# Properties

 Label 4.2.af_n_az_bn Base Field $\F_{2}$ Dimension $4$ Ordinary Yes $p$-rank $4$ 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 6 7 8 9 10 $A(\F_{q^r})$ 1 241 1891 43621 929296 14127661 256112011 4581295525 57170567071 863591055616

 $r$ 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)