# Properties

 Label 2.41.as_ft 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: $1 - 18 x + 149 x^{2} - 738 x^{3} + 1681 x^{4}$ Frobenius angles: $\pm0.0319832925581$, $\pm0.365316625891$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{14})$$ Galois group: $C_2^2$ Jacobians: 6

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 6 curves, and hence is principally polarizable:

• $y^2=38x^6+16x^5+15x^4+31x^3+14x^2+37x+11$
• $y^2=28x^6+14x^5+26x^4+16x^3+7x^2+31x+28$
• $y^2=27x^6+8x^5+37x^4+8x^3+3x^2+5x+24$
• $y^2=13x^6+35x^5+37x^4+29x^3+16x^2+2x+13$
• $y^2=37x^6+19x^5+20x^4+9x^3+15x^2+24x+14$
• $y^2=34x^6+36x^5+36x^4+33x^3+15x^2+34x+11$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1075 2781025 4749990700 7977339621225 13418341001726875 22562411650086490000 37929126827025765085675 63759043395676530206174025 107178930967531938606221248300 180167779849021299747820853550625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 24 1656 68922 2823076 115818924 4749877158 194753758524 7984926792196 327381934393962 13422659078648376

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{14})$$.
Endomorphism algebra over $\overline{\F}_{41}$
 The base change of $A$ to $\F_{41^{6}}$ is 1.4750104241.aglza 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-42})$$$)$
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.aba_abmr and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{14})$$.
• Endomorphism algebra over $\F_{41^{3}}$  The base change of $A$ to $\F_{41^{3}}$ is the simple isogeny class 2.68921.a_aglza and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{14})$$.

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