Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 5 x + 8 x^{2} )( 1 - 7 x + 24 x^{2} - 56 x^{3} + 64 x^{4} )$ |
$1 - 12 x + 67 x^{2} - 232 x^{3} + 536 x^{4} - 768 x^{5} + 512 x^{6}$ | |
Frobenius angles: | $\pm0.0585111942353$, $\pm0.154919815756$, $\pm0.418160225599$ |
Angle rank: | $3$ (numerical) |
This isogeny class is not simple, not primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $104$ | $221312$ | $131247896$ | $66851273216$ | $34871453017544$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $55$ | $501$ | $3983$ | $32477$ | $262711$ | $2101957$ | $16786207$ | $134210157$ | $1073693895$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.af $\times$ 2.8.ah_y 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.
Subfield | Primitive Model |
$\F_{2}$ | 3.2.a_b_ae |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
3.8.ac_ad_i | $2$ | (not in LMFDB) |
3.8.c_ad_ai | $2$ | (not in LMFDB) |
3.8.m_cp_iy | $2$ | (not in LMFDB) |