Invariants
| Base field: | $\F_{89}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 30 x + 400 x^{2} - 2670 x^{3} + 7921 x^{4}$ |
| Frobenius angles: | $\pm0.152925175432$, $\pm0.251753890986$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1970496.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 28 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5622$ | $61965684$ | $497680359462$ | $3937924423317456$ | $31182910194749502102$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $60$ | $7822$ | $705960$ | $62763526$ | $5584272600$ | $496982607262$ | $44231338111020$ | $3936588776166718$ | $350356403361388140$ | $31181719929556363102$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=83 x^6+51 x^5+65 x^4+69 x^3+63 x^2+88 x+23$
- $y^2=2 x^6+19 x^5+42 x^4+60 x^3+88 x^2+64 x+88$
- $y^2=52 x^6+30 x^5+36 x^4+x^3+70 x^2+49 x+83$
- $y^2=48 x^6+8 x^5+41 x^4+60 x^3+3 x^2+59 x+73$
- $y^2=73 x^6+16 x^5+15 x^4+37 x^2+55 x+45$
- $y^2=x^6+67 x^5+41 x^4+3 x^3+61 x^2+51 x+74$
- $y^2=3 x^6+62 x^5+15 x^4+76 x^3+49 x^2+66 x+52$
- $y^2=86 x^6+32 x^5+78 x^4+8 x^3+83 x^2+64 x+5$
- $y^2=80 x^6+63 x^5+55 x^4+6 x^3+30 x^2+9 x+33$
- $y^2=4 x^6+32 x^5+86 x^4+9 x^3+42 x^2+88 x+46$
- $y^2=40 x^6+7 x^5+17 x^3+18 x^2+33 x+42$
- $y^2=15 x^6+81 x^5+87 x^4+80 x^3+72 x^2+32 x+12$
- $y^2=35 x^6+86 x^5+3 x^4+51 x^3+22 x^2+87 x+7$
- $y^2=40 x^6+35 x^5+69 x^4+17 x^3+15 x^2+49 x+70$
- $y^2=23 x^6+88 x^5+33 x^4+11 x^3+36 x^2+18 x+65$
- $y^2=39 x^6+57 x^5+58 x^4+26 x^3+46 x^2+51 x+6$
- $y^2=64 x^6+28 x^5+2 x^4+60 x^3+76 x^2+88 x+30$
- $y^2=60 x^6+73 x^5+85 x^4+47 x^3+67 x^2+68 x+27$
- $y^2=76 x^6+27 x^5+8 x^4+70 x^3+20 x^2+54 x+10$
- $y^2=82 x^6+78 x^5+47 x^4+35 x^3+25 x^2+73 x+65$
- $y^2=76 x^6+75 x^5+4 x^4+59 x^3+49 x^2+57 x+50$
- $y^2=34 x^6+59 x^5+24 x^4+74 x^3+31 x^2+74 x+66$
- $y^2=56 x^6+12 x^5+47 x^4+5 x^3+66 x^2+3 x+13$
- $y^2=6 x^6+54 x^5+57 x^4+50 x^3+78 x^2+33 x+85$
- $y^2=86 x^6+86 x^5+88 x^4+35 x^3+71 x^2+43 x+26$
- $y^2=27 x^6+62 x^5+44 x^4+55 x^3+12 x^2+70 x+23$
- $y^2=76 x^6+71 x^5+54 x^4+25 x^3+80 x^2+9 x+12$
- $y^2=26 x^6+43 x^5+83 x^4+7 x^3+7 x^2+36 x+18$
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 4.0.1970496.2. |
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 |
|---|---|---|
| 2.89.be_pk | $2$ | (not in LMFDB) |