Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $1 - x^{2} + 2 x^{4} - 6 x^{6} + 8 x^{8} - 16 x^{10} + 64 x^{12}$ |
Frobenius angles: | $\pm0.0705005207583$, $\pm0.242162957214$, $\pm0.374012366975$, $\pm0.625987633025$, $\pm0.757837042786$, $\pm0.929499479242$ |
Angle rank: | $3$ (numerical) |
Number field: | 12.0.3339589787385856000000.1 |
Galois group: | 12T139 |
Jacobians: | $4$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/4, 1/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $52$ | $2704$ | $220324$ | $25969216$ | $1053213772$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $3$ | $9$ | $23$ | $33$ | $39$ | $129$ | $271$ | $513$ | $983$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which 0 are hyperelliptic), and hence is principally polarizable:
- $x^{5} + x^{2} y^{3} + y^{5} + x^{4} + x^{2} y + y^{3} + x y + y^{2} + y=0$
- $ x^{9} + x^{6} y + x^{6} + x^{5} y + x^{3} y^{3} + x^{5} + x^{3} y^{2} + x^{2} y^{3} + x y^{4} + x^{4} + x^{3} y + x y^{3} + x y^{2} + y^{3} + x^{2} + x + y + 1=0$
- $ x^{8} + x^{6} y^{2} + x^{5} y^{3} + x^{4} y^{4} + x^{3} y^{5} + x^{4} y^{3} + x^{2} y^{5} + x^{5} y + x^{4} y^{2} + x^{3} y^{3} + x^{2} y^{4} + x^{4} y + x^{3} y^{2} + x^{2} y^{3} + x y^{4} + x^{2} y^{2} + x y^{3} + y^{4} + x^{3} + x^{2} y + y^{3} + x^{2} + y^{2} + x + y + 1=0$
- $ x^{10} + x^{9} y + x^{8} y^{2} + x^{7} y^{3} + x^{6} y^{4} + x^{7} y^{2} + x^{6} y^{3} + x^{7} y + x^{5} y^{3} + x^{4} y^{4} + x^{7} + x^{5} y^{2} + x^{4} y^{3} + x^{3} y^{4} + x^{6} + x^{2} y^{4} + x^{3} y^{2} + x y^{4} + x^{2} y^{2} + x y^{3} + y^{4} + x y^{2} + y^{2}=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is 12.0.3339589787385856000000.1. |
The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 6.4.ac_f_aq_bg_acu_iu and its endomorphism algebra is the quaternion algebra over 6.0.1805912000.3 with the following ramification data at primes above $2$, and unramified at all archimedean places: | ||||||||||
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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 |
---|---|---|
6.2.a_b_a_c_a_g | $4$ | (not in LMFDB) |