Invariants
Base field: | $\F_{53}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 22 x + 225 x^{2} - 1166 x^{3} + 2809 x^{4}$ |
Frobenius angles: | $\pm0.175017347257$, $\pm0.271252782182$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.444992.2 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1847$ | $7799881$ | $22269508028$ | $62328544875641$ | $174910739437262767$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $32$ | $2776$ | $149582$ | $7899204$ | $418251132$ | $22164539086$ | $1174710728636$ | $62259681698628$ | $3299763543851990$ | $174887470291635096$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=24x^6+44x^5+7x^4+9x^3+47x^2+8x+2$
- $y^2=20x^6+40x^5+40x^4+41x^3+18x^2+34x+21$
- $y^2=18x^6+4x^4+8x^3+2x^2+13x+30$
- $y^2=40x^6+31x^5+11x^4+6x^3+2x^2+38x+18$
- $y^2=17x^6+x^5+47x^4+20x^3+50x^2+45x+30$
- $y^2=40x^6+23x^5+10x^4+14x^3+19x^2+32x+3$
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 4.0.444992.2. |
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.w_ir | $2$ | (not in LMFDB) |