Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 21 x + 121 x^{2} )( 1 - 19 x + 121 x^{2} )$ |
$1 - 40 x + 641 x^{2} - 4840 x^{3} + 14641 x^{4}$ | |
Frobenius angles: | $\pm0.0963413489042$, $\pm0.168181340661$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 8 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10403$ | $209755689$ | $3135597358400$ | $45950759487607689$ | $672756847646784455603$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $82$ | $14324$ | $1769962$ | $214363684$ | $25937688802$ | $3138432778478$ | $379749886968562$ | $45949730385881284$ | $5559917317600953082$ | $672749994956047565204$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+3ax^5+(a+10)x^4+(10a+1)x^3+(a+10)x^2+3ax+a$
- $y^2=(6a+5)x^6+(10a+3)x^5+(4a+6)x^4+6x^3+(4a+6)x^2+(10a+3)x+6a+5$
- $y^2=(3a+2)x^6+(7a+2)x^5+(8a+2)x^4+(7a+8)x^3+(8a+2)x^2+(7a+2)x+3a+2$
- $y^2=(10a+3)x^6+(2a+10)x^5+5ax^4+(8a+8)x^3+5ax^2+(2a+10)x+10a+3$
- $y^2=(3a+5)x^6+(10a+6)x^5+(6a+6)x^4+6x^3+(6a+6)x^2+(10a+6)x+3a+5$
- $y^2=(2a+2)x^6+(9a+2)x^5+(7a+1)x^4+(10a+5)x^3+(7a+1)x^2+(9a+2)x+2a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11^{2}}$The isogeny class factors as 1.121.av $\times$ 1.121.at and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ac_agb | $2$ | (not in LMFDB) |
2.121.c_agb | $2$ | (not in LMFDB) |
2.121.bo_yr | $2$ | (not in LMFDB) |