Invariants
Base field: | $\F_{53}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 22 x + 224 x^{2} - 1166 x^{3} + 2809 x^{4}$ |
Frobenius angles: | $\pm0.161228510555$, $\pm0.280371056196$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.906048.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1846$ | $7793812$ | $22259616262$ | $62320349535184$ | $174906691046785126$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $32$ | $2774$ | $149516$ | $7898166$ | $418241452$ | $22164496022$ | $1174710980272$ | $62259688687774$ | $3299763617814032$ | $174887470758892934$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+33x^5+5x^4+18x^3+31x^2+12x+20$
- $y^2=43x^6+33x^5+37x^4+9x^3+13x^2+33x+27$
- $y^2=7x^6+20x^5+47x^4+32x^3+39x^2+41x+19$
- $y^2=22x^6+2x^5+23x^3+24x^2+39x+48$
- $y^2=50x^6+39x^5+24x^4+29x^3+17x^2+2x+2$
- $y^2=43x^6+41x^5+10x^4+41x^3+28x^2+45x+28$
- $y^2=43x^6+44x^5+2x^4+24x^3+43x^2+13x+26$
- $y^2=51x^6+41x^5+46x^4+50x^3+47x^2+24x+14$
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 4.0.906048.1. |
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 |
---|---|---|
2.53.w_iq | $2$ | (not in LMFDB) |