Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 47 x + 885 x^{2} - 7943 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0403592669200$, $\pm0.196341187291$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1078245.4 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21457$ | $803285709$ | $23284251363121$ | $665412276240794805$ | $19004983497851678762752$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $123$ | $28123$ | $4823943$ | $815725411$ | $137858634918$ | $23298086312611$ | $3937376354719287$ | $665416607961932131$ | $112455406927343082867$ | $19004963774524665642478$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(6a+10)x^6+3x^5+2x^4+(11a+1)x^3+(3a+7)x^2+(6a+8)x+8a$
- $y^2=ax^6+(4a+9)x^5+(3a+8)x^4+11x^3+(a+12)x^2+(11a+7)x+3a+9$
- $y^2=(8a+12)x^6+(11a+4)x^5+5ax^4+(a+8)x^3+(9a+1)x^2+(9a+5)x+5a+2$
- $y^2=(2a+7)x^6+(7a+6)x^5+(9a+10)x^4+(12a+11)x^3+(2a+2)x^2+(8a+10)x+3a+3$
- $y^2=(11a+2)x^6+(11a+12)x^5+(a+9)x^4+(5a+10)x^3+(6a+9)x^2+8x+7a+2$
- $y^2=(2a+10)x^6+(4a+5)x^5+(12a+10)x^4+(2a+1)x^3+(12a+2)x^2+(6a+11)x+10a$
- $y^2=(5a+2)x^6+(11a+3)x^5+(5a+9)x^4+12ax^3+(11a+3)x^2+(5a+3)x+12a+5$
- $y^2=(5a+7)x^6+(8a+10)x^5+(12a+9)x^4+(12a+2)x^3+(6a+5)x^2+(a+8)x+11a+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.1078245.4. |
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 |
---|---|---|
2.169.bv_bib | $2$ | (not in LMFDB) |