Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 13 x )^{2}( 1 - 22 x + 169 x^{2} )$ |
$1 - 48 x + 910 x^{2} - 8112 x^{3} + 28561 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.178912375022$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $20$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21312$ | $802013184$ | $23279325915456$ | $665399220653260800$ | $19004958209563460165952$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $122$ | $28078$ | $4822922$ | $815709406$ | $137858451482$ | $23298084866446$ | $3937376348274218$ | $665416607901110206$ | $112455406923550852538$ | $19004963774387689319278$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+(a+6)x^5+(9a+5)x^4+(9a+12)x^3+4ax^2+(11a+9)x+a+12$
- $y^2=(a+5)x^6+(11a+9)x^5+3x^4+12ax^3+(4a+12)x^2+3ax+4a+9$
- $y^2=(10a+8)x^6+(9a+4)x^5+(8a+10)x^4+12ax^3+(10a+1)x^2+3ax+2a+4$
- $y^2=8ax^6+4ax^5+12ax^4+3ax^3+12ax^2+4ax+8a$
- $y^2=(9a+10)x^6+(5a+10)x^5+(6a+12)x^4+(3a+11)x^3+(6a+12)x^2+(5a+10)x+9a+10$
- $y^2=9x^6+3ax^5+(6a+8)x^4+(11a+2)x^3+(6a+8)x^2+3ax+9$
- $y^2=(6a+1)x^6+(5a+3)x^5+(7a+6)x^4+(4a+11)x^3+(7a+6)x^2+(5a+3)x+6a+1$
- $y^2=5ax^6+2ax^4+11ax^3+11ax^2+8a$
- $y^2=(2a+1)x^6+(4a+3)x^5+(5a+3)x^4+(8a+2)x^3+(2a+5)x^2+(a+9)x+10a+11$
- $y^2=(6a+4)x^6+(9a+2)x^5+(9a+4)x^4+(2a+8)x^3+(9a+4)x^2+(9a+2)x+6a+4$
- $y^2=(6a+8)x^6+(5a+10)x^5+(12a+8)x^4+9x^3+(12a+8)x^2+(5a+10)x+6a+8$
- $y^2=(11a+3)x^6+(12a+7)x^5+(4a+1)x^4+(4a+8)x^3+(9a+4)x^2+(2a+7)x+12a$
- $y^2=(7a+8)x^6+(5a+5)x^5+(6a+6)x^4+(11a+4)x^3+(6a+6)x^2+(5a+5)x+7a+8$
- $y^2=(12a+11)x^6+(10a+1)x^5+(4a+11)x^4+10x^3+(4a+11)x^2+(10a+1)x+12a+11$
- $y^2=9ax^6+5ax^5+9ax^4+8ax^3+10ax^2+2ax+4a$
- $y^2=(2a+2)x^6+(5a+11)x^5+(10a+12)x^4+(11a+9)x^3+(9a+10)x^2+4ax+3a+11$
- $y^2=(11a+3)x^6+(9a+7)x^5+(8a+8)x^4+(5a+10)x^3+(11a+7)x^2+(12a+10)x+3a+8$
- $y^2=9ax^6+7ax^5+8ax^4+11ax^2+11ax+9a$
- $y^2=12x^6+(11a+4)x^5+(6a+2)x^4+(2a+11)x^3+(6a+2)x^2+(11a+4)x+12$
- $y^2=(3a+11)x^6+(4a+2)x^5+(3a+1)x^4+(9a+4)x^3+(a+3)x^2+4ax+2a+2$
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.aba $\times$ 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:
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Base change
This is a primitive isogeny class.