Properties

Label 2.1024.aer_ilc
Base Field $\F_{2^{10}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $2$
L-polynomial:  $( 1 - 32 x )^{2}( 1 - 57 x + 1024 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.150267280813$
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
$A(\F_{q^r})$ 930248 1096109357904 1152840305690007704 1208924233678783668420000 1267650578321682914084440710248 1329227995667562387786429070824101616 1393796574907089124010088438339208958721848 1461501637330384653068481681655718505866236680000 1532495540865840706528080954026754188503537671269028104 1606938044258987397078019150369087600619233732875638292303504

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 904 1045328 1073666200 1099510185376 1125899887385704 1152921504505059056 1180591620716500893496 1208925819614200475834176 1237940039285341378160398600 1267650600228227130789031530128

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The isogeny class factors as 1.1024.acm $\times$ 1.1024.acf 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 $\F_{2^{10}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.ah_acjo$2$(not in LMFDB)
2.1024.h_acjo$2$(not in LMFDB)
2.1024.er_ilc$2$(not in LMFDB)
2.1024.az_iq$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.ah_acjo$2$(not in LMFDB)
2.1024.h_acjo$2$(not in LMFDB)
2.1024.er_ilc$2$(not in LMFDB)
2.1024.az_iq$3$(not in LMFDB)
2.1024.acf_dau$4$(not in LMFDB)
2.1024.cf_dau$4$(not in LMFDB)
2.1024.adl_fsy$6$(not in LMFDB)
2.1024.z_iq$6$(not in LMFDB)
2.1024.dl_fsy$6$(not in LMFDB)