Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 21 x + 202 x^{2} - 1113 x^{3} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.0631169073278$, $\pm0.347174862571$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-8 + \sqrt{57}})\) |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1878$ | $7786188$ | $22182928488$ | $62243783504064$ | $174866561627229438$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $33$ | $2773$ | $149004$ | $7888465$ | $418145493$ | $22163962714$ | $1174710274449$ | $62259702732865$ | $3299763722378076$ | $174887470774636813$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=30 x^6+33 x^5+44 x^4+10 x+20$
- $y^2=40 x^6+50 x^5+10 x^4+49 x^3+46 x^2+17 x+39$
- $y^2=20 x^6+21 x^5+46 x^4+34 x^3+18 x^2+13 x+38$
- $y^2=30 x^6+28 x^5+31 x^4+35 x^3+47 x^2+28 x+38$
- $y^2=43 x^6+27 x^5+x^4+29 x^3+24 x+5$
- $y^2=22 x^6+48 x^5+27 x^4+x^3+10 x^2+7 x+30$
- $y^2=21 x^6+9 x^5+9 x^4+51 x^3+38 x^2+41 x+8$
- $y^2=18 x^6+21 x^5+38 x^4+6 x^3+37 x^2+23 x+13$
- $y^2=13 x^6+52 x^5+x^4+28 x^3+48 x^2+36 x+7$
- $y^2=48 x^6+43 x^5+25 x^4+x^3+5 x^2+3 x+26$
- $y^2=4 x^6+30 x^5+28 x^4+41 x^3+17 x^2+36 x+8$
- $y^2=20 x^6+33 x^5+41 x^4+23 x^3+32 x^2+2$
- $y^2=51 x^6+12 x^5+16 x^4+2 x^3+29 x^2+35 x+51$
- $y^2=40 x^6+35 x^5+13 x^4+28 x^3+35 x^2+31 x+18$
- $y^2=3 x^6+5 x^5+29 x^4+26 x^3+11 x^2+19 x+40$
- $y^2=43 x^6+x^5+8 x^4+31 x^3+29 x^2+19 x+45$
- $y^2=22 x^6+18 x^5+10 x^4+6 x^3+40 x^2+11 x+48$
- $y^2=12 x^6+7 x^5+44 x^3+52 x^2+25 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-8 + \sqrt{57}})\). |
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 |
|---|---|---|
| 2.53.v_hu | $2$ | (not in LMFDB) |