# Properties

 Label 2.137.abn_yu Base Field $\F_{137}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{137}$ Dimension: $2$ L-polynomial: $1 - 39 x + 644 x^{2} - 5343 x^{3} + 18769 x^{4}$ Frobenius angles: $\pm0.0784760750967$, $\pm0.254857258237$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{41})$$ Galois group: $C_2^2$ Jacobians: 32

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 32 curves, and hence is principally polarizable:

• $y^2=35x^6+134x^5+47x^4+14x^3+109x^2+85x+19$
• $y^2=100x^6+95x^5+134x^4+116x^3+52x^2+36x+80$
• $y^2=47x^6+119x^5+20x^4+136x^3+72x^2+74x+111$
• $y^2=94x^6+120x^5+14x^4+85x^3+56x^2+120x+62$
• $y^2=95x^6+11x^5+x^4+124x^3+117x^2+10x+58$
• $y^2=110x^6+63x^5+66x^4+77x^3+26x^2+14x+3$
• $y^2=57x^6+46x^5+104x^4+22x^3+6x^2+40x+104$
• $y^2=32x^6+14x^5+88x^4+19x^3+13x^2+49x+83$
• $y^2=22x^6+116x^5+88x^4+30x^3+31x^2+47x+108$
• $y^2=131x^6+38x^5+113x^4+116x^3+122x^2+78x+92$
• $y^2=104x^6+x^5+103x^4+17x^3+62x^2+105x+25$
• $y^2=24x^6+81x^5+108x^4+40x^3+132x^2+79x+110$
• $y^2=13x^6+81x^5+100x^4+97x^3+60x^2+12x+51$
• $y^2=85x^6+107x^5+24x^4+89x^3+3x^2+12x+62$
• $y^2=67x^6+95x^5+29x^3+73x^2+134x+127$
• $y^2=73x^6+88x^5+98x^4+111x^3+136x^2+45x+116$
• $y^2=39x^6+29x^5+107x^4+119x^3+89x^2+72x+94$
• $y^2=84x^6+51x^5+29x^3+90x^2+73x+128$
• $y^2=100x^6+56x^5+119x^4+120x^3+86x^2+78x+131$
• $y^2=23x^6+76x^5+37x^4+61x^3+2x^2+62x+23$
• and 12 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14032 347937472 6611855780416 124103830860589824 2329200821682127388752 43716636861020472409133056 820517639915323894020391512784 15400296169651386256229847210470400 289048159795659680197698116558117099584 5425144911414262399520881728503982797225152

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 99 18537 2571354 352292113 48261864819 6611855310222 905824268766411 124097929543725601 17001416405572203978 2329194047653348323657

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{137}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{41})$$.
Endomorphism algebra over $\overline{\F}_{137}$
 The base change of $A$ to $\F_{137^{6}}$ is 1.6611856250609.abatok 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-123})$$$)$
All geometric endomorphisms are defined over $\F_{137^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{137^{2}}$  The base change of $A$ to $\F_{137^{2}}$ is the simple isogeny class 2.18769.aiz_caoe and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{41})$$.
• Endomorphism algebra over $\F_{137^{3}}$  The base change of $A$ to $\F_{137^{3}}$ is the simple isogeny class 2.2571353.a_abatok and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{41})$$.

## 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 2.137.bn_yu $2$ (not in LMFDB) 2.137.a_iz $3$ (not in LMFDB) 2.137.bn_yu $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.137.bn_yu $2$ (not in LMFDB) 2.137.a_iz $3$ (not in LMFDB) 2.137.bn_yu $3$ (not in LMFDB) 2.137.a_aiz $12$ (not in LMFDB)