Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 241 x^{2} - 1357 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.104529320573$, $\pm0.314395470244$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-378 -46 \sqrt{37}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $18$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2343$ | $11956329$ | $42260801121$ | $146867205288573$ | $511110900884744688$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $37$ | $3435$ | $205771$ | $12120395$ | $714916112$ | $42180308559$ | $2488651035929$ | $146830459855843$ | $8662996161687451$ | $511116756002141430$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=31 x^6+52 x^5+31 x^4+39 x^3+25 x^2+41 x+19$
- $y^2=37 x^6+20 x^5+46 x^4+4 x^3+7 x^2+11 x+25$
- $y^2=7 x^6+52 x^5+23 x^4+7 x^3+20 x^2+28 x+39$
- $y^2=50 x^6+15 x^5+30 x^4+3 x^3+38 x^2+40 x+32$
- $y^2=23 x^6+27 x^5+19 x^4+17 x^3+45 x^2+40 x+3$
- $y^2=38 x^6+50 x^5+10 x^4+26 x^3+12 x^2+11 x+24$
- $y^2=57 x^6+18 x^5+37 x^4+52 x^3+29 x^2+37 x+17$
- $y^2=11 x^6+3 x^5+38 x^4+22 x^3+42 x^2+9 x+33$
- $y^2=18 x^6+6 x^5+41 x^4+17 x^3+21 x^2+57 x+57$
- $y^2=44 x^6+44 x^5+10 x^4+9 x^3+46 x^2+14 x+6$
- $y^2=18 x^6+50 x^5+33 x^4+31 x^3+20 x^2+12 x+50$
- $y^2=30 x^6+10 x^5+5 x^4+2 x^3+4 x^2+32 x+44$
- $y^2=48 x^6+22 x^5+48 x^4+44 x^3+51 x^2+23 x+31$
- $y^2=55 x^6+35 x^5+17 x^4+57 x^3+55 x^2+44 x+23$
- $y^2=7 x^6+10 x^5+9 x^4+51 x^3+5 x^2+51 x+55$
- $y^2=38 x^6+7 x^5+40 x^4+38 x^3+48 x^2+48 x+58$
- $y^2=4 x^6+2 x^5+39 x^4+34 x^3+45 x^2+6 x+27$
- $y^2=58 x^6+x^5+21 x^4+48 x^3+20 x^2+50 x+29$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-378 -46 \sqrt{37}})\). |
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 |
|---|---|---|
| 2.59.x_jh | $2$ | (not in LMFDB) |