Abelian variety isogeny class 2.157.abq_bcr over $\F_{157}$

• Label: 2.157.abq_bcr
• Base field: $\F_{157}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{157}$
Dimension:	$2$
L-polynomial:	$1 - 42 x + 745 x^{2} - 6594 x^{3} + 24649 x^{4}$
Frobenius angles:	$\pm0.0854542818411$, $\pm0.247879051492$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{10})\)
Galois group:	$C_2^2$
Jacobians:	$36$

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18759$	$600869529$	$14976072140796$	$369160856652319641$	$9099089584122993853359$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$116$	$24376$	$3869894$	$607598980$	$95389303736$	$14976072450142$	$2351243241675080$	$369145194022745284$	$57955795548021664958$	$9099059901197902134136$

Jacobians and polarizations

This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=109x^6+94x^5+49x^4+104x^3+122x^2+122x+112$
• $y^2=104x^6+98x^5+71x^4+48x^3+109x^2+88x+36$
• $y^2=51x^6+99x^5+139x^4+2x^3+49x^2+117x+116$
• $y^2=89x^6+146x^5+51x^4+15x^3+149x^2+11x+87$
• $y^2=24x^6+107x^5+65x^4+81x^3+89x^2+32x+76$
• $y^2=103x^6+95x^5+45x^4+108x^3+94x^2+154x+6$
• $y^2=80x^6+87x^5+10x^4+36x^3+104x^2+36x+127$
• $y^2=135x^6+15x^5+23x^4+23x^3+89x^2+53x+132$
• $y^2=32x^6+58x^5+31x^4+72x^3+120x^2+148x+36$
• $y^2=97x^6+78x^5+71x^4+110x^3+152x^2+53x+72$
• $y^2=114x^6+73x^5+23x^4+42x^3+101x^2+23x+79$
• $y^2=141x^6+130x^5+100x^4+43x^3+67x^2+90x+55$
• $y^2=6x^6+149x^5+149x^4+89x^3+25x^2+27x+65$
• $y^2=144x^6+126x^5+155x^4+100x^3+101x^2+18x+68$
• $y^2=30x^6+154x^5+40x^4+19x^3+103x^2+152x+81$
• $y^2=26x^6+71x^5+109x^4+116x^3+95x^2+72x+98$
• $y^2=29x^6+89x^5+34x^4+47x^3+104x^2+91x+24$
• $y^2=60x^6+8x^5+x^4+96x^3+62x^2+63x+73$
• $y^2=24x^6+8x^5+155x^4+129x^3+101x^2+13x+152$
• $y^2=x^6+x^3+133$
• and

• $y^2=63x^6+54x^5+96x^4+142x^3+103x^2+138x+135$
• $y^2=2x^6+77x^5+73x^4+136x^3+102x^2+40x+151$
• $y^2=152x^6+8x^5+130x^4+41x^3+43x^2+117x+129$
• $y^2=51x^6+5x^5+33x^4+38x^3+92x^2+117x+53$
• $y^2=x^6+x^3+94$
• $y^2=19x^6+37x^5+99x^4+128x^3+107x+125$
• $y^2=112x^6+148x^5+139x^4+21x^3+81x^2+15x+113$
• $y^2=122x^6+44x^5+30x^4+111x^3+60x^2+134x+90$
• $y^2=133x^6+78x^5+136x^4+15x^3+31x^2+2x+116$
• $y^2=100x^6+79x^5+46x^4+47x^3+49x^2+140x+102$
• $y^2=59x^6+17x^5+96x^4+38x^3+16x^2+29x+5$
• $y^2=87x^6+14x^5+130x^4+67x^3+58x^2+80x+60$
• $y^2=88x^6+153x^5+38x^4+41x^3+86x^2+124x+86$
• $y^2=18x^6+15x^5+149x^4+130x^3+28x^2+134x+76$
• $y^2=120x^6+75x^5+77x^4+151x^2+93x+56$
• $y^2=15x^6+9x^5+112x^4+135x^3+115x^2+9x+73$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{157^{6}}$.

Endomorphism algebra over $\F_{157}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{10})\).

Endomorphism algebra over $\overline{\F}_{157}$

The base change of $A$ to $\F_{157^{6}}$ is 1.14976071831449.rppy^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{157^{2}}$

  The base change of $A$ to $\F_{157^{2}}$ is the simple isogeny class 2.24649.ako_cwpn and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{10})\).

• Endomorphism algebra over $\F_{157^{3}}$

  The base change of $A$ to $\F_{157^{3}}$ is the simple isogeny class 2.3869893.a_rppy and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{10})\).

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
2.157.bq_bcr	$2$	(not in LMFDB)
2.157.a_ko	$3$	(not in LMFDB)
2.157.bq_bcr	$3$	(not in LMFDB)