Invariants
Base field: | $\F_{53}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 25 x + 261 x^{2} - 1325 x^{3} + 2809 x^{4}$ |
Frobenius angles: | $\pm0.115138254762$, $\pm0.214344340489$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.54725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1721$ | $7608541$ | $22160816189$ | $62294176191941$ | $174908172262931536$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $29$ | $2707$ | $148853$ | $7894851$ | $418244994$ | $22164714523$ | $1174712914313$ | $62259695852643$ | $3299763593896409$ | $174887470356194022$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+20x^5+51x^4+29x^3+12x^2+49x+41$
- $y^2=3x^6+11x^5+11x^4+31x^3+46x^2+41x+5$
- $y^2=11x^6+51x^5+6x^4+32x^3+24x^2+10x+7$
- $y^2=44x^6+33x^5+12x^4+15x^3+16x^2+40x+20$
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 4.0.54725.1. |
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.z_kb | $2$ | (not in LMFDB) |