# Properties

 Label 2.113.abk_uz Base Field $\F_{113}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{113}$ Dimension: $2$ L-polynomial: $1 - 36 x + 545 x^{2} - 4068 x^{3} + 12769 x^{4}$ Frobenius angles: $\pm0.0992057967989$, $\pm0.234127536534$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Galois group: $C_2^2$ Jacobians: 22

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 22 curves, and hence is principally polarizable:

• $y^2=53x^6+63x^5+45x^4+27x^3+66x^2+112x+81$
• $y^2=46x^6+21x^5+93x^4+63x^3+59x^2+28x+33$
• $y^2=28x^6+65x^5+108x^4+49x^3+109x^2+35x+26$
• $y^2=23x^6+22x^5+89x^4+32x^3+91x^2+x+41$
• $y^2=62x^6+6x^5+4x^4+45x^3+81x^2+106x+90$
• $y^2=47x^6+13x^5+17x^4+5x^3+33x^2+70x+80$
• $y^2=37x^6+50x^5+38x^4+83x^3+4x^2+59x+37$
• $y^2=102x^6+42x^5+54x^4+32x^3+107x^2+29x+92$
• $y^2=86x^6+56x^5+96x^4+7x^3+41x^2+60x+88$
• $y^2=27x^6+104x^5+97x^4+57x^3+95x^2+58x+27$
• $y^2=48x^6+8x^5+27x^4+60x^3+26x^2+95x+104$
• $y^2=94x^6+31x^5+36x^4+75x^3+59x^2+59x+71$
• $y^2=39x^6+92x^5+53x^4+94x^3+3x^2+27x+41$
• $y^2=54x^6+56x^5+50x^4+75x^3+2x^2+103x+34$
• $y^2=39x^6+38x^5+34x^4+53x^3+100x^2+75x+54$
• $y^2=108x^6+74x^5+29x^4+77x^3+19x^2+106x+77$
• $y^2=76x^6+24x^5+2x^4+92x^3+9x^2+40x+93$
• $y^2=102x^6+101x^5+21x^4+78x^3+38x^2+104x+32$
• $y^2=x^6+51x^5+102x^4+90x^3+32x^2+11x+51$
• $y^2=86x^6+39x^5+2x^4+8x^3+10x^2+47x+90$
• $y^2=75x^6+89x^5+67x^4+40x^3+31x^2+55x+57$
• $y^2=64x^6+25x^5+23x^4+29x^3+2x^2+24x+27$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 9211 160446409 2081952603184 26587197225882441 339460976702890640491 4334526641904634166937856 55347526555654103051969171371 706732551601732348919733293101449 9024267965168278303972721530516379056 115230877653756324445990574469069096754249

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 78 12564 1442898 163064260 18424581798 2081953453758 235260552675030 26584441888512004 3004041937984268274 339456739011437071764

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{113}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{113}$
 The base change of $A$ to $\F_{113^{6}}$ is 1.2081951752609.bwkgk 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{113^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{113^{2}}$  The base change of $A$ to $\F_{113^{2}}$ is the simple isogeny class 2.12769.ahy_brxb and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{113^{3}}$  The base change of $A$ to $\F_{113^{3}}$ is the simple isogeny class 2.1442897.a_bwkgk and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{5})$$.

## 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 2.113.a_hy $3$ (not in LMFDB) 2.113.bk_uz $3$ (not in LMFDB) 2.113.a_hy $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.113.a_hy $3$ (not in LMFDB) 2.113.bk_uz $3$ (not in LMFDB) 2.113.a_hy $6$ (not in LMFDB) 2.113.a_ahy $12$ (not in LMFDB)