Invariants
Base field: | $\F_{113}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 34 x + 497 x^{2} - 3842 x^{3} + 12769 x^{4}$ |
Frobenius angles: | $\pm0.0129716274501$, $\pm0.295145069788$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.13888.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9391$ | $160989913$ | $2081753042812$ | $26583844632986809$ | $339452221851545632351$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $80$ | $12608$ | $1442762$ | $163043700$ | $18424106620$ | $2081946782774$ | $235260488589052$ | $26584441482378340$ | $3004041936382977482$ | $339456738994812143888$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+21x^5+31x^4+53x^3+36x^2+19x+74$
- $y^2=39x^6+10x^5+92x^4+8x^3+93x^2+26x+3$
- $y^2=111x^6+72x^5+25x^4+32x^3+110x^2+87x+84$
- $y^2=17x^6+65x^5+63x^4+71x^3+66x^2+68x+12$
- $y^2=30x^6+94x^5+59x^4+73x^3+14x^2+89x+65$
- $y^2=28x^6+93x^5+56x^4+64x^3+93x^2+87x+79$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{113}$.
Endomorphism algebra over $\F_{113}$The endomorphism algebra of this simple isogeny class is 4.0.13888.1. |
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 |
---|---|---|
2.113.bi_td | $2$ | (not in LMFDB) |