Invariants
Base field: | $\F_{79}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 29 x + 361 x^{2} - 2291 x^{3} + 6241 x^{4}$ |
Frobenius angles: | $\pm0.0818038217454$, $\pm0.268764181476$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3031805.2 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
This isogeny class is simple and 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4283$ | $38217209$ | $243158771849$ | $1517330396571245$ | $9468340693021932048$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $51$ | $6123$ | $493185$ | $38955771$ | $3077077396$ | $243087085551$ | $19203902805999$ | $1517108776838211$ | $119851596276034665$ | $9468276091223836078$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=77x^6+32x^5+36x^4+8x^3+58x^2+6x+21$
- $y^2=48x^6+21x^5+34x^4+18x^3+56x^2+74x+77$
- $y^2=58x^6+6x^5+70x^4+67x^3+48x^2+5x+55$
- $y^2=75x^6+47x^5+18x^4+65x^3+30x^2+27x+41$
- $y^2=3x^6+30x^5+64x^4+4x^2+49x+36$
- $y^2=6x^6+74x^5+42x^4+60x^3+52x^2+44x+27$
- $y^2=17x^6+16x^5+10x^4+49x^3+40x^2+72x+54$
- $y^2=30x^6+55x^5+67x^4+73x^3+22x^2+64x+33$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{79}$.
Endomorphism algebra over $\F_{79}$The endomorphism algebra of this simple isogeny class is 4.0.3031805.2. |
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 |
---|---|---|
2.79.bd_nx | $2$ | (not in LMFDB) |