Invariants
| Base field: | $\F_{11^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 39 x + 619 x^{2} - 4719 x^{3} + 14641 x^{4}$ |
| Frobenius angles: | $\pm0.0803514571320$, $\pm0.202476610547$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.96837.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 18 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $10503$ | $210259557$ | $3136564759887$ | $45951580475250453$ | $672755805351053145648$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $83$ | $14359$ | $1770509$ | $214367515$ | $25937648618$ | $3138430943815$ | $379749851190065$ | $45949729891943443$ | $5559917312534381615$ | $672749994923143830574$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(3a+2)x^6+(a+4)x^5+(3a+3)x^4+6ax^3+(7a+5)x^2+(5a+7)x+7a+2$
- $y^2=(3a+3)x^6+(4a+9)x^5+(6a+8)x^4+(8a+1)x^3+(3a+4)x^2+(5a+1)x+8a+10$
- $y^2=(a+1)x^6+(7a+5)x^5+10x^4+(3a+2)x^3+(2a+2)x^2+(2a+10)x+3a+7$
- $y^2=(a+8)x^6+(5a+3)x^5+(9a+8)x^4+(8a+6)x^3+(10a+10)x^2+(10a+4)x+10a+9$
- $y^2=(9a+9)x^6+(2a+5)x^5+(3a+1)x^4+(a+4)x^3+(6a+9)x^2+(4a+6)x+1$
- $y^2=(6a+8)x^6+(3a+10)x^5+(2a+10)x^4+(8a+9)x^3+(5a+7)x^2+(9a+1)x+7a+7$
- $y^2=(9a+9)x^6+(5a+8)x^5+(8a+2)x^4+9ax^3+(5a+9)x^2+6ax+2a+7$
- $y^2=7x^6+(a+3)x^5+(9a+9)x^4+(5a+8)x^3+(4a+5)x^2+(4a+3)x+6a+5$
- $y^2=(a+10)x^6+(10a+4)x^5+(8a+2)x^4+2x^3+(3a+3)x^2+(6a+9)x+7a$
- $y^2=(9a+7)x^6+4x^5+9ax^4+(9a+7)x^3+(a+6)x^2+(3a+10)x+9a+10$
- $y^2=9ax^6+(7a+7)x^5+8ax^4+(7a+8)x^3+6ax^2+(10a+1)x+2a+5$
- $y^2=(a+2)x^6+(4a+10)x^5+(4a+2)x^4+(4a+7)x^3+(9a+7)x^2+(7a+8)x+8a+1$
- $y^2=5ax^6+(9a+7)x^5+(2a+7)x^4+(a+6)x^3+(a+6)x^2+(7a+5)x+4a$
- $y^2=(10a+6)x^6+(5a+5)x^5+(7a+10)x^4+(2a+9)x^3+7x^2+7ax+4a+6$
- $y^2=5ax^6+(a+8)x^5+(8a+1)x^4+(9a+5)x^3+(10a+9)x^2+(10a+10)x+7a+8$
- $y^2=(2a+9)x^6+(9a+5)x^5+(10a+1)x^4+(9a+1)x^3+(3a+7)x^2+(2a+6)x+3a+6$
- $y^2=(7a+9)x^6+(5a+2)x^5+(9a+6)x^4+(5a+5)x^3+(7a+7)x^2+(3a+9)x+7a+10$
- $y^2=(6a+4)x^6+(10a+1)x^5+(6a+2)x^4+(2a+9)x^3+(9a+2)x^2+(5a+5)x+4a$
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 4.0.96837.1. |
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.bn_xv | $2$ | (not in LMFDB) |