# Properties

 Label 2.41.av_hg 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: $1 - 21 x + 188 x^{2} - 861 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0623248522748$, $\pm0.271008481058$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Galois group: $C_2^2$ Jacobians: 8

This isogeny class is simple but not geometrically 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:

• $y^2=8x^6+2x^5+19x^4+6x^3+12x^2+22x+24$
• $y^2=19x^6+34x^4+23x^3+18x^2+x+11$
• $y^2=26x^5+25x^4+2x^3+18x^2+15x$
• $y^2=3x^6+7x^5+27x^4+27x^3+34x^2+3x+29$
• $y^2=22x^6+37x^5+35x^4+22x^3+18x^2+37x+38$
• $y^2=15x^6+9x^5+37x^4+20x^3+39x^2+21x+38$
• $y^2=16x^6+5x^5+15x^4+7x^3+2x^2+21x+40$
• $y^2=28x^6+7x^5+37x^4+15x^3+21x^2+40x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 988 2718976 4750051072 7987361780736 13422365037551548 22562985186608349184 37929030028367375781052 63758991734613322876145664 107178930967532391731290738432 180167786037728988982209161733376

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 21 1617 68922 2826625 115853661 4749997902 194753261493 7984920322369 327381934393962 13422659539712577

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{17})$$.
Endomorphism algebra over $\overline{\F}_{41}$
 The base change of $A$ to $\F_{41^{6}}$ is 1.4750104241.adara 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$
All geometric endomorphisms are defined over $\F_{41^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{41^{2}}$  The base change of $A$ to $\F_{41^{2}}$ is the simple isogeny class 2.1681.acn_dtw and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{17})$$.
• Endomorphism algebra over $\F_{41^{3}}$  The base change of $A$ to $\F_{41^{3}}$ is the simple isogeny class 2.68921.a_adara and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{17})$$.

## 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.v_hg $2$ (not in LMFDB) 2.41.a_cn $3$ (not in LMFDB) 2.41.v_hg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.41.v_hg $2$ (not in LMFDB) 2.41.a_cn $3$ (not in LMFDB) 2.41.v_hg $3$ (not in LMFDB) 2.41.a_cn $6$ (not in LMFDB) 2.41.a_acn $12$ (not in LMFDB)