# Properties

 Label 4.2.af_n_az_bn 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: $1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 50 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8}$ Frobenius angles: $\pm0.00978468837242$, $\pm0.190215311628$, $\pm0.409784688372$, $\pm0.609784688372$ 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $241$ $1891$ $43621$ $929296$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $6$ $4$ $10$ $33$ $54$ $124$ $274$ $418$ $781$

## 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 $\F_{2^{10}}$ is 1.1024.acj 4 and its endomorphism algebra is $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{10}}$.
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.b_ad_ah_f 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.a_acj 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.f_n_z_bn$2$4.4.b_ad_ah_f
4.2.b_e_c_j$3$(not in LMFDB)
4.2.e_e_ah_av$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
4.2.f_n_z_bn$2$4.4.b_ad_ah_f
4.2.b_e_c_j$3$(not in LMFDB)
4.2.e_e_ah_av$3$(not in LMFDB)
4.2.a_ac_a_j$5$(not in LMFDB)
4.2.f_n_z_bn$5$(not in LMFDB)
4.2.ae_e_h_av$6$(not in LMFDB)
4.2.ab_e_ac_j$6$(not in LMFDB)
4.2.ag_t_abq_cr$15$(not in LMFDB)
4.2.ae_e_h_av$15$(not in LMFDB)
4.2.ad_e_ad_d$15$(not in LMFDB)
4.2.ab_e_ac_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.g_t_bq_cr$15$(not in LMFDB)
4.2.a_a_a_h$20$(not in LMFDB)
4.2.a_c_a_j$20$(not in LMFDB)
4.2.a_a_a_ah$40$(not in LMFDB)
4.2.ad_g_aj_n$60$(not in LMFDB)
4.2.a_ab_a_ad$60$(not in LMFDB)
4.2.d_g_j_n$60$(not in LMFDB)