Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 36 x + 554 x^{2} - 4356 x^{3} + 14641 x^{4}$ |
Frobenius angles: | $\pm0.0704012798914$, $\pm0.270249945596$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.228672.1 |
Galois group: | $D_{4}$ |
Jacobians: | $42$ |
Isomorphism classes: | 54 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10804$ | $211628752$ | $3138618623764$ | $45951827911308288$ | $672750012704677556884$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $86$ | $14454$ | $1771670$ | $214368670$ | $25937425286$ | $3138426201750$ | $379749795960614$ | $45949729573765438$ | $5559917314417932854$ | $672749994994397781174$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 42 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(3a+9)x^6+5x^5+(3a+8)x^4+(4a+7)x^3+(2a+8)x^2+(10a+4)x+7a+2$
- $y^2=(7a+1)x^6+8x^5+(4a+2)x^4+(10a+1)x^3+(3a+2)x^2+(a+8)x+10a+1$
- $y^2=10x^6+(8a+3)x^5+(10a+8)x^4+(6a+10)x^3+(5a+8)x^2+3ax+6a$
- $y^2=(3a+10)x^6+(10a+1)x^4+(2a+9)x^3+(9a+10)x^2+(2a+3)x+3a+7$
- $y^2=(6a+6)x^6+(3a+10)x^5+(2a+5)x^4+x^3+6ax^2+ax+4a+2$
- $y^2=2x^6+4ax^5+(8a+8)x^4+3ax^3+(7a+2)x^2+(2a+1)x+6a+1$
- $y^2=(2a+6)x^6+6x^5+(2a+6)x^4+(2a+9)x^3+(4a+4)x^2+ax+10a+3$
- $y^2=(9a+10)x^6+(9a+5)x^5+(5a+10)x^4+(3a+4)x^3+(4a+9)x^2+(9a+6)x+4a$
- $y^2=(3a+9)x^6+(2a+9)x^5+(5a+2)x^4+3ax^3+(9a+7)x^2+(7a+8)x+a+6$
- $y^2=(9a+10)x^6+(6a+7)x^5+(9a+8)x^4+(6a+9)x^3+(9a+3)x^2+(3a+10)x+2a+7$
- $y^2=(3a+10)x^6+(5a+7)x^5+(10a+7)x^4+(10a+2)x^3+(4a+7)x^2+3x+6a+4$
- $y^2=(4a+8)x^6+(6a+3)x^5+(3a+4)x^4+(4a+7)x^3+(3a+1)x^2+(4a+4)x+6a+1$
- $y^2=(5a+7)x^6+(6a+3)x^5+(4a+4)x^4+(4a+5)x^3+(7a+6)x^2+8ax+9a+1$
- $y^2=6ax^6+(6a+5)x^5+x^4+(6a+5)x^3+(3a+5)x^2+(7a+6)x+4a+7$
- $y^2=(2a+3)x^6+(6a+10)x^5+(a+2)x^4+(5a+10)x^3+(7a+8)x^2+(10a+6)x+6a+1$
- $y^2=(3a+1)x^6+(10a+2)x^5+(7a+4)x^4+(10a+10)x^3+9ax^2+(8a+10)x+a+2$
- $y^2=(7a+1)x^6+7x^5+(3a+6)x^4+(3a+4)x^3+(9a+2)x^2+6x+a$
- $y^2=(5a+8)x^6+(6a+1)x^5+(a+10)x^4+2x^3+(4a+6)x^2+(a+4)x+6a+8$
- $y^2=(9a+10)x^6+(6a+8)x^5+(4a+2)x^4+(a+7)x^3+(8a+8)x^2+(9a+10)x+2a+1$
- $y^2=(7a+7)x^6+(3a+5)x^5+(9a+6)x^4+(4a+1)x^3+9x^2+(4a+4)x+1$
- and 22 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.228672.1. |
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 |
---|---|---|
2.121.bk_vi | $2$ | (not in LMFDB) |