Properties

Label 3.5.aj_bo_aeg
Base field $\F_{5}$
Dimension $3$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4} )$
  $1 - 9 x + 40 x^{2} - 110 x^{3} + 200 x^{4} - 225 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.147583617650$, $\pm0.200000000000$, $\pm0.400000000000$
Angle rank:  $1$ (numerical)
Jacobians:  $0$
Isomorphism classes:  2

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $22$ $15620$ $2336422$ $260416640$ $31249521952$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $25$ $147$ $665$ $3202$ $15985$ $79307$ $392305$ $1951437$ $9753600$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{10}}$.

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 2.5.af_p and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{10}}$ is 1.9765625.ajgk 2 $\times$ 1.9765625.sg. The endomorphism algebra for each factor is:
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.5.ab_a_k$2$3.25.ab_u_dw
3.5.b_a_ak$2$3.25.ab_u_dw
3.5.j_bo_eg$2$3.25.ab_u_dw

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

TwistExtension degreeCommon base change
3.5.ab_a_k$2$3.25.ab_u_dw
3.5.b_a_ak$2$3.25.ab_u_dw
3.5.j_bo_eg$2$3.25.ab_u_dw
3.5.ah_be_adc$4$(not in LMFDB)
3.5.ad_k_au$4$(not in LMFDB)
3.5.d_k_u$4$(not in LMFDB)
3.5.h_be_dc$4$(not in LMFDB)
3.5.ae_af_bo$5$(not in LMFDB)
3.5.b_a_ak$5$(not in LMFDB)
3.5.e_af_abo$10$(not in LMFDB)
3.5.ae_k_au$15$(not in LMFDB)
3.5.ae_p_abo$20$(not in LMFDB)
3.5.ac_af_u$20$(not in LMFDB)
3.5.ac_p_au$20$(not in LMFDB)
3.5.c_af_au$20$(not in LMFDB)
3.5.c_p_u$20$(not in LMFDB)
3.5.e_p_bo$20$(not in LMFDB)
3.5.e_k_u$30$(not in LMFDB)
3.5.ae_f_a$40$(not in LMFDB)
3.5.ac_f_a$40$(not in LMFDB)
3.5.c_f_a$40$(not in LMFDB)
3.5.e_f_a$40$(not in LMFDB)
3.5.ae_a_u$60$(not in LMFDB)
3.5.ac_a_k$60$(not in LMFDB)
3.5.ac_k_ak$60$(not in LMFDB)
3.5.c_a_ak$60$(not in LMFDB)
3.5.c_k_k$60$(not in LMFDB)
3.5.e_a_au$60$(not in LMFDB)