# Properties

 Label 2.137.abs_bde 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: $( 1 - 22 x + 137 x^{2} )^{2}$ Frobenius angles: $\pm0.111017258455$, $\pm0.111017258455$ Angle rank: $1$ (numerical) Jacobians: 10

This isogeny class is not 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 10 curves, and hence is principally polarizable:

• $y^2=24x^6+6x^5+42x^4+13x^3+110x^2+57x+10$
• $y^2=45x^6+102x^5+93x^4+109x^3+93x^2+102x+45$
• $y^2=78x^6+29x^4+29x^2+78$
• $y^2=86x^6+24x^5+39x^4+31x^3+68x^2+20x+91$
• $y^2=50x^6+32x^4+32x^2+50$
• $y^2=95x^6+79x^5+104x^4+40x^3+102x^2+12x+111$
• $y^2=117x^6+93x^5+101x^4+100x^3+101x^2+93x+117$
• $y^2=79x^6+24x^4+24x^2+79$
• $y^2=49x^6+126x^5+29x^4+72x^3+29x^2+126x+49$
• $y^2=78x^6+68x^5+58x^4+14x^3+128x^2+112x+55$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 13456 344473600 6603604783504 124093307453440000 2329201350436737613456 43716676977327384006246400 820517757336797747775065578384 15400296386442587467088398909440000 289048160076064916232496775763861440656 5425144911627714989937535120907093616640000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 94 18350 2568142 352262238 48261875774 6611861377550 905824398395822 124097931290662078 17001416422065255454 2329194047744990576750

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{137}$
 The isogeny class factors as 1.137.aw 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{137}$.

## 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.a_aic $2$ (not in LMFDB) 2.137.bs_bde $2$ (not in LMFDB) 2.137.w_nj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.137.a_aic $2$ (not in LMFDB) 2.137.bs_bde $2$ (not in LMFDB) 2.137.w_nj $3$ (not in LMFDB) 2.137.abe_ri $4$ (not in LMFDB) 2.137.aq_na $4$ (not in LMFDB) 2.137.ao_du $4$ (not in LMFDB) 2.137.a_ic $4$ (not in LMFDB) 2.137.o_du $4$ (not in LMFDB) 2.137.q_na $4$ (not in LMFDB) 2.137.be_ri $4$ (not in LMFDB) 2.137.aw_nj $6$ (not in LMFDB) 2.137.a_agu $8$ (not in LMFDB) 2.137.a_gu $8$ (not in LMFDB) 2.137.ai_acv $12$ (not in LMFDB) 2.137.i_acv $12$ (not in LMFDB)