# Properties

 Label 2.49.aw_il 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: $( 1 - 11 x + 49 x^{2} )^{2}$ Frobenius angles: $\pm0.212295615010$, $\pm0.212295615010$ Angle rank: $1$ (numerical) Jacobians: 8

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 8 curves, and hence is principally polarizable:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1521 5659641 13908900096 33282226355625 79810905102881121 191585480762348015616 459986634292967305197921 1104427227000104792784455625 2651730593295364763909071677696 6366805677394672646699363854447641

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 28 2356 118222 5773348 282541228 13841594206 678223216972 33232917111748 1628413442812078 79792265250964756

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{7^{2}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.

 Subfield Primitive Model $\F_{7}$ 2.7.ak_bn $\F_{7}$ 2.7.a_al $\F_{7}$ 2.7.k_bn

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.a_ax $2$ (not in LMFDB) 2.49.w_il $2$ (not in LMFDB) 2.49.an_eq $3$ (not in LMFDB) 2.49.ae_dy $3$ (not in LMFDB) 2.49.c_abt $3$ (not in LMFDB) 2.49.l_cu $3$ (not in LMFDB) 2.49.ba_kh $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.49.a_ax $2$ (not in LMFDB) 2.49.w_il $2$ (not in LMFDB) 2.49.an_eq $3$ (not in LMFDB) 2.49.ae_dy $3$ (not in LMFDB) 2.49.c_abt $3$ (not in LMFDB) 2.49.l_cu $3$ (not in LMFDB) 2.49.ba_kh $3$ (not in LMFDB) 2.49.a_x $4$ (not in LMFDB) 2.49.aba_kh $6$ (not in LMFDB) 2.49.ay_jh $6$ (not in LMFDB) 2.49.ap_eu $6$ (not in LMFDB) 2.49.al_cu $6$ (not in LMFDB) 2.49.aj_cy $6$ (not in LMFDB) 2.49.ac_abt $6$ (not in LMFDB) 2.49.a_act $6$ (not in LMFDB) 2.49.a_dq $6$ (not in LMFDB) 2.49.e_dy $6$ (not in LMFDB) 2.49.j_cy $6$ (not in LMFDB) 2.49.n_eq $6$ (not in LMFDB) 2.49.p_eu $6$ (not in LMFDB) 2.49.y_jh $6$ (not in LMFDB) 2.49.a_adq $12$ (not in LMFDB) 2.49.a_ct $12$ (not in LMFDB)