Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 18 x^{2} - 78 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.0497783698316$, $\pm0.549778369832$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{17})\) |
Galois group: | $C_2^2$ |
Jacobians: | $8$ |
Isomorphism classes: | 14 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $104$ | $28288$ | $4557800$ | $800210944$ | $137856264104$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $170$ | $2072$ | $28014$ | $371288$ | $4826810$ | $62727176$ | $815694814$ | $10604669096$ | $137858491850$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+10x^5+4x^4+2x^3+6x^2+2x+7$
- $y^2=9x^6+10x^5+7x^4+3x^3+5x$
- $y^2=6x^6+6x^5+4x^4+4x^2+7x+6$
- $y^2=8x^6+11x^5+2x^4+2x^2+2x+8$
- $y^2=3x^6+4x^5+8x^4+11x^3+6x^2+3x+6$
- $y^2=7x^6+x^5+7x^4+6x^3+11x^2+6x+8$
- $y^2=12x^6+4x^5+3x^4+5x^3+9x^2+x+7$
- $y^2=11x^6+11x^4+x^3+6x^2+11x+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Endomorphism algebra over $\F_{13}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{17})\). |
The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ako 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is the simple isogeny class 2.169.a_ako and its endomorphism algebra is \(\Q(i, \sqrt{17})\).
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 |
---|---|---|
2.13.g_s | $2$ | 2.169.a_ako |
2.13.a_ai | $8$ | (not in LMFDB) |
2.13.a_i | $8$ | (not in LMFDB) |