# Properties

 Label 2.64.abd_mz Base Field $\F_{2^{6}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $1 - 29 x + 337 x^{2} - 1856 x^{3} + 4096 x^{4}$ Frobenius angles: $\pm0.0696923787592$, $\pm0.184671871212$ Angle rank: $2$ (numerical) Number field: 4.0.23225.1 Galois group: $D_{4}$ Jacobians: 3

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 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2549 16107131 68552612276 281480634897299 1152957594379246749 4722391294383123317696 19342823591656104908403869 79228164883005478003360636259 324518553183424854658279455623156 1329227995026455015981906837933016651

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 36 3930 261507 16777554 1073775436 68719837791 4398048893484 281474985126114 18014398483114083 1152921503948987050

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The endomorphism algebra of this simple isogeny class is 4.0.23225.1.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

## 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 2.64.bd_mz $2$ (not in LMFDB)