# Properties

 Label 2.41.at_go Base Field $\F_{41}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{41}$ Dimension: $2$ L-polynomial: $( 1 - 11 x + 41 x^{2} )( 1 - 8 x + 41 x^{2} )$ Frobenius angles: $\pm0.171113726078$, $\pm0.285223287477$ Angle rank: $2$ (numerical) Jacobians: 5

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 5 curves, and hence is principally polarizable:

• $y^2=2x^6+7x^5+39x^4+40x^3+3x^2+11x+11$
• $y^2=22x^6+25x^5+14x^4+15x^3+26x^2+19x+27$
• $y^2=11x^6+28x^5+31x^4+12x^3+9x^2+11x+35$
• $y^2=4x^6+19x^5+19x^4+28x^3+38x^2+17x+15$
• $y^2=18x^6+36x^5+36x^4+30x^3+10x^2+11x+29$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1054 2793100 4784299936 7998723366400 13425475059245854 22563739279662880000 37929195072096531135454 63759020409046519698585600 107178931957461386093389910176 180167783822433442599517794527500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 23 1661 69416 2830641 115880503 4750156658 194754108943 7984923913441 327381937417736 13422659374671101

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The isogeny class factors as 1.41.al $\times$ 1.41.ai 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 $\F_{41}$.

## 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 2.41.ad_ag $2$ (not in LMFDB) 2.41.d_ag $2$ (not in LMFDB) 2.41.t_go $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.ad_ag $2$ (not in LMFDB) 2.41.d_ag $2$ (not in LMFDB) 2.41.t_go $2$ (not in LMFDB) 2.41.av_hk $4$ (not in LMFDB) 2.41.ab_abc $4$ (not in LMFDB) 2.41.b_abc $4$ (not in LMFDB) 2.41.v_hk $4$ (not in LMFDB)