Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 47 x + 887 x^{2} - 7943 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0738822392977$, $\pm0.185751622566$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1240629.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21459$ | $803403501$ | $23285614822227$ | $665420910811195941$ | $19005022762936426138224$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $123$ | $28127$ | $4824225$ | $815735995$ | $137858919738$ | $23298092447447$ | $3937376466149613$ | $665416609715213299$ | $112455406951564816935$ | $19004963774820243271502$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(5a+3)x^6+6ax^5+(8a+2)x^4+(5a+8)x^3+(10a+9)x^2+(6a+3)x+8$
- $y^2=(a+7)x^6+(5a+3)x^5+(7a+11)x^4+(4a+10)x^3+(2a+6)x^2+7x+2a+4$
- $y^2=(3a+2)x^6+(2a+4)x^5+(7a+8)x^4+(5a+12)x^3+12x^2+8ax+9a+5$
- $y^2=5x^6+(2a+8)x^5+(8a+1)x^4+(10a+6)x^3+(12a+9)x^2+(7a+12)x+5a+10$
- $y^2=(a+6)x^6+(9a+2)x^5+(6a+8)x^4+(4a+4)x^3+(8a+4)x^2+(3a+3)x+6a+8$
- $y^2=(11a+1)x^6+(4a+6)x^5+(3a+3)x^4+(11a+1)x^3+(2a+7)x^2+(3a+10)x+9a+12$
- $y^2=4ax^6+(4a+3)x^5+(7a+12)x^4+(2a+4)x^3+(2a+3)x^2+(7a+6)x+5a+11$
- $y^2=(4a+8)x^6+(4a+1)x^5+(5a+1)x^4+(6a+11)x^3+(11a+7)x^2+(11a+4)x+a+12$
- $y^2=(3a+1)x^6+(12a+5)x^5+(8a+3)x^4+(4a+3)x^3+7ax^2+(11a+3)x+3a+9$
- $y^2=(8a+5)x^6+(6a+8)x^5+(a+9)x^4+(8a+10)x^3+(10a+3)x^2+(a+6)x+12a+11$
- $y^2=(9a+6)x^6+(8a+10)x^5+(10a+10)x^4+(11a+6)x^3+(12a+1)x^2+(11a+5)x+5a+4$
- $y^2=12x^6+(5a+3)x^5+(5a+1)x^4+(11a+3)x^3+(a+6)x^2+(9a+7)x+2a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.1240629.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.169.bv_bid | $2$ | (not in LMFDB) |