Invariants
Base field: | $\F_{23}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 23 x^{2} )( 1 - 4 x + 23 x^{2} )$ |
$1 - 12 x + 78 x^{2} - 276 x^{3} + 529 x^{4}$ | |
Frobenius angles: | $\pm0.186011988595$, $\pm0.363071407864$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $18$ |
Isomorphism classes: | 86 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $320$ | $286720$ | $151135040$ | $78561280000$ | $41431060001600$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $542$ | $12420$ | $280734$ | $6437052$ | $148038014$ | $3404908596$ | $78311540926$ | $1801153055340$ | $41426494425182$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+15x^5+5x^4+20x^3+15x^2+20x+17$
- $y^2=19x^6+7x^5+10x^4+12x^3+10x^2+7x+19$
- $y^2=8x^6+8x^4+12x^3+9x^2+6$
- $y^2=7x^6+12x^5+10x^4+7x^3+10x^2+12x+7$
- $y^2=11x^6+17x^4+7x^3+17x^2+11$
- $y^2=8x^6+19x^5+7x^4+20x^3+7x^2+19x+8$
- $y^2=15x^6+12x^5+x^4+7x^3+17x^2+17x+22$
- $y^2=15x^6+7x^5+5x^4+20x^3+5x^2+7x+15$
- $y^2=21x^6+15x^5+6x^4+13x^3+9x^2+5x+22$
- $y^2=11x^6+19x^5+18x^4+21x^3+x^2+21x+15$
- $y^2=18x^5+3x^4+18x^3+2x^2+8x$
- $y^2=10x^6+5x^5+2x^4+17x^3+22x^2+7x+21$
- $y^2=17x^6+x^5+11x^4+14x^3+17x^2+10x+18$
- $y^2=10x^6+7x^5+10x^4+9x^3+15x^2+9$
- $y^2=8x^6+3x^5+8x^4+x^2+3x+10$
- $y^2=11x^6+4x^5+20x^4+5x^3+17x^2+16x+19$
- $y^2=11x^6+22x^5+15x^4+20x^3+15x^2+22x+11$
- $y^2=7x^5+6x^4+13x^3+17x^2+9x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$The isogeny class factors as 1.23.ai $\times$ 1.23.ae 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 is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.23.ae_o | $2$ | (not in LMFDB) |
2.23.e_o | $2$ | (not in LMFDB) |
2.23.m_da | $2$ | (not in LMFDB) |