Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 2 x + 2 x^{2} - 4 x^{3} + 4 x^{4} )$ |
$1 - 9 x + 41 x^{2} - 126 x^{3} + 294 x^{4} - 552 x^{5} + 856 x^{6} - 1104 x^{7} + 1176 x^{8} - 1008 x^{9} + 656 x^{10} - 288 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.0833333333333$, $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.583333333333$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $6175$ | $321100$ | $18061875$ | $2133997561$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $6$ | $9$ | $18$ | $54$ | $87$ | $120$ | $242$ | $513$ | $1086$ |
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 $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 2.2.ad_f $\times$ 2.2.ac_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey 4 . The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 2 $\times$ 2.4.a_ae $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.4.a_ae : \(\Q(\zeta_{12})\).
- 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{3}}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.ae 2 $\times$ 1.8.e 2 $\times$ 2.8.a_l. The endomorphism algebra for each factor is: - 1.8.ae 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.8.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae 2 $\times$ 1.16.i 2 $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: - 1.16.ae 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 2.16.ah_bh : \(\Q(\sqrt{-3}, \sqrt{5})\).
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 4 $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: - 1.64.a 4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
- 1.64.l 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
Base change
This is a primitive isogeny class.