Properties

Label 2.73.abe_oh
Base field $\F_{73}$
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:  $\F_{73}$
Dimension:  $2$
L-polynomial:  $( 1 - 15 x + 73 x^{2} )^{2}$
  $1 - 30 x + 371 x^{2} - 2190 x^{3} + 5329 x^{4}$
Frobenius angles:  $\pm0.159004799845$, $\pm0.159004799845$
Angle rank:  $1$ (numerical)
Jacobians:  $3$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3481$ $27573001$ $151264989184$ $806711038270281$ $4297927782030847561$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $44$ $5172$ $388838$ $28407076$ $2073217244$ $151335766158$ $11047410984668$ $806460166467268$ $58871586916879094$ $4297625827388882772$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73}$.

Endomorphism algebra over $\F_{73}$
The isogeny class factors as 1.73.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.73.a_adb$2$(not in LMFDB)
2.73.be_oh$2$(not in LMFDB)
2.73.p_fw$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.73.a_adb$2$(not in LMFDB)
2.73.be_oh$2$(not in LMFDB)
2.73.p_fw$3$(not in LMFDB)
2.73.a_db$4$(not in LMFDB)
2.73.ap_fw$6$(not in LMFDB)