Properties

Label 2.53.aba_ko
Base field $\F_{53}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{53}$
Dimension:  $2$
L-polynomial:  $( 1 - 14 x + 53 x^{2} )( 1 - 12 x + 53 x^{2} )$
  $1 - 26 x + 274 x^{2} - 1378 x^{3} + 2809 x^{4}$
Frobenius angles:  $\pm0.0885855327829$, $\pm0.191645762723$
Angle rank:  $2$ (numerical)
Jacobians:  $8$
Isomorphism classes:  28

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1680$ $7539840$ $22114244880$ $62273046528000$ $174901377579104400$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $28$ $2682$ $148540$ $7892174$ $418228748$ $22164655914$ $1174712973644$ $62259698390686$ $3299763609603580$ $174887470340952282$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53}$.

Endomorphism algebra over $\F_{53}$
The isogeny class factors as 1.53.ao $\times$ 1.53.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.53.ac_ack$2$(not in LMFDB)
2.53.c_ack$2$(not in LMFDB)
2.53.ba_ko$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.53.ac_ack$2$(not in LMFDB)
2.53.c_ack$2$(not in LMFDB)
2.53.ba_ko$2$(not in LMFDB)
2.53.aq_fy$4$(not in LMFDB)
2.53.ai_cg$4$(not in LMFDB)
2.53.i_cg$4$(not in LMFDB)
2.53.q_fy$4$(not in LMFDB)