Invariants
| Base field: | $\F_{79}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 79 x^{2} )^{2}$ |
| $1 - 158 x^{2} + 6241 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $1$, $1$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{79}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $15$ |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6084$ | $37015056$ | $243086469444$ | $1516136693760000$ | $9468276076472734404$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $80$ | $5926$ | $493040$ | $38925118$ | $3077056400$ | $243085483366$ | $19203908986160$ | $1517108654106238$ | $119851595982618320$ | $9468276070318621606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):
- $y^2=28 x^6+74 x^5+26 x^4+26 x^2+5 x+28$
- $y^2=x^6+56 x^3+78$
- $y^2=8 x^6+36 x^5+50 x^4+30 x^3+51 x^2+17 x+26$
- $y^2=51 x^6+63 x^5+58 x^4+27 x^2+20 x+45$
- $y^2=x^6+x^3+14$
- $y^2=40 x^6+6 x^5+52 x^4+32 x^2+x+4$
- $y^2=6 x^6+2 x^5+29 x^4+22 x^3+32 x^2+51 x+1$
- $y^2=71 x^6+76 x^5+41 x^4+49 x^3+26 x^2+52 x+65$
- $y^2=55 x^6+70 x^5+44 x^4+68 x^3+78 x^2+77 x+37$
- $y^2=18 x^6+49 x^5+53 x^4+31 x^3+75 x^2+18 x+46$
- $y^2=30 x^6+31 x^5+62 x^4+41 x^3+17 x^2+66 x+47$
- $y^2=4 x^6+24 x^5+73 x^4+76 x^2+73 x+40$
- $y^2=x^6+50 x^5+74 x^4+74 x^2+29 x+1$
- $y^2=59 x^6+61 x^5+13 x^4+17 x^3+18 x^2+32 x+63$
- $y^2=19 x^6+25 x^5+39 x^4+51 x^3+54 x^2+17 x+31$
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 the quaternion algebra over \(\Q(\sqrt{79}) \) ramified at both real infinite places. |
| The base change of $A$ to $\F_{79^{2}}$ is 1.6241.agc 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $79$ and $\infty$. |
Base change
This is a primitive isogeny class.