Properties

 Label 2.16.aj_bp Base Field $\F_{2^{4}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{2^{4}}$ Dimension: $2$ L-polynomial: $1 - 9 x + 41 x^{2} - 144 x^{3} + 256 x^{4}$ Frobenius angles: $\pm0.0608845495576$, $\pm0.454248801212$ Angle rank: $2$ (numerical) Number field: 4.0.3625.1 Galois group: $D_{4}$ Jacobians: 6

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

• $y^2+(x^3+x+1)y=(a^3+a^2+a)x^6+(a^2+a)x^5+(a^2+a)x^4+x^3+(a^3+a^2+a)x^2+x+a^3+1$
• $y^2+(x^3+ax+a)y=x^5+x^4+(a^3+a^2)x^3+(a^3+a^2+1)x+a^3+a^2+a+1$
• $y^2+(x^3+x+1)y=(a^3+a^2+1)x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+x^3+(a^3+a^2+1)x^2+x+a^3+a+1$
• $y^2+(x^3+(a^2+1)x+a^2+1)y=x^5+x^4+a^3x^3+(a^3+1)x+a^3+a^2$
• $y^2+(x^3+(a+1)x+a+1)y=x^5+x^4+(a^3+a)x^3+(a^3+a+1)x+a^3$
• $y^2+(x^3+a^2x+a^2)y=x^5+x^4+(a^3+a^2+a+1)x^3+(a^3+a^2+a)x+a^3+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 145 65395 16558420 4242892995 1096863465625 281507976344320 72063003609248545 18446491199204947395 4722337237744806168820 1208926899690386175371875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 8 258 4043 64738 1046048 16779183 268455608 4294908418 68719051163 1099512610098

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The endomorphism algebra of this simple isogeny class is 4.0.3625.1.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

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 2.16.j_bp $2$ 2.256.b_apj