Invariants
Base field: | $\F_{211}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 29 x + 211 x^{2} )( 1 - 24 x + 211 x^{2} )$ |
$1 - 53 x + 1118 x^{2} - 11183 x^{3} + 44521 x^{4}$ | |
Frobenius angles: | $\pm0.0189887838440$, $\pm0.190546645091$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $34404$ | $1956761904$ | $88202136870000$ | $3928755469645116864$ | $174914010466990731384684$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $159$ | $43949$ | $9389268$ | $1982098249$ | $418227244929$ | $88245938952038$ | $18619893149142339$ | $3928797474565606609$ | $828976267854847699308$ | $174913992533916160958429$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=210x^6+146x^5+203x^4+x^3+170x^2+104x+60$
- $y^2=159x^6+200x^5+163x^4+194x^3+113x^2+189x+97$
- $y^2=66x^6+67x^5+111x^4+6x^3+13x^2+28x+18$
- $y^2=31x^6+128x^5+204x^4+138x^3+40x^2+198x+41$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{211}$.
Endomorphism algebra over $\F_{211}$The isogeny class factors as 1.211.abd $\times$ 1.211.ay 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.