Abelian variety isogeny class 2.59.a_aec over $\F_{59}$

• Label: 2.59.a_aec
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 - 106 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.0723995420184$, $\pm0.927600457982$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{14})\)
Galois group:	$C_2^2$
Jacobians:	$26$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3376$	$11397376$	$42180449584$	$146726900895744$	$511116754225817776$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3270$	$205380$	$12108814$	$714924300$	$42180365526$	$2488651484820$	$146830449539614$	$8662995818654940$	$511116755150994150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

• $y^2=50 x^6+41 x^5+37 x^4+17 x^3+x^2+28 x+53$
• $y^2=49 x^6+28 x^5+32 x^4+47 x^3+10 x^2+18 x+50$
• $y^2=40 x^6+11 x^5+14 x^4+51 x^3+3 x^2+58 x+36$
• $y^2=33 x^6+44 x^5+32 x^4+35 x^3+39 x^2+42 x+47$
• $y^2=43 x^6+40 x^5+15 x^4+39 x^3+7 x^2+9 x+41$
• $y^2=43 x^6+2 x^5+28 x^4+53 x^3+18 x^2+52 x+36$
• $y^2=27 x^6+34 x^5+5 x^4+14 x^3+48 x^2+4 x+49$
• $y^2=36 x^6+38 x^5+52 x^4+58 x^3+3 x^2+13 x+24$
• $y^2=41 x^6+46 x^5+45 x^4+48 x^3+30 x^2+32 x+44$
• $y^2=53 x^6+51 x^5+19 x^4+12 x^3+40 x^2+35 x+24$
• $y^2=17 x^6+53 x^5+35 x^4+30 x^3+34 x^2+15 x+43$
• $y^2=9 x^6+45 x^5+42 x^4+56 x^3+29 x^2+42 x+50$
• $y^2=19 x^6+57 x^5+54 x^4+10 x^3+37 x^2+6 x+22$
• $y^2=7 x^6+33 x^5+57 x^4+38 x^3+50 x^2+49 x+50$
• $y^2=10 x^6+9 x^5+28 x^4+41 x^3+43 x^2+21 x+26$
• $y^2=32 x^6+49 x^5+39 x^4+12 x^3+25 x^2+36 x+26$
• $y^2=58 x^6+19 x^5+3 x^4+6 x^3+55 x^2+23 x+26$
• $y^2=57 x^6+38 x^5+6 x^4+12 x^3+51 x^2+46 x+52$
• $y^2=52 x^6+24 x^5+55 x^4+2 x^3+49 x^2+32 x+16$
• $y^2=17 x^6+7 x^5+4 x^4+58 x^3+50 x^2+17 x+23$
• $y^2=4 x^6+7 x^5+42 x^4+32 x^3+15 x^2+3 x+56$
• $y^2=13 x^6+41 x^5+14 x^4+8 x^3+15 x+16$
• $y^2=39 x^6+8 x^5+54 x^4+32 x^3+57 x^2+6 x+53$
• $y^2=38 x^6+10 x^5+28 x^4+9 x^3+31 x^2+10 x+21$
• $y^2=13 x^6+26 x^5+11 x^4+48 x^3+14 x^2+36 x+48$
• $y^2=17 x^6+55 x^5+12 x^4+38 x^3+26 x^2+28 x+58$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{2}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{14})\).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.aec^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$

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
2.59.ag_ct	$3$	(not in LMFDB)
2.59.g_ct	$3$	(not in LMFDB)
2.59.a_ec	$4$	(not in LMFDB)