# Properties

 Label 2.61.ay_jt Base Field $\F_{61}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

# Learn more about

## Invariants

 Base field: $\F_{61}$ Dimension: $2$ L-polynomial: $1 - 24 x + 253 x^{2} - 1464 x^{3} + 3721 x^{4}$ Frobenius angles: $\pm0.0139265954990$, $\pm0.319406737834$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Galois group: $C_2^2$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=x^6+2x^3+29$
• $y^2=57x^6+14x^5+11x^4+18x^3+57x^2+19x+30$
• $y^2=4x^6+7x^5+30x^4+35x^3+47x^2+40x+3$
• $y^2=x^6+x^3+26$
• $y^2=x^6+x^3+40$
• $y^2=6x^6+44x^5+42x^4+51x^3+41x^2+42x+6$
• $y^2=2x^6+37x^5+12x^4+3x^3+8x^2+15x+2$
• $y^2=2x^6+4x^3+52$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2487 13586481 51519935952 191672109482841 713280258809833287 2654303800498182146304 9876813687554472930864063 36751689786707188701387599529 136753052840547989392630633704912 508858109516080801856437157732296401

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 38 3652 226982 13843300 844522118 51519497542 3142736839406 191707291767364 11694146092834142 713342911517653252

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{13})$$.
Endomorphism algebra over $\overline{\F}_{61}$
 The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.ayyny 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-39})$$$)$
All geometric endomorphisms are defined over $\F_{61^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{61^{2}}$  The base change of $A$ to $\F_{61^{2}}$ is the simple isogeny class 2.3721.acs_btj and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{13})$$.
• Endomorphism algebra over $\F_{61^{3}}$  The base change of $A$ to $\F_{61^{3}}$ is the simple isogeny class 2.226981.a_ayyny and its endomorphism algebra is $$\Q(\sqrt{-3}, \sqrt{13})$$.

## 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 2.61.a_cs $3$ (not in LMFDB) 2.61.y_jt $3$ (not in LMFDB) 2.61.a_cs $6$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.61.a_cs $3$ (not in LMFDB) 2.61.y_jt $3$ (not in LMFDB) 2.61.a_cs $6$ (not in LMFDB) 2.61.a_acs $12$ (not in LMFDB)