Properties

Label 2.256.acl_cfw
Base Field $\F_{2^{8}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x )^{2}( 1 - 31 x + 256 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.0797861753495$
Angle rank:  $1$ (numerical)

This isogeny class is not simple.

Newton polygon

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 50850 4232347200 281237242226850 18445878221841820800 1208922793868854295921250 79228152438505363164788280000 5192296826919398151796185684847650 340282366829178089486626309886229715200 22300745198292266885198135706553632988423650 1461501637330384653068481681655718505866236680000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 194 64576 16763042 4294765696 1099508875874 281474940914368 72057593599175714 18446744068735211776 4722366482819171313122 1208925819614200475834176

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The isogeny class factors as 1.256.abg $\times$ 1.256.abf and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{8}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.ab_asm$2$(not in LMFDB)
2.256.b_asm$2$(not in LMFDB)
2.256.cl_cfw$2$(not in LMFDB)
2.256.ap_q$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.ab_asm$2$(not in LMFDB)
2.256.b_asm$2$(not in LMFDB)
2.256.cl_cfw$2$(not in LMFDB)
2.256.ap_q$3$(not in LMFDB)
2.256.abf_ts$4$(not in LMFDB)
2.256.bf_ts$4$(not in LMFDB)
2.256.abv_bmu$6$(not in LMFDB)
2.256.p_q$6$(not in LMFDB)
2.256.bv_bmu$6$(not in LMFDB)