Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 44 x + 808 x^{2} - 7436 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0449093466818$, $\pm0.252181585824$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.14362880.1 |
Galois group: | $D_{4}$ |
Jacobians: | $48$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21890$ | $806646500$ | $23294046862130$ | $665423815855370000$ | $19004955701325285042450$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $126$ | $28242$ | $4825974$ | $815739558$ | $137858433286$ | $23298078329202$ | $3937376223696174$ | $665416606856166078$ | $112455406931625121566$ | $19004963774855774310802$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 48 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(7a+3)x^6+(11a+10)x^5+(10a+6)x^4+(7a+2)x^3+(4a+12)x^2+7x+9a+1$
- $y^2=(10a+9)x^6+(a+10)x^5+(4a+3)x^4+(9a+5)x^3+5x^2+(9a+3)x+8a+5$
- $y^2=(10a+7)x^6+2ax^5+(4a+9)x^4+(2a+2)x^3+(a+3)x^2+(10a+9)x+a+9$
- $y^2=5ax^6+(4a+4)x^5+(10a+9)x^4+(12a+5)x^3+(12a+7)x^2+(a+11)x+4a+10$
- $y^2=10ax^6+(8a+11)x^5+2x^4+(2a+1)x^3+(12a+12)x^2+(6a+8)x+a+3$
- $y^2=(2a+3)x^6+(a+2)x^5+11ax^4+(9a+4)x^3+(12a+8)x^2+3x+3a+6$
- $y^2=(9a+4)x^6+(7a+11)x^5+(7a+8)x^4+(3a+7)x^3+(10a+1)x^2+(5a+2)x+3a+9$
- $y^2=(7a+12)x^6+(8a+12)x^5+(5a+6)x^4+(7a+11)x^3+(9a+8)x^2+(10a+9)x+9a+2$
- $y^2=(2a+11)x^6+(2a+2)x^5+(7a+8)x^4+(a+3)x^3+(10a+4)x^2+6x+10a+2$
- $y^2=(3a+10)x^6+(2a+1)x^5+(2a+8)x^4+(2a+12)x^3+(11a+4)x^2+(9a+6)x+8a+11$
- $y^2=(6a+2)x^6+9x^5+(11a+11)x^4+(5a+4)x^3+(5a+12)x^2+(8a+9)x+8a+12$
- $y^2=(5a+7)x^6+(5a+11)x^5+(7a+8)x^4+(10a+12)x^3+6ax^2+(8a+1)x+a$
- $y^2=(6a+7)x^6+4ax^5+(5a+10)x^4+(9a+7)x^3+(10a+4)x^2+(7a+5)x+2a+1$
- $y^2=(7a+5)x^6+(4a+9)x^5+(9a+1)x^4+(2a+2)x^3+(7a+5)x^2+(5a+8)x+5a+10$
- $y^2=4x^6+(6a+1)x^5+(4a+4)x^4+(3a+4)x^3+(a+7)x^2+(9a+6)x+3a+2$
- $y^2=(7a+11)x^6+(11a+6)x^5+(11a+11)x^4+(a+6)x^3+(9a+10)x^2+12ax+3a+1$
- $y^2=(4a+8)x^6+(11a+12)x^5+(6a+7)x^4+(8a+2)x^3+(a+2)x^2+12ax+12a+1$
- $y^2=(9a+5)x^6+12ax^4+(2a+4)x^3+(8a+7)x^2+(7a+11)x+9a$
- $y^2=(9a+11)x^6+(9a+2)x^5+(8a+10)x^4+(4a+6)x^3+7x^2+(12a+6)x+3a+7$
- $y^2=(2a+7)x^6+(9a+2)x^5+(12a+5)x^4+(a+1)x^3+(4a+5)x^2+(3a+11)x+2a+7$
- and 28 more
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 4.0.14362880.1. |
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.bs_bfc | $2$ | (not in LMFDB) |