ | $665412853037812425$ | $19005022727784799137075$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $122$ | $28084$ | $4823354$ | $815726116$ | $137858919482$ | $23298095471182$ | $3937376553351818$ | $665416611382640836$ | $112455406976199819626$ | $19004963775100756109524$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+4)x^6+(11a+8)x^5+(6a+8)x^4+(4a+11)x^3+(6a+8)x^2+(11a+8)x+4a+4$
- $y^2=3ax^6+(10a+6)x^5+(a+6)x^4+(8a+1)x^3+(a+6)x^2+(10a+6)x+3a$
- $y^2=(9a+4)x^6+(a+7)x^5+(11a+2)x^4+12ax^3+(11a+2)x^2+(a+7)x+9a+4$
- $y^2=3x^6+(6a+3)x^5+(8a+8)x^4+(5a+5)x^3+(8a+8)x^2+(6a+3)x+3$
- $y^2=3ax^6+(12a+1)x^5+(10a+9)x^4+(7a+5)x^3+(10a+9)x^2+(12a+1)x+3a$
- $y^2=(3a+8)x^6+(11a+2)x^5+(3a+1)x^4+(11a+10)x^3+(3a+1)x^2+(11a+2)x+3a+8$
- $y^2=(6a+2)x^6+12ax^5+(2a+7)x^4+(8a+5)x^3+(2a+7)x^2+12ax+6a+2$
- $y^2=(8a+5)x^6+11ax^5+(4a+9)x^4+(10a+4)x^3+(4a+9)x^2+11ax+8a+5$
- $y^2=(12a+8)x^6+(12a+12)x^5+(2a+8)x^4+(4a+9)x^3+(2a+8)x^2+(12a+12)x+12a+8$
- $y^2=(10a+4)x^6+5x^5+(5a+1)x^4+(4a+6)x^3+(5a+1)x^2+5x+10a+4$
- $y^2=(10a+2)x^6+(12a+8)x^5+(3a+7)x^4+(8a+6)x^3+(3a+7)x^2+(12a+8)x+10a+2$
- $y^2=(3a+6)x^6+(2a+5)x^5+(10a+12)x^4+(12a+2)x^3+(10a+12)x^2+(2a+5)x+3a+6$
- $y^2=(6a+3)x^6+(a+3)x^5+(5a+3)x^4+(2a+11)x^3+(5a+3)x^2+(a+3)x+6a+3$
- $y^2=(3a+3)x^6+5ax^5+(a+2)x^4+(6a+4)x^3+(a+2)x^2+5ax+3a+3$
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.ax 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.