# Properties

 Label 2.61.abb_ls Base Field $\F_{61}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $( 1 - 14 x + 61 x^{2} )( 1 - 13 x + 61 x^{2} )$ Frobenius angles: $\pm0.146275019398$, $\pm0.187058313935$ Angle rank: $1$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2352 13406400 51520795200 191807027193600 713423619261382512 2654392338040343040000 9876849924977249890820592 36751698483286618954697702400 136753052840548015382510835124800 508858108653623078737901572967160000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 35 3601 226982 13853041 844691855 51521216038 3142748369915 191707337131201 11694146092834142 713342910308616601

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ao $\times$ 1.61.an 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 $\overline{\F}_{61}$
 The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.xyoc 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{61^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{61^{2}}$  The base change of $A$ to $\F_{61^{2}}$ is 1.3721.acw $\times$ 1.3721.abv. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{61^{3}}$  The base change of $A$ to $\F_{61^{3}}$ is 1.226981.aha $\times$ 1.226981.ha. 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 2.61.ab_aci $2$ (not in LMFDB) 2.61.b_aci $2$ (not in LMFDB) 2.61.bb_ls $2$ (not in LMFDB) 2.61.ap_fg $3$ (not in LMFDB) 2.61.am_ef $3$ (not in LMFDB) 2.61.a_acw $3$ (not in LMFDB) 2.61.a_abv $3$ (not in LMFDB) 2.61.a_er $3$ (not in LMFDB) 2.61.m_ef $3$ (not in LMFDB) 2.61.p_fg $3$ (not in LMFDB) 2.61.bb_ls $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.ab_aci $2$ (not in LMFDB) 2.61.b_aci $2$ (not in LMFDB) 2.61.bb_ls $2$ (not in LMFDB) 2.61.ap_fg $3$ (not in LMFDB) 2.61.am_ef $3$ (not in LMFDB) 2.61.a_acw $3$ (not in LMFDB) 2.61.a_abv $3$ (not in LMFDB) 2.61.a_er $3$ (not in LMFDB) 2.61.m_ef $3$ (not in LMFDB) 2.61.p_fg $3$ (not in LMFDB) 2.61.bb_ls $3$ (not in LMFDB) 2.61.abc_mg $6$ (not in LMFDB) 2.61.aba_lf $6$ (not in LMFDB) 2.61.ao_ff $6$ (not in LMFDB) 2.61.an_ee $6$ (not in LMFDB) 2.61.ac_et $6$ (not in LMFDB) 2.61.c_et $6$ (not in LMFDB) 2.61.n_ee $6$ (not in LMFDB) 2.61.o_ff $6$ (not in LMFDB) 2.61.ba_lf $6$ (not in LMFDB) 2.61.bc_mg $6$ (not in LMFDB) 2.61.a_aer $12$ (not in LMFDB) 2.61.a_bv $12$ (not in LMFDB) 2.61.a_cw $12$ (not in LMFDB)