Invariants
Base field: | $\F_{113}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 19 x + 113 x^{2} )( 1 - 17 x + 113 x^{2} )$ |
$1 - 36 x + 549 x^{2} - 4068 x^{3} + 12769 x^{4}$ | |
Frobenius angles: | $\pm0.148111132014$, $\pm0.205038125192$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 8 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9215$ | $160552945$ | $2082577615040$ | $26589151546802425$ | $339465155397052785575$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $78$ | $12572$ | $1443330$ | $163076244$ | $18424808598$ | $2081956626974$ | $235260584712294$ | $26584442062552036$ | $3004041936655364130$ | $339456738958019549132$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=43x^6+19x^5+107x^4+28x^3+107x^2+19x+43$
- $y^2=4x^6+53x^5+46x^4+28x^3+46x^2+53x+4$
- $y^2=12x^6+79x^5+46x^4+51x^3+46x^2+79x+12$
- $y^2=35x^6+76x^5+93x^4+9x^3+93x^2+76x+35$
- $y^2=33x^6+106x^5+49x^4+2x^3+49x^2+106x+33$
- $y^2=111x^6+69x^5+89x^4+67x^3+89x^2+69x+111$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{113}$.
Endomorphism algebra over $\F_{113}$The isogeny class factors as 1.113.at $\times$ 1.113.ar and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ac_adt | $2$ | (not in LMFDB) |
2.113.c_adt | $2$ | (not in LMFDB) |
2.113.bk_vd | $2$ | (not in LMFDB) |