Properties

Label 2.7.ag_s
Base field F7\F_{7}
Dimension 22
pp-rank 22
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  F7\F_{7}
Dimension:  22
L-polynomial:  16x+18x242x3+49x41 - 6 x + 18 x^{2} - 42 x^{3} + 49 x^{4}
Frobenius angles:  ±0.0461154155528\pm0.0461154155528, ±0.453884584447\pm0.453884584447
Angle rank:  11 (numerical)
Number field:  Q(i,5)\Q(i, \sqrt{5})
Galois group:  C22C_2^2
Jacobians:  22
Isomorphism classes:  3

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

pp-rank:  22
Slopes:  [0,0,1,1][0, 0, 1, 1]

Point counts

Point counts of the abelian variety

rr 11 22 33 44 55
A(Fqr)A(\F_{q^r}) 2020 23202320 111620111620 53824005382400 276390500276390500

Point counts of the curve

rr 11 22 33 44 55 66 77 88 99 1010
C(Fqr)C(\F_{q^r}) 22 5050 326326 22382238 1644216442 117650117650 824126824126 57609585760958 4033800240338002 282475250282475250

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over F74\F_{7^{4}}.

Endomorphism algebra over F7\F_{7}
The endomorphism algebra of this simple isogeny class is Q(i,5)\Q(i, \sqrt{5}).
Endomorphism algebra over F7\overline{\F}_{7}
The base change of AA to F74\F_{7^{4}} is 1.2401.ade 2 and its endomorphism algebra is M2(\mathrm{M}_{2}(Q(5)\Q(\sqrt{-5}) ))
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.7.g_s222.49.a_ade
2.7.a_ae88(not in LMFDB)
2.7.a_e88(not in LMFDB)