Invariants
| Base field: | $\F_{13^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 25 x + 169 x^{2} )^{2}$ |
| $1 - 50 x + 963 x^{2} - 8450 x^{3} + 28561 x^{4}$ | |
| Frobenius angles: | $\pm0.0885687144757$, $\pm0.0885687144757$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
This isogeny class is not simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $21025$ | $799475625$ | $23269625299600$ | $665375421944975625$ | $19004927208209142600625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $120$ | $27988$ | $4820910$ | $815680228$ | $137858226600$ | $23298087024718$ | $3937376478082440$ | $665416611171280708$ | $112455406986047162430$ | $19004963775397054624948$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+ax^3+a$
- $y^2=11ax^6+6ax^5+3ax^4+9ax^3+ax^2+5ax+11a$
- $y^2=5ax^6+3ax^5+4ax^4+ax^3+4ax^2+3ax+5a$
- $y^2=7ax^6+11ax^5+5ax^4+11ax^3+2ax^2+8ax+7a$
- $y^2=(4a+7)x^6+(6a+9)x^5+(a+12)x^4+(8a+3)x^3+(5a+8)x^2+(3a+6)x+9a+5$
- $y^2=2ax^6+3ax^5+ax^4+5ax^3+9ax^2+9ax+2a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$| The isogeny class factors as 1.169.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.
| Subfield | Primitive Model |
| $\F_{13}$ | 2.13.a_az |