Properties

 Label 2.16.ai_bv 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: $( 1 - 5 x + 16 x^{2} )( 1 - 3 x + 16 x^{2} )$ Frobenius angles: $\pm0.285098958592$, $\pm0.377642706461$ Angle rank: $2$ (numerical) Jacobians: 16

This isogeny class is not 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 16 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 168 73920 17749368 4324320000 1097994472008 281298373836480 72052631272283928 18446948022222720000 4722385152182431137768 1208926117306605521208000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 9 287 4329 65983 1047129 16766687 268416969 4295014783 68719748409 1099511898527

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.af $\times$ 1.16.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ac_r $2$ 2.256.be_zx 2.16.c_r $2$ 2.256.be_zx 2.16.i_bv $2$ 2.256.be_zx