Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 37 x + 574 x^{2} - 4477 x^{3} + 14641 x^{4}$ |
Frobenius angles: | $\pm0.0524895598519$, $\pm0.255234509565$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.5480060.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10702$ | $211150460$ | $3137771819800$ | $45951032703823040$ | $672749725267483149262$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $85$ | $14421$ | $1771192$ | $214364961$ | $25937414205$ | $3138426081858$ | $379749787274005$ | $45949729331832321$ | $5559917310277475752$ | $672749994945105365701$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+4)x^6+(6a+3)x^5+(7a+4)x^4+(8a+3)x^3+7x^2+(2a+2)x+7a+4$
- $y^2=(6a+4)x^6+(4a+2)x^5+(4a+1)x^4+(2a+2)x^3+(3a+9)x^2+(2a+2)x+5a+4$
- $y^2=(3a+2)x^6+(7a+5)x^5+(6a+2)x^4+(4a+5)x^3+(8a+5)x^2+(8a+9)x+9$
- $y^2=(6a+2)x^6+(10a+8)x^5+9x^4+(7a+10)x^3+(10a+3)x^2+(4a+10)x+9a+5$
- $y^2=(8a+4)x^6+5x^5+(a+2)x^4+(a+2)x^3+(6a+9)x^2+(9a+9)x+9a+5$
- $y^2=(8a+1)x^6+(a+8)x^5+(4a+2)x^4+(8a+2)x^3+7ax^2+3ax+5a+6$
- $y^2=(10a+3)x^6+9x^5+(9a+10)x^4+3ax^3+(10a+7)x^2+(9a+3)x+10a+1$
- $y^2=(5a+3)x^6+5x^5+(5a+8)x^4+(5a+9)x^3+(3a+6)x^2+(8a+7)x+10a+9$
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 4.0.5480060.1. |
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.bl_wc | $2$ | (not in LMFDB) |