Invariants
| Base field: | $\F_{109}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 35 x + 517 x^{2} - 3815 x^{3} + 11881 x^{4}$ |
| Frobenius angles: | $\pm0.0819402301809$, $\pm0.249080388952$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.206045.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $20$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $8549$ | $138912701$ | $1677054773849$ | $19927252691589125$ | $236738036884620098304$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $75$ | $11691$ | $1294995$ | $141169683$ | $15386348050$ | $1677100087731$ | $182803899546735$ | $19925626267174083$ | $2171893279305139455$ | $236736367484273873206$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=11 x^6+48 x^5+42 x^4+63 x^3+76 x^2+68 x+39$
- $y^2=103 x^6+54 x^5+13 x^4+19 x^3+54 x^2+70 x+73$
- $y^2=90 x^6+5 x^5+18 x^4+x^3+21 x^2+33 x+46$
- $y^2=93 x^6+54 x^5+42 x^4+10 x^3+55 x^2+52 x+107$
- $y^2=9 x^6+84 x^5+107 x^4+102 x^3+45 x^2+94 x+102$
- $y^2=107 x^6+34 x^5+106 x^4+14 x^3+45 x^2+91 x+107$
- $y^2=33 x^6+59 x^5+103 x^4+14 x^3+11 x^2+7 x+23$
- $y^2=32 x^6+32 x^5+34 x^4+23 x^3+83 x^2+5 x+50$
- $y^2=44 x^6+48 x^5+75 x^4+20 x^3+43 x^2+51 x+58$
- $y^2=45 x^6+101 x^5+14 x^4+15 x^3+28 x^2+35 x+30$
- $y^2=43 x^6+89 x^5+51 x^4+2 x^3+16 x^2+41 x+72$
- $y^2=66 x^6+105 x^5+92 x^4+21 x^3+62 x^2+31 x+7$
- $y^2=85 x^6+57 x^5+53 x^4+24 x^3+61 x^2+29 x+92$
- $y^2=40 x^6+14 x^5+19 x^4+105 x^3+94 x^2+2 x+23$
- $y^2=85 x^6+30 x^5+71 x^4+16 x^3+26 x^2+67 x+8$
- $y^2=79 x^6+79 x^5+43 x^4+55 x^3+107 x^2+61 x+55$
- $y^2=89 x^6+28 x^5+30 x^4+44 x^3+64 x^2+92 x+72$
- $y^2=72 x^6+59 x^5+29 x^3+18 x^2+21 x+96$
- $y^2=68 x^6+75 x^5+2 x^4+53 x^3+104 x^2+4 x+34$
- $y^2=19 x^6+26 x^5+102 x^4+90 x^3+78 x^2+100 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{109}$.
Endomorphism algebra over $\F_{109}$| The endomorphism algebra of this simple isogeny class is 4.0.206045.1. |
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 |
|---|---|---|
| 2.109.bj_tx | $2$ | (not in LMFDB) |