Invariants
| Base field: | $\F_{13^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 24 x + 169 x^{2} )( 1 - 22 x + 169 x^{2} )$ |
| $1 - 46 x + 866 x^{2} - 7774 x^{3} + 28561 x^{4}$ | |
| Frobenius angles: | $\pm0.125665916378$, $\pm0.178912375022$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $20$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $21608$ | $804854784$ | $23292543106664$ | $665446208805273600$ | $19005100745217109340648$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $124$ | $28178$ | $4825660$ | $815767006$ | $137859485404$ | $23298101431346$ | $3937376590406236$ | $665416611163804606$ | $112455406964198129980$ | $19004963774854246621778$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(5a+4)x^6+7x^5+(6a+1)x^4+(10a+1)x^3+(12a+2)x^2+2x+a+6$
- $y^2=(8a+9)x^6+7x^5+(7a+7)x^4+(3a+11)x^3+(a+1)x^2+2x+12a+7$
- $y^2=(2a+3)x^6+(5a+7)x^5+(2a+2)x^4+(5a+9)x^3+(5a+6)x^2+(4a+5)x+9a+1$
- $y^2=(6a+8)x^6+(11a+2)x^5+(4a+2)x^4+(8a+4)x^3+(11a+12)x^2+(6a+7)x+9a+12$
- $y^2=(2a+3)x^6+(9a+11)x^5+2ax^4+(3a+2)x^3+2ax^2+(9a+11)x+2a+3$
- $y^2=(a+3)x^6+(6a+7)x^5+12x^4+(2a+5)x^3+12x^2+(6a+7)x+a+3$
- $y^2=(2a+4)x^6+(8a+3)x^5+(6a+11)x^4+(11a+5)x^3+(6a+11)x^2+(8a+3)x+2a+4$
- $y^2=(12a+8)x^6+(10a+2)x^5+(12a+8)x^4+(11a+7)x^3+(12a+8)x^2+(10a+2)x+12a+8$
- $y^2=(10a+10)x^6+(2a+6)x^5+(3a+11)x^4+(3a+4)x^3+(8a+7)x^2+8x+3a+12$
- $y^2=(6a+11)x^6+(2a+3)x^5+(12a+9)x^4+(7a+9)x^3+(12a+9)x^2+(2a+3)x+6a+11$
- $y^2=(4a+12)x^6+(8a+5)x^5+9x^4+(7a+11)x^3+9x^2+(8a+5)x+4a+12$
- $y^2=(7a+1)x^6+2ax^5+(9a+6)x^4+(5a+12)x^3+(2a+10)x^2+7ax+4a+8$
- $y^2=(4a+1)x^6+(12a+9)x^5+(8a+10)x^4+(10a+1)x^3+(a+3)x^2+(3a+11)x+2a+10$
- $y^2=(11a+12)x^6+(10a+4)x^5+3x^4+(4a+12)x^3+(9a+10)x^2+(2a+2)x+7a+3$
- $y^2=(4a+8)x^6+(5a+4)x^5+(2a+8)x^4+(4a+5)x^3+(2a+8)x^2+(5a+4)x+4a+8$
- $y^2=(6a+7)x^6+(10a+6)x^5+(6a+5)x^4+(10a+5)x^3+8x^2+(5a+1)x+8a+1$
- $y^2=(12a+8)x^6+(10a+11)x^5+(6a+9)x^4+(9a+11)x^3+(6a+9)x^2+(10a+11)x+12a+8$
- $y^2=(7a+3)x^6+3ax^5+(11a+10)x^4+(8a+6)x^3+(11a+10)x^2+3ax+7a+3$
- $y^2=(2a+1)x^6+(a+4)x^5+(8a+12)x^4+(5a+7)x^3+(8a+12)x^2+(a+4)x+2a+1$
- $y^2=7ax^6+(3a+3)x^5+(7a+11)x^4+(3a+2)x^3+8x^2+(8a+6)x+5a+9$
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.ay $\times$ 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.