# Properties

 Label 2.61.aba_lf 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: $( 1 - 13 x + 61 x^{2} )^{2}$ Frobenius angles: $\pm0.187058313935$, $\pm0.187058313935$ 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:

• $y^2=2x^6+2x^3+2$
• $y^2=29x^6+53x^5+28x^4+29x^3+7x^2+30x+30$
• $y^2=55x^6+58x^5+44x^4+5x^3+38x^2+42x+6$
• $y^2=7x^6+52x^5+53x^4+60x^3+28x^2+27x+43$
• $y^2=53x^6+4x^5+51x^4+35x^3+51x^2+4x+53$
• $y^2=32x^6+38x^5+52x^4+25x^3+27x^2+37x+51$
• $y^2=15x^6+25x^5+35x^4+6x^3+49x^2+32x+15$
• $y^2=60x^6+56x^5+28x^4+40x^3+37x^2+15x+41$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2401 13505625 51603482896 191852278655625 713439077260126441 2654392338040343040000 9876845093953033243628161 36751693974496820269376855625 136753050082996873096978549415056 508858107406189308538120649816015625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 36 3628 227346 13856308 844710156 51521216038 3142746832716 191707313612068 11694145857028026 713342908559901148

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.an 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_abv $2$ (not in LMFDB) 2.61.ba_lf $2$ (not in LMFDB) 2.61.ao_ff $3$ (not in LMFDB) 2.61.ac_et $3$ (not in LMFDB) 2.61.b_aci $3$ (not in LMFDB) 2.61.n_ee $3$ (not in LMFDB) 2.61.bc_mg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.a_abv $2$ (not in LMFDB) 2.61.ba_lf $2$ (not in LMFDB) 2.61.ao_ff $3$ (not in LMFDB) 2.61.ac_et $3$ (not in LMFDB) 2.61.b_aci $3$ (not in LMFDB) 2.61.n_ee $3$ (not in LMFDB) 2.61.bc_mg $3$ (not in LMFDB) 2.61.a_bv $4$ (not in LMFDB) 2.61.abc_mg $6$ (not in LMFDB) 2.61.abb_ls $6$ (not in LMFDB) 2.61.ap_fg $6$ (not in LMFDB) 2.61.an_ee $6$ (not in LMFDB) 2.61.am_ef $6$ (not in LMFDB) 2.61.ab_aci $6$ (not in LMFDB) 2.61.a_acw $6$ (not in LMFDB) 2.61.a_er $6$ (not in LMFDB) 2.61.c_et $6$ (not in LMFDB) 2.61.m_ef $6$ (not in LMFDB) 2.61.o_ff $6$ (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_cw $12$ (not in LMFDB)