Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 36 x + 551 x^{2} - 4356 x^{3} + 14641 x^{4}$ |
Frobenius angles: | $\pm0.0342210467229$, $\pm0.278047975429$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.5706000.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 16 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10801$ | $211537585$ | $3138043433476$ | $45949915195625385$ | $672745642758211956241$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $86$ | $14448$ | $1771346$ | $214359748$ | $25937256806$ | $3138423758358$ | $379749766712486$ | $45949729257345028$ | $5559917310852948866$ | $672749994947905440048$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+2)x^6+(8a+10)x^5+(9a+1)x^4+(4a+9)x^3+(5a+6)x^2+(2a+9)x+2$
- $y^2=(9a+2)x^6+(10a+2)x^5+(8a+3)x^4+(5a+7)x^3+(a+6)x^2+(9a+1)x+9a+1$
- $y^2=(a+10)x^6+(6a+1)x^5+(3a+7)x^4+2ax^3+(3a+7)x^2+(5a+8)x+6a+9$
- $y^2=(7a+4)x^6+10ax^5+(a+5)x^4+(9a+7)x^3+(7a+5)x^2+(8a+6)x+7a$
- $y^2=(3a+2)x^6+4x^5+(9a+6)x^4+4ax^3+8ax^2+(a+6)x+3a$
- $y^2=(6a+6)x^6+(4a+8)x^5+5ax^4+(3a+6)x^3+(10a+3)x^2+(a+10)x+7a+1$
- $y^2=(6a+4)x^6+ax^5+(8a+1)x^4+(7a+10)x^3+2ax^2+(10a+8)x+10a$
- $y^2=(10a+4)x^6+(5a+7)x^5+(2a+9)x^4+6x^3+(5a+6)x^2+(10a+6)x+2a+2$
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 4.0.5706000.1. |
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_vf | $2$ | (not in LMFDB) |