Properties

 Label 2.49.aba_kh 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: $( 1 - 13 x + 49 x^{2} )^{2}$ Frobenius angles: $\pm0.121037718324$, $\pm0.121037718324$ Angle rank: $1$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=2ax^6+2ax^5+ax^4+5ax^3+4ax^2+4ax+2a$
• $y^2=ax^6+ax^3+a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1369 5433561 13774308496 33230186580969 79798428896628649 191585480762348015616 459988518428635068448489 1104428436772447798142978889 2651731098423960721055865252496 6366805832081790770045470407222201

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 2260 117078 5764324 282497064 13841594206 678225995016 33232953514564 1628413753008822 79792267189587700

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
 The isogeny class factors as 1.49.an 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.a_an

Twists

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