# Properties

 Label 2.49.ay_jh Base Field $\F_{7^{2}}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{7^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 13 x + 49 x^{2} )( 1 - 11 x + 49 x^{2} )$ Frobenius angles: $\pm0.121037718324$, $\pm0.212295615010$ Angle rank: $1$ (numerical) Jacobians: 10

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 10 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1443 5545449 13841440704 33256196289225 79804666755947523 191585480762348015616 459987576359836495764963 1104427831886110649837126025 2651730845859650714729969939904 6366805754738231238590052255431529

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 26 2308 117650 5768836 282519146 13841594206 678224605994 33232935313156 1628413597910450 79792266220276228

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.an $\times$ 1.49.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{7^{2}}$
 The base change of $A$ to $\F_{7^{12}}$ is 1.13841287201.itby 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{7^{12}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{7^{4}}$  The base change of $A$ to $\F_{7^{4}}$ is 1.2401.act $\times$ 1.2401.ax. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{7^{6}}$  The base change of $A$ to $\F_{7^{6}}$ is 1.117649.ala $\times$ 1.117649.la. The endomorphism algebra for each factor is:

## 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 2.49.ac_abt $2$ (not in LMFDB) 2.49.c_abt $2$ (not in LMFDB) 2.49.y_jh $2$ (not in LMFDB) 2.49.ap_eu $3$ (not in LMFDB) 2.49.aj_cy $3$ (not in LMFDB) 2.49.a_act $3$ (not in LMFDB) 2.49.a_ax $3$ (not in LMFDB) 2.49.a_dq $3$ (not in LMFDB) 2.49.j_cy $3$ (not in LMFDB) 2.49.p_eu $3$ (not in LMFDB) 2.49.y_jh $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.ac_abt $2$ (not in LMFDB) 2.49.c_abt $2$ (not in LMFDB) 2.49.y_jh $2$ (not in LMFDB) 2.49.ap_eu $3$ (not in LMFDB) 2.49.aj_cy $3$ (not in LMFDB) 2.49.a_act $3$ (not in LMFDB) 2.49.a_ax $3$ (not in LMFDB) 2.49.a_dq $3$ (not in LMFDB) 2.49.j_cy $3$ (not in LMFDB) 2.49.p_eu $3$ (not in LMFDB) 2.49.y_jh $3$ (not in LMFDB) 2.49.aba_kh $6$ (not in LMFDB) 2.49.aw_il $6$ (not in LMFDB) 2.49.an_eq $6$ (not in LMFDB) 2.49.al_cu $6$ (not in LMFDB) 2.49.ae_dy $6$ (not in LMFDB) 2.49.e_dy $6$ (not in LMFDB) 2.49.l_cu $6$ (not in LMFDB) 2.49.n_eq $6$ (not in LMFDB) 2.49.w_il $6$ (not in LMFDB) 2.49.ba_kh $6$ (not in LMFDB) 2.49.a_adq $12$ (not in LMFDB) 2.49.a_x $12$ (not in LMFDB) 2.49.a_ct $12$ (not in LMFDB)