Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x + 179 x^{2} + 267 x^{3} + 7921 x^{4}$ |
| Frobenius angles: | $\pm0.506444355892$, $\pm0.544310223239$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.3106125.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8371$ | $65553301$ | $496430778811$ | $3934754623132245$ | $31182339371128214896$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $93$ | $8271$ | $704187$ | $62713003$ | $5584170378$ | $496983636591$ | $44231322011487$ | $3936588626470963$ | $350356405046478633$ | $31181719942891885806$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=51 x^6+57 x^5+41 x^4+75 x^3+27 x^2+68 x+8$
- $y^2=25 x^6+6 x^5+29 x^4+40 x^3+84 x^2+33 x+75$
- $y^2=24 x^6+6 x^5+36 x^4+42 x^3+2 x^2+41 x+6$
- $y^2=76 x^6+30 x^5+57 x^4+57 x^3+22 x^2+7 x+44$
- $y^2=68 x^6+3 x^5+46 x^4+56 x^3+34 x^2+5 x+64$
- $y^2=7 x^6+24 x^5+80 x^4+70 x^3+26 x^2+79 x+20$
- $y^2=57 x^6+24 x^5+31 x^4+73 x^3+70 x^2+26 x+1$
- $y^2=55 x^6+14 x^5+9 x^4+26 x^3+71 x^2+15 x+60$
- $y^2=3 x^6+21 x^5+56 x^4+54 x^3+40 x^2+87 x+9$
- $y^2=22 x^6+78 x^5+37 x^4+67 x^3+74 x^2+87 x+55$
- $y^2=39 x^6+26 x^5+37 x^4+14 x^3+55 x^2+39 x+78$
- $y^2=35 x^6+78 x^5+61 x^4+25 x^3+17 x^2+30 x+4$
- $y^2=3 x^6+49 x^5+71 x^4+87 x^3+37 x+10$
- $y^2=29 x^6+6 x^5+63 x^4+76 x^3+52 x^2+34 x+47$
- $y^2=50 x^6+12 x^5+44 x^4+87 x^3+43 x^2+35 x+43$
- $y^2=34 x^6+71 x^5+12 x^4+81 x^3+11 x^2+47 x+6$
- $y^2=x^6+61 x^5+52 x^4+78 x^3+85 x^2+40 x+17$
- $y^2=10 x^6+60 x^5+57 x^4+35 x^3+88 x^2+64 x+67$
- $y^2=52 x^6+62 x^5+35 x^4+12 x^3+47 x^2+24 x+6$
- $y^2=44 x^6+36 x^5+31 x^4+7 x^3+47 x^2+86 x+22$
- $y^2=56 x^6+19 x^5+81 x^4+17 x^3+82 x^2+33 x+26$
- $y^2=40 x^6+17 x^5+55 x^4+71 x^3+13 x^2+31 x+47$
- $y^2=21 x^6+30 x^5+76 x^4+25 x^3+31 x^2+80 x+71$
- $y^2=39 x^6+45 x^5+78 x^4+55 x^3+23 x^2+47 x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{89}$.
Endomorphism algebra over $\F_{89}$| The endomorphism algebra of this simple isogeny class is 4.0.3106125.1. |
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 |
|---|---|---|
| 2.89.ad_gx | $2$ | (not in LMFDB) |