Properties

Label 2.4.a_ae
Base field $\F_{2^{2}}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 4 x^{2} + 16 x^{4}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.833333333333$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\zeta_{12})\)
Galois group:  $C_2^2$
Jacobians:  $1$

This isogeny class is simple but not geometrically simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $13$ $169$ $4225$ $74529$ $1047553$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $9$ $65$ $289$ $1025$ $4353$ $16385$ $66049$ $262145$ $1046529$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Remainder of endomorphism lattice by field

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.ac_c
$\F_{2}$2.2.c_c

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.4.a_i$3$2.64.a_ey
2.4.ae_m$4$2.256.bg_bdo
2.4.a_e$4$2.256.bg_bdo

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.4.a_i$3$2.64.a_ey
2.4.ae_m$4$2.256.bg_bdo
2.4.a_e$4$2.256.bg_bdo
2.4.e_m$4$2.256.bg_bdo
2.4.a_i$6$(not in LMFDB)
2.4.ai_y$12$(not in LMFDB)
2.4.ag_q$12$(not in LMFDB)
2.4.ae_i$12$(not in LMFDB)
2.4.ae_m$12$(not in LMFDB)
2.4.ac_a$12$(not in LMFDB)
2.4.ac_i$12$(not in LMFDB)
2.4.a_ai$12$(not in LMFDB)
2.4.a_e$12$(not in LMFDB)
2.4.c_a$12$(not in LMFDB)
2.4.c_i$12$(not in LMFDB)
2.4.e_i$12$(not in LMFDB)
2.4.e_m$12$(not in LMFDB)
2.4.g_q$12$(not in LMFDB)
2.4.i_y$12$(not in LMFDB)
2.4.a_a$24$(not in LMFDB)
2.4.ac_e$60$(not in LMFDB)
2.4.c_e$60$(not in LMFDB)