# Properties

 Label 4.2.ae_e_h_av Base field $\F_{2}$ Dimension $4$ $p$-rank $4$ Ordinary yes Supersingular no Simple yes Geometrically simple no Primitive yes Principally polarizable no Contains a Jacobian no

## Invariants

 Base field: $\F_{2}$ Dimension: $4$ L-polynomial: $1 - 4 x + 4 x^{2} + 7 x^{3} - 21 x^{4} + 14 x^{5} + 16 x^{6} - 32 x^{7} + 16 x^{8}$ Frobenius angles: $\pm0.0764513550391$, $\pm0.143118021706$, $\pm0.323548644961$, $\pm0.943118021706$ Angle rank: $1$ (numerical) Number field: $$\Q(\zeta_{15})$$ Galois group: $C_4\times C_2$

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1, 1, 1, 1]$

## Point counts

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $31$ $7471$ $55831$ $1343281$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $-3$ $14$ $13$ $39$ $54$ $146$ $301$ $608$ $1147$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{15})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{30}}$ is 1.1073741824.acgor 4 and its endomorphism algebra is $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{30}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.ai_be_acv_ft and its endomorphism algebra is $$\Q(\zeta_{15})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 4.8.f_h_z_ez and its endomorphism algebra is $$\Q(\zeta_{15})$$.
• Endomorphism algebra over $\F_{2^{5}}$  The base change of $A$ to $\F_{2^{5}}$ is 2.32.d_bj 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 4.64.al_cf_cz_agqx and its endomorphism algebra is $$\Q(\zeta_{15})$$.
• Endomorphism algebra over $\F_{2^{10}}$  The base change of $A$ to $\F_{2^{10}}$ is 2.1024.cj_dzt 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
• Endomorphism algebra over $\F_{2^{15}}$  The base change of $A$ to $\F_{2^{15}}$ is 2.32768.a_acgor 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
4.2.e_e_ah_av$2$4.4.ai_be_acv_ft
4.2.ab_e_ac_j$3$(not in LMFDB)
4.2.f_n_z_bn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
4.2.e_e_ah_av$2$4.4.ai_be_acv_ft
4.2.ab_e_ac_j$3$(not in LMFDB)
4.2.f_n_z_bn$3$(not in LMFDB)
4.2.b_e_c_j$5$(not in LMFDB)
4.2.g_t_bq_cr$5$(not in LMFDB)
4.2.af_n_az_bn$6$(not in LMFDB)
4.2.b_e_c_j$6$(not in LMFDB)
4.2.ag_t_abq_cr$10$(not in LMFDB)
4.2.ab_e_ac_j$10$(not in LMFDB)
4.2.a_b_a_ad$10$(not in LMFDB)
4.2.ag_t_abq_cr$15$(not in LMFDB)
4.2.af_n_az_bn$15$(not in LMFDB)
4.2.ad_e_ad_d$15$(not in LMFDB)
4.2.a_ac_a_j$15$(not in LMFDB)
4.2.a_b_a_ad$15$(not in LMFDB)
4.2.d_e_d_d$15$(not in LMFDB)
4.2.e_e_ah_av$15$(not in LMFDB)
4.2.a_ab_a_ad$20$(not in LMFDB)
4.2.ad_g_aj_n$60$(not in LMFDB)
4.2.a_a_a_h$60$(not in LMFDB)
4.2.a_c_a_j$60$(not in LMFDB)
4.2.d_g_j_n$60$(not in LMFDB)
4.2.a_a_a_ah$120$(not in LMFDB)