Properties

Label 2.79.a_ac
Base field $\F_{79}$
Dimension $2$
$p$-rank $2$
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:  $\F_{79}$
Dimension:  $2$
L-polynomial:  $1 - 2 x^{2} + 6241 x^{4}$
Frobenius angles:  $\pm0.247985326663$, $\pm0.752014673337$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{10}, \sqrt{-39})\)
Galois group:  $C_2^2$
Jacobians:  $712$
Cyclic group of points:    no
Non-cyclic primes:   $2$

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.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6240$ $38937600$ $243087492960$ $1518081081753600$ $9468276082237596000$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $80$ $6238$ $493040$ $38975038$ $3077056400$ $243087530398$ $19203908986160$ $1517108654305918$ $119851595982618320$ $9468276081848344798$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 712 curves (of which all are hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{2}}$.

Endomorphism algebra over $\F_{79}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{10}, \sqrt{-39})\).
Endomorphism algebra over $\overline{\F}_{79}$
The base change of $A$ to $\F_{79^{2}}$ is 1.6241.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-390}) \)$)$

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.79.a_c$4$(not in LMFDB)