# Properties

 Label 2.83.abe_ot Base Field $\F_{83}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{83}$ Dimension: $2$ L-polynomial: $1 - 30 x + 383 x^{2} - 2490 x^{3} + 6889 x^{4}$ Frobenius angles: $\pm0.0661835162234$, $\pm0.267149817110$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Galois group: $C_2^2$ Jacobians: 21

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 21 curves, and hence is principally polarizable:

• $y^2=46x^6+56x^5+3x^4+40x^3+27x^2+43x+2$
• $y^2=57x^6+56x^5+60x^4+78x^3+54x^2+37x+57$
• $y^2=35x^6+19x^5+3x^4+35x^3+40x^2+6x+60$
• $y^2=24x^6+54x^5+65x^4+30x^3+31x^2+17x+47$
• $y^2=35x^6+51x^5+61x^4+28x^3+64x^2+21x+13$
• $y^2=55x^6+19x^5+46x^4+60x^3+47x^2+61x+46$
• $y^2=60x^6+28x^5+24x^4+20x^3+54x^2+43x+61$
• $y^2=20x^6+29x^5+22x^4+32x^3+45x^2+3x+61$
• $y^2=29x^6+40x^5+72x^4+20x^3+45x^2+65x+32$
• $y^2=24x^6+47x^5+38x^4+76x^3+12x^2+48x+60$
• $y^2=54x^6+72x^5+65x^4+81x^3+14x^2+31x+5$
• $y^2=43x^6+14x^5+34x^4+x^3+49x^2+55x+66$
• $y^2=45x^6+10x^5+24x^4+5x^3+63x^2+18x+56$
• $y^2=20x^6+64x^5+26x^4+46x^3+40x^2+18x+62$
• $y^2=55x^6+9x^5+17x^4+7x^3+48x^2+78x+17$
• $y^2=4x^6+77x^5+15x^4+76x^3+41x^2+81x+43$
• $y^2=20x^6+57x^5+78x^4+71x^3+10x^2+63x+20$
• $y^2=82x^6+63x^5+9x^4+52x^3+82x^2+14x+9$
• $y^2=71x^6+63x^5+73x^4+2x^3+7x^2+20x+61$
• $y^2=6x^6+79x^5+24x^4+81x^3+45x^2+63x+22$
• $y^2=74x^6+2x^5+2x^4+70x^3+23x^2+27x+74$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4753 46546129 326940010096 2252490483005721 15516034691728738993 106889770201572581929216 736364971293330790602485497 5072820124489153502937556016169 34946659039493167699189665926817904 240747534245290983104270476618202577649

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 54 6756 571788 47462500 3939038994 326939646822 27136040228358 2252292154678084 186940255267540404 15516041195083027236

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{83}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{83}$
 The base change of $A$ to $\F_{83^{6}}$ is 1.326940373369.aurkc 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$
All geometric endomorphisms are defined over $\F_{83^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{83^{2}}$  The base change of $A$ to $\F_{83^{2}}$ is the simple isogeny class 2.6889.afe_qjr and its endomorphism algebra is $$\Q(\sqrt{2}, \sqrt{-3})$$.
• Endomorphism algebra over $\F_{83^{3}}$  The base change of $A$ to $\F_{83^{3}}$ is the simple isogeny class 2.571787.a_aurkc and its endomorphism algebra is $$\Q(\sqrt{2}, \sqrt{-3})$$.

## 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 2.83.be_ot $2$ (not in LMFDB) 2.83.a_fe $3$ (not in LMFDB) 2.83.be_ot $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.83.be_ot $2$ (not in LMFDB) 2.83.a_fe $3$ (not in LMFDB) 2.83.be_ot $3$ (not in LMFDB) 2.83.a_afe $12$ (not in LMFDB) 2.83.ai_bg $24$ (not in LMFDB) 2.83.i_bg $24$ (not in LMFDB)