Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x + 4 x^{2} )( 1 - 5 x + 13 x^{2} - 20 x^{3} + 16 x^{4} )$ |
$1 - 8 x + 32 x^{2} - 79 x^{3} + 128 x^{4} - 128 x^{5} + 64 x^{6}$ | |
Frobenius angles: | $\pm0.140237960897$, $\pm0.230053456163$, $\pm0.387712212190$ |
Angle rank: | $3$ (numerical) |
Isomorphism classes: | 1 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $4400$ | $356680$ | $19087200$ | $1106480250$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $17$ | $84$ | $289$ | $1057$ | $4190$ | $16713$ | $66049$ | $261876$ | $1046097$ |
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 $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$The isogeny class factors as 1.4.ad $\times$ 2.4.af_n 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 |
---|---|---|
3.4.ac_c_ab | $2$ | 3.16.a_q_bf |
3.4.c_c_b | $2$ | 3.16.a_q_bf |
3.4.i_bg_db | $2$ | 3.16.a_q_bf |