# Properties

 Label 2.61.abc_mg Base Field $\F_{61}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $( 1 - 14 x + 61 x^{2} )^{2}$ Frobenius angles: $\pm0.146275019398$, $\pm0.146275019398$ 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:

• $y^2=26x^6+53x^5+27x^4+44x^3+25x^2+40x+31$
• $y^2=25x^6+5x^4+5x^2+25$
• $y^2=38x^6+45x^5+14x^4+26x^3+14x^2+45x+38$
• $y^2=x^6+60$
• $y^2=53x^6+56x^4+56x^2+53$
• $y^2=x^6+57x^3+58$
• $y^2=55x^6+6x^5+39x^4+17x^3+39x^2+6x+55$
• $y^2=37x^5+8x^4+60x^3+8x^2+37x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2304 13307904 51438240000 191761786404864 713408161597565184 2654392338040343040000 9876854756003829518778624 36751702992076970789635620864 136753055598099213272567463840000 508858109901056851995688231679299584

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 34 3574 226618 13849774 844673554 51521216038 3142749907114 191707360650334 11694146328640258 713342912057332054

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{61}$.

## 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.a_acw $2$ (not in LMFDB) 2.61.bc_mg $2$ (not in LMFDB) 2.61.an_ee $3$ (not in LMFDB) 2.61.ab_aci $3$ (not in LMFDB) 2.61.c_et $3$ (not in LMFDB) 2.61.o_ff $3$ (not in LMFDB) 2.61.ba_lf $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.a_acw $2$ (not in LMFDB) 2.61.bc_mg $2$ (not in LMFDB) 2.61.an_ee $3$ (not in LMFDB) 2.61.ab_aci $3$ (not in LMFDB) 2.61.c_et $3$ (not in LMFDB) 2.61.o_ff $3$ (not in LMFDB) 2.61.ba_lf $3$ (not in LMFDB) 2.61.a_cw $4$ (not in LMFDB) 2.61.abb_ls $6$ (not in LMFDB) 2.61.aba_lf $6$ (not in LMFDB) 2.61.ap_fg $6$ (not in LMFDB) 2.61.ao_ff $6$ (not in LMFDB) 2.61.am_ef $6$ (not in LMFDB) 2.61.ac_et $6$ (not in LMFDB) 2.61.a_abv $6$ (not in LMFDB) 2.61.a_er $6$ (not in LMFDB) 2.61.b_aci $6$ (not in LMFDB) 2.61.m_ef $6$ (not in LMFDB) 2.61.n_ee $6$ (not in LMFDB) 2.61.p_fg $6$ (not in LMFDB) 2.61.bb_ls $6$ (not in LMFDB) 2.61.a_aer $12$ (not in LMFDB) 2.61.a_bv $12$ (not in LMFDB)