Invariants
Base field: | $\F_{53}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 21 x + 208 x^{2} - 1113 x^{3} + 2809 x^{4}$ |
Frobenius angles: | $\pm0.129472067798$, $\pm0.324486313334$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1389564.2 |
Galois group: | $D_{4}$ |
Jacobians: | $36$ |
Isomorphism classes: | 36 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1884$ | $7822368$ | $22239489600$ | $62288483831424$ | $174888691634249004$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $33$ | $2785$ | $149382$ | $7894129$ | $418198413$ | $22164295030$ | $1174711787961$ | $62259710775169$ | $3299763802863726$ | $174887471469170425$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+x^5+45x^4+30x^3+48x^2+21x+21$
- $y^2=7x^6+5x^5+40x^4+13x^3+8x^2+16x+21$
- $y^2=48x^6+9x^5+10x^4+43x^3+16x^2+17x$
- $y^2=x^6+47x^5+20x^4+23x^3+50x^2+38x+48$
- $y^2=41x^6+14x^5+18x^4+17x^3+43x^2+2x+16$
- $y^2=18x^6+32x^5+31x^4+39x^3+19x^2+22x+33$
- $y^2=27x^6+7x^5+48x^4+48x^3+30x^2+49x+37$
- $y^2=39x^6+9x^5+49x^4+49x^3+42x^2+12x+32$
- $y^2=51x^6+14x^5+52x^4+3x^3+39x^2+34x+36$
- $y^2=20x^6+41x^5+28x^4+15x^3+50x^2+32$
- $y^2=26x^6+52x^5+42x^4+17x^3+33x^2+22x$
- $y^2=34x^6+24x^5+51x^4+4x^3+20x^2+46x+14$
- $y^2=21x^6+30x^5+17x^4+33x^3+52x^2+3x+11$
- $y^2=9x^6+50x^5+49x^4+12x^3+6x^2+6x+27$
- $y^2=50x^6+27x^5+34x^4+5x^3+18x^2+52x+48$
- $y^2=21x^6+43x^5+25x^4+49x^3+17x^2+12x+48$
- $y^2=20x^6+19x^5+17x^4+29x^3+51x^2+13x+30$
- $y^2=36x^6+28x^5+45x^4+37x^3+5x^2+46x+23$
- $y^2=45x^6+7x^5+20x^4+41x^3+37x^2+24x+26$
- $y^2=28x^6+3x^5+14x^4+47x^3+28x+22$
- and 16 more
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 4.0.1389564.2. |
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_ia | $2$ | (not in LMFDB) |