# Properties

 Label 2.61.abc_mf 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: $( 1 - 15 x + 61 x^{2} )( 1 - 13 x + 61 x^{2} )$ Frobenius angles: $\pm0.0900194921159$, $\pm0.187058313935$ Angle rank: $2$ (numerical) Jacobians: 7

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 7 curves, and hence is principally polarizable:

• $y^2=20x^6+x^5+34x^4+18x^3+34x^2+x+20$
• $y^2=52x^6+56x^5+55x^4+14x^3+55x^2+56x+52$
• $y^2=24x^6+35x^5+45x^4+45x^2+35x+24$
• $y^2=48x^6+60x^5+57x^4+50x^3+57x^2+60x+48$
• $y^2=6x^6+49x^4+22x^3+49x^2+6$
• $y^2=44x^6+45x^5+25x^4+45x^3+25x^2+45x+44$
• $y^2=55x^6+44x^5+11x^4+32x^3+11x^2+44x+55$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2303 13299825 51419025728 191735929625625 713383328457705983 2654373596011548364800 9876843243372879484117463 36751697301462218284028675625 136753053554405836488875124615488 508858109659159983994911334629020625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 34 3572 226534 13847908 844644154 51520852262 3142746243874 191707330966468 11694146153878174 713342911718228852

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The isogeny class factors as 1.61.ap $\times$ 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:
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.ac_acv $2$ (not in LMFDB) 2.61.c_acv $2$ (not in LMFDB) 2.61.bc_mf $2$ (not in LMFDB) 2.61.aq_fh $3$ (not in LMFDB) 2.61.ab_adk $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.ac_acv $2$ (not in LMFDB) 2.61.c_acv $2$ (not in LMFDB) 2.61.bc_mf $2$ (not in LMFDB) 2.61.aq_fh $3$ (not in LMFDB) 2.61.ab_adk $3$ (not in LMFDB) 2.61.abd_mu $6$ (not in LMFDB) 2.61.ao_ed $6$ (not in LMFDB) 2.61.b_adk $6$ (not in LMFDB) 2.61.o_ed $6$ (not in LMFDB) 2.61.q_fh $6$ (not in LMFDB) 2.61.bd_mu $6$ (not in LMFDB)