Invariants
| Base field: | $\F_{11^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 37 x + 575 x^{2} - 4477 x^{3} + 14641 x^{4}$ |
| Frobenius angles: | $\pm0.0651084016179$, $\pm0.251993505151$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.6698517.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $10703$ | $211180893$ | $3137968912463$ | $45951714026541477$ | $672751361538452773808$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $85$ | $14423$ | $1771303$ | $214368139$ | $25937477290$ | $3138427050239$ | $379749799443379$ | $45949729463477395$ | $5559917311609076929$ | $672749994959514652718$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+7)x^6+(8a+1)x^5+(10a+5)x^4+x^3+(2a+1)x^2+(4a+10)x+7a+2$
- $y^2=(6a+7)x^6+(10a+4)x^5+(4a+2)x^4+ax^3+(10a+2)x^2+(5a+1)x+7a+5$
- $y^2=(6a+9)x^6+(10a+6)x^5+(3a+8)x^4+(10a+5)x^3+(7a+7)x^2+(6a+4)x+5a+8$
- $y^2=(10a+9)x^6+(5a+9)x^5+(7a+6)x^4+(8a+7)x^3+(10a+10)x^2+(10a+8)x+3a+5$
- $y^2=(7a+10)x^6+(7a+10)x^5+3x^4+3x^2+(10a+1)x+6a+6$
- $y^2=(10a+4)x^6+(a+3)x^5+(6a+1)x^4+(6a+1)x^3+(5a+2)x^2+(9a+3)x+3a+8$
- $y^2=(6a+5)x^6+6x^5+(6a+4)x^4+10x^3+(4a+7)x^2+(4a+1)x+2$
- $y^2=10x^6+(a+10)x^5+(8a+5)x^4+(2a+3)x^3+(a+1)x^2+(8a+3)x+9a$
- $y^2=(3a+9)x^6+(a+6)x^5+(7a+10)x^4+(9a+2)x^3+(2a+4)x^2+(8a+9)x+7$
- $y^2=(3a+8)x^6+(10a+2)x^5+(2a+1)x^4+(7a+2)x^3+(2a+6)x^2+(9a+7)x+7a+5$
- $y^2=(a+3)x^6+(2a+5)x^5+(2a+5)x^4+(2a+3)x^3+(7a+3)x^2+(10a+6)x+6a+8$
- $y^2=(5a+8)x^6+(6a+1)x^5+(2a+5)x^4+(4a+4)x^3+(6a+1)x^2+(4a+9)x+9a+3$
- $y^2=(6a+9)x^6+(2a+8)x^5+(6a+2)x^4+(3a+8)x^3+(9a+6)x^2+(8a+10)x+5a+5$
- $y^2=(10a+1)x^6+(9a+5)x^5+(4a+10)x^4+4x^3+(9a+6)x^2+(a+1)x+a+1$
- $y^2=(6a+1)x^6+(a+2)x^5+(5a+8)x^4+(5a+8)x^3+(4a+6)x^2+(7a+7)x+8a$
- $y^2=(3a+10)x^6+2x^5+(5a+1)x^4+(a+9)x^3+9ax^2+8x+2a+9$
- $y^2=(a+7)x^5+(5a+1)x^4+(10a+6)x^3+(7a+4)x^2+(5a+8)x+10a+10$
- $y^2=(9a+9)x^6+(7a+6)x^5+(10a+6)x^4+(8a+7)x^2+(2a+4)x+5a+4$
- $y^2=(8a+1)x^6+(a+10)x^5+(5a+3)x^4+10ax^3+(4a+4)x^2+5x+a+10$
- $y^2=4x^6+(7a+1)x^5+(7a+7)x^4+(2a+9)x^3+(a+9)x^2+(4a+10)x+5a+6$
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 4.0.6698517.1. |
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_wd | $2$ | (not in LMFDB) |