Properties

Label 2.4.ac_e
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 2 x + 4 x^{2} - 8 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.200000000000$, $\pm0.600000000000$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\zeta_{5})\)
Galois group:  $C_4$
Jacobians:  2

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 11 341 3641 69905 1185921 17043521 266354561 4311810305 68585520641 1095222947841

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 21 57 273 1153 4161 16257 65793 261633 1044481

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{10}}$ is 1.1024.cm 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.c_e$2$2.16.e_q
2.4.i_y$5$2.1024.ey_jci
2.4.ai_y$10$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.c_e$2$2.16.e_q
2.4.i_y$5$2.1024.ey_jci
2.4.ai_y$10$(not in LMFDB)
2.4.a_ai$10$(not in LMFDB)
2.4.ae_m$15$(not in LMFDB)
2.4.c_a$15$(not in LMFDB)
2.4.ae_i$20$(not in LMFDB)
2.4.a_i$20$(not in LMFDB)
2.4.e_i$20$(not in LMFDB)
2.4.ag_q$30$(not in LMFDB)
2.4.ac_a$30$(not in LMFDB)
2.4.a_e$30$(not in LMFDB)
2.4.e_m$30$(not in LMFDB)
2.4.g_q$30$(not in LMFDB)
2.4.a_a$40$(not in LMFDB)
2.4.ac_i$60$(not in LMFDB)
2.4.a_ae$60$(not in LMFDB)
2.4.c_i$60$(not in LMFDB)