Properties

 Label 2.64.ax_jg Base Field $\F_{2^{6}}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

Invariants

 Base field: $\F_{2^{6}}$ Dimension: $2$ L-polynomial: $( 1 - 8 x )^{2}( 1 - 7 x + 64 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.355864001265$ Angle rank: $1$ (numerical) Jacobians: 3

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2842 16574544 68712946666 281370287671200 1152797017209387802 4722297626273133409776 19342794148380484204114762 79228161442857269538792532800 324518552709952644592019441491066 1329227992855497229005498750000265104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 42 4048 262122 16770976 1073625882 68718474736 4398042198858 281474972904256 18014398456831098 1152921502065981328

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
 The isogeny class factors as 1.64.aq $\times$ 1.64.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.64.aq : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.64.ah : $$\Q(\sqrt{-23})$$.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.aj_q $2$ (not in LMFDB) 2.64.j_q $2$ (not in LMFDB) 2.64.x_jg $2$ (not in LMFDB) 2.64.b_cu $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.64.aj_q $2$ (not in LMFDB) 2.64.j_q $2$ (not in LMFDB) 2.64.x_jg $2$ (not in LMFDB) 2.64.b_cu $3$ (not in LMFDB) 2.64.ah_ey $4$ (not in LMFDB) 2.64.h_ey $4$ (not in LMFDB) 2.64.ap_hc $6$ (not in LMFDB) 2.64.ab_cu $6$ (not in LMFDB) 2.64.p_hc $6$ (not in LMFDB)