Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 44 x + 807 x^{2} - 7436 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0314764500801$, $\pm0.254432080806$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.8240400.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21889$ | $806587761$ | $23293408834276$ | $665420133396508329$ | $19004940840147860278849$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $126$ | $28240$ | $4825842$ | $815735044$ | $137858325486$ | $23298076313206$ | $3937376192132334$ | $665416606414796164$ | $112455406925602790658$ | $19004963774767520405200$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(10a+4)x^6+(9a+12)x^5+6ax^4+(a+7)x^3+3ax^2+(3a+10)x+5a+12$
- $y^2=(2a+8)x^6+8ax^5+(a+8)x^4+(10a+4)x^3+(a+12)x^2+(5a+11)x+9a+2$
- $y^2=(7a+4)x^6+12x^5+(8a+3)x^4+(12a+7)x^3+(11a+2)x^2+(2a+8)x+7a+6$
- $y^2=(a+9)x^6+(6a+12)x^5+(10a+5)x^4+(6a+5)x^3+(10a+7)x^2+(2a+12)x+2a+10$
- $y^2=(11a+7)x^6+(5a+1)x^5+(9a+12)x^4+9x^3+(7a+11)x^2+(2a+12)x+5a+11$
- $y^2=(5a+4)x^6+(4a+6)x^5+(6a+5)x^4+(a+12)x^3+5x^2+(12a+7)x+3a+8$
- $y^2=(3a+11)x^6+(5a+11)x^5+(3a+2)x^4+(9a+8)x^3+(2a+9)x^2+(4a+9)x+8a+1$
- $y^2=(11a+7)x^6+(7a+11)x^5+10x^3+(8a+11)x^2+ax+11a+7$
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 4.0.8240400.2. |
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 |
---|---|---|
2.169.bs_bfb | $2$ | (not in LMFDB) |