LMFDB - Number field 24.0.59344171108947314029391991313268736.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096)

gp: K = bnfinit(y^24 - 4*y^21 + 4*y^20 - 4*y^19 + 8*y^18 - 16*y^17 + 40*y^16 - 24*y^15 + 40*y^14 - 104*y^13 + 100*y^12 - 208*y^11 + 160*y^10 - 192*y^9 + 640*y^8 - 512*y^7 + 512*y^6 - 512*y^5 + 1024*y^4 - 2048*y^3 + 4096, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096)

$ x^{24} - 4 x^{21} + 4 x^{20} - 4 x^{19} + 8 x^{18} - 16 x^{17} + 40 x^{16} - 24 x^{15} + 40 x^{14} + \cdots + 4096 $ x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$59344171108947314029391991313268736$ 59344171108947314029391991313268736 $\medspace = 2^{32}\cdot 3^{12}\cdot 7^{4}\cdot 101^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$28.11$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{4/3}3^{1/2}7^{1/2}101^{1/2}\approx 116.04960372572671$
Ramified primes:	$2$, $3$, $7$, $101$ 2, 3, 7, 101	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{2048}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{13}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{15}+\frac{1}{4}a^{3}$, $\frac{1}{32}a^{16}+\frac{1}{8}a^{4}$, $\frac{1}{64}a^{17}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{5}{16}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{256}a^{18}+\frac{1}{64}a^{15}+\frac{1}{64}a^{14}-\frac{1}{64}a^{13}+\frac{1}{32}a^{12}-\frac{1}{16}a^{11}+\frac{1}{32}a^{10}-\frac{3}{32}a^{9}-\frac{3}{32}a^{8}-\frac{5}{32}a^{7}+\frac{9}{64}a^{6}+\frac{3}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}$, $\frac{1}{512}a^{19}+\frac{1}{128}a^{16}+\frac{1}{128}a^{15}-\frac{1}{128}a^{14}+\frac{1}{64}a^{13}+\frac{1}{32}a^{12}+\frac{1}{64}a^{11}+\frac{5}{64}a^{10}+\frac{5}{64}a^{9}-\frac{5}{64}a^{8}-\frac{23}{128}a^{7}+\frac{7}{32}a^{6}-\frac{5}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{1536}a^{20}+\frac{1}{1536}a^{19}+\frac{1}{768}a^{18}+\frac{1}{384}a^{17}+\frac{1}{192}a^{16}+\frac{1}{192}a^{15}+\frac{1}{128}a^{14}+\frac{1}{96}a^{13}+\frac{5}{192}a^{12}-\frac{1}{32}a^{11}-\frac{1}{48}a^{10}-\frac{1}{32}a^{9}+\frac{19}{384}a^{8}-\frac{79}{384}a^{7}+\frac{1}{64}a^{6}+\frac{5}{48}a^{5}-\frac{23}{48}a^{4}-\frac{1}{24}a^{3}+\frac{1}{12}a^{2}+\frac{1}{12}a+\frac{1}{6}$, $\frac{1}{3072}a^{21}-\frac{1}{1536}a^{19}-\frac{1}{768}a^{18}+\frac{1}{768}a^{17}-\frac{1}{256}a^{16}-\frac{1}{96}a^{15}-\frac{1}{384}a^{14}+\frac{1}{128}a^{13}-\frac{23}{384}a^{12}-\frac{13}{384}a^{11}+\frac{25}{384}a^{10}-\frac{59}{768}a^{9}-\frac{1}{24}a^{8}+\frac{11}{384}a^{7}+\frac{11}{96}a^{6}-\frac{5}{48}a^{5}-\frac{5}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{12}a-\frac{1}{3}$, $\frac{1}{18432}a^{22}+\frac{1}{9216}a^{21}-\frac{1}{4608}a^{20}-\frac{1}{4608}a^{19}+\frac{7}{4608}a^{18}-\frac{3}{512}a^{17}-\frac{5}{384}a^{16}+\frac{29}{1152}a^{15}-\frac{35}{2304}a^{14}+\frac{5}{768}a^{13}-\frac{23}{768}a^{12}+\frac{89}{2304}a^{11}+\frac{127}{1536}a^{10}-\frac{79}{768}a^{9}-\frac{41}{1152}a^{8}+\frac{71}{288}a^{7}+\frac{1}{12}a^{6}+\frac{3}{8}a^{5}+\frac{37}{144}a^{4}+\frac{5}{72}a^{3}-\frac{13}{36}a^{2}-\frac{7}{18}a+\frac{1}{9}$, $\frac{1}{3575808}a^{23}-\frac{7}{1787904}a^{22}-\frac{1}{18624}a^{21}-\frac{35}{297984}a^{20}-\frac{469}{893952}a^{19}-\frac{1195}{893952}a^{18}-\frac{95}{37248}a^{17}+\frac{11}{3492}a^{16}+\frac{9}{49664}a^{15}-\frac{9673}{446976}a^{14}+\frac{1295}{49664}a^{13}-\frac{26995}{446976}a^{12}+\frac{41189}{893952}a^{11}-\frac{4577}{49664}a^{10}-\frac{1529}{13968}a^{9}+\frac{277}{3492}a^{8}+\frac{1885}{55872}a^{7}-\frac{659}{6208}a^{6}-\frac{4457}{27936}a^{5}+\frac{233}{1552}a^{4}-\frac{515}{6984}a^{3}+\frac{1081}{3492}a^{2}+\frac{16}{291}a-\frac{277}{1746}$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6, 1/2*a^7, 1/4*a^8 - 1/2*a^5 - 1/2*a^2, 1/4*a^9 - 1/2*a^3, 1/4*a^10 - 1/2*a^4, 1/8*a^11 - 1/4*a^5 - 1/2*a^2, 1/8*a^12 - 1/4*a^6 - 1/2*a^3, 1/8*a^13 - 1/4*a^7 - 1/2*a^4, 1/8*a^14 - 1/2*a^2, 1/16*a^15 + 1/4*a^3, 1/32*a^16 + 1/8*a^4, 1/64*a^17 - 1/16*a^14 - 1/16*a^13 - 1/16*a^12 - 1/8*a^9 - 1/8*a^8 - 1/8*a^7 - 1/8*a^6 + 5/16*a^5 - 1/4*a^4, 1/256*a^18 + 1/64*a^15 + 1/64*a^14 - 1/64*a^13 + 1/32*a^12 - 1/16*a^11 + 1/32*a^10 - 3/32*a^9 - 3/32*a^8 - 5/32*a^7 + 9/64*a^6 + 3/16*a^5 + 3/8*a^4 - 1/8*a^3 - 1/2, 1/512*a^19 + 1/128*a^16 + 1/128*a^15 - 1/128*a^14 + 1/64*a^13 + 1/32*a^12 + 1/64*a^11 + 5/64*a^10 + 5/64*a^9 - 5/64*a^8 - 23/128*a^7 + 7/32*a^6 - 5/16*a^5 + 3/16*a^4 - 1/2*a^3 - 1/2*a^2 + 1/4*a, 1/1536*a^20 + 1/1536*a^19 + 1/768*a^18 + 1/384*a^17 + 1/192*a^16 + 1/192*a^15 + 1/128*a^14 + 1/96*a^13 + 5/192*a^12 - 1/32*a^11 - 1/48*a^10 - 1/32*a^9 + 19/384*a^8 - 79/384*a^7 + 1/64*a^6 + 5/48*a^5 - 23/48*a^4 - 1/24*a^3 + 1/12*a^2 + 1/12*a + 1/6, 1/3072*a^21 - 1/1536*a^19 - 1/768*a^18 + 1/768*a^17 - 1/256*a^16 - 1/96*a^15 - 1/384*a^14 + 1/128*a^13 - 23/384*a^12 - 13/384*a^11 + 25/384*a^10 - 59/768*a^9 - 1/24*a^8 + 11/384*a^7 + 11/96*a^6 - 5/48*a^5 - 5/16*a^4 - 3/8*a^3 - 1/2*a^2 - 1/12*a - 1/3, 1/18432*a^22 + 1/9216*a^21 - 1/4608*a^20 - 1/4608*a^19 + 7/4608*a^18 - 3/512*a^17 - 5/384*a^16 + 29/1152*a^15 - 35/2304*a^14 + 5/768*a^13 - 23/768*a^12 + 89/2304*a^11 + 127/1536*a^10 - 79/768*a^9 - 41/1152*a^8 + 71/288*a^7 + 1/12*a^6 + 3/8*a^5 + 37/144*a^4 + 5/72*a^3 - 13/36*a^2 - 7/18*a + 1/9, 1/3575808*a^23 - 7/1787904*a^22 - 1/18624*a^21 - 35/297984*a^20 - 469/893952*a^19 - 1195/893952*a^18 - 95/37248*a^17 + 11/3492*a^16 + 9/49664*a^15 - 9673/446976*a^14 + 1295/49664*a^13 - 26995/446976*a^12 + 41189/893952*a^11 - 4577/49664*a^10 - 1529/13968*a^9 + 277/3492*a^8 + 1885/55872*a^7 - 659/6208*a^6 - 4457/27936*a^5 + 233/1552*a^4 - 515/6984*a^3 + 1081/3492*a^2 + 16/291*a - 277/1746

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{1745}{1191936} a^{23} - \frac{395}{595968} a^{22} - \frac{109}{74496} a^{21} + \frac{671}{297984} a^{20} + \frac{307}{297984} a^{19} - \frac{75}{99328} a^{18} + \frac{295}{24832} a^{17} + \frac{61}{37248} a^{16} - \frac{113}{148992} a^{15} - \frac{2125}{49664} a^{14} + \frac{975}{49664} a^{13} - \frac{8953}{148992} a^{12} + \frac{9009}{99328} a^{11} - \frac{4419}{49664} a^{10} + \frac{3635}{9312} a^{9} - \frac{9275}{37248} a^{8} + \frac{3229}{12416} a^{7} - \frac{3335}{6208} a^{6} + \frac{6601}{9312} a^{5} - \frac{2119}{1164} a^{4} + \frac{1037}{2328} a^{3} - \frac{433}{291} a^{2} + \frac{4169}{1164} a - \frac{545}{194} $ -(1745)/(1191936)a^(23) - (395)/(595968)a^(22) - (109)/(74496)a^(21) + (671)/(297984)a^(20) + (307)/(297984)a^(19) - (75)/(99328)a^(18) + (295)/(24832)a^(17) + (61)/(37248)a^(16) - (113)/(148992)a^(15) - (2125)/(49664)a^(14) + (975)/(49664)a^(13) - (8953)/(148992)a^(12) + (9009)/(99328)a^(11) - (4419)/(49664)a^(10) + (3635)/(9312)a^(9) - (9275)/(37248)a^(8) + (3229)/(12416)a^(7) - (3335)/(6208)a^(6) + (6601)/(9312)a^(5) - (2119)/(1164)a^(4) + (1037)/(2328)a^(3) - (433)/(291)a^(2) + (4169)/(1164)*a - (545)/(194) (order $12$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{971}{1191936}a^{23}+\frac{1157}{595968}a^{22}-\frac{103}{37248}a^{21}+\frac{283}{297984}a^{20}-\frac{2797}{297984}a^{19}+\frac{1089}{99328}a^{18}-\frac{675}{24832}a^{17}+\frac{1031}{37248}a^{16}-\frac{4381}{148992}a^{15}+\frac{5247}{49664}a^{14}-\frac{4845}{49664}a^{13}+\frac{25579}{148992}a^{12}-\frac{26299}{99328}a^{11}+\frac{11877}{49664}a^{10}-\frac{11839}{18624}a^{9}+\frac{22541}{37248}a^{8}-\frac{7053}{12416}a^{7}+\frac{7529}{6208}a^{6}-\frac{10471}{9312}a^{5}+\frac{5753}{2328}a^{4}-\frac{3425}{2328}a^{3}+\frac{343}{291}a^{2}-\frac{4949}{1164}a-\frac{157}{194}$, $\frac{3079}{1787904}a^{23}+\frac{563}{893952}a^{22}-\frac{473}{148992}a^{21}-\frac{95}{148992}a^{20}-\frac{79}{6984}a^{19}+\frac{3491}{446976}a^{18}-\frac{973}{37248}a^{17}+\frac{935}{27936}a^{16}-\frac{419}{24832}a^{15}+\frac{23963}{223488}a^{14}-\frac{7109}{74496}a^{13}+\frac{39719}{223488}a^{12}-\frac{148189}{446976}a^{11}+\frac{16967}{74496}a^{10}-\frac{75257}{111744}a^{9}+\frac{10463}{13968}a^{8}-\frac{66377}{111744}a^{7}+\frac{13729}{9312}a^{6}-\frac{15257}{13968}a^{5}+\frac{13453}{4656}a^{4}-\frac{2189}{873}a^{3}+\frac{2269}{1746}a^{2}-\frac{6983}{1164}a+\frac{329}{873}$, $\frac{2917}{1787904}a^{23}-\frac{3299}{1787904}a^{22}-\frac{107}{99328}a^{21}-\frac{57}{24832}a^{20}+\frac{791}{446976}a^{19}-\frac{1709}{223488}a^{18}+\frac{209}{148992}a^{17}+\frac{563}{111744}a^{16}+\frac{2323}{74496}a^{15}-\frac{1141}{55872}a^{14}+\frac{65}{9312}a^{13}-\frac{53}{111744}a^{12}-\frac{22465}{446976}a^{11}-\frac{17537}{148992}a^{10}+\frac{10487}{223488}a^{9}+\frac{17881}{111744}a^{8}+\frac{3019}{111744}a^{7}-\frac{65}{9312}a^{6}+\frac{2879}{6984}a^{5}+\frac{749}{1164}a^{4}-\frac{9829}{6984}a^{3}-\frac{2035}{3492}a^{2}-\frac{1141}{1164}a+\frac{98}{873}$, $\frac{459}{198656}a^{23}-\frac{3661}{595968}a^{22}+\frac{2929}{297984}a^{21}-\frac{2771}{148992}a^{20}+\frac{6221}{148992}a^{19}-\frac{4417}{74496}a^{18}+\frac{12649}{148992}a^{17}-\frac{1523}{12416}a^{16}+\frac{15895}{74496}a^{15}-\frac{13081}{37248}a^{14}+\frac{5625}{12416}a^{13}-\frac{12287}{18624}a^{12}+\frac{47981}{49664}a^{11}-\frac{199477}{148992}a^{10}+\frac{134555}{74496}a^{9}-\frac{6107}{3104}a^{8}+\frac{99707}{37248}a^{7}-\frac{77647}{18624}a^{6}+\frac{5771}{1164}a^{5}-\frac{1555}{291}a^{4}+\frac{3013}{582}a^{3}-\frac{1904}{291}a^{2}+\frac{2723}{388}a+\frac{145}{194}$, $\frac{2513}{893952}a^{23}-\frac{69}{198656}a^{22}+\frac{379}{446976}a^{21}-\frac{1189}{223488}a^{20}+\frac{437}{74496}a^{19}-\frac{4001}{446976}a^{18}-\frac{385}{148992}a^{17}-\frac{2549}{223488}a^{16}+\frac{4897}{111744}a^{15}-\frac{3881}{223488}a^{14}+\frac{1559}{74496}a^{13}-\frac{1903}{223488}a^{12}+\frac{2005}{111744}a^{11}-\frac{8373}{49664}a^{10}-\frac{7483}{55872}a^{9}+\frac{14}{97}a^{8}+\frac{13175}{111744}a^{7}-\frac{2063}{18624}a^{6}+\frac{1895}{6984}a^{5}+\frac{2737}{1746}a^{4}-\frac{901}{2328}a^{3}+\frac{13}{582}a^{2}-\frac{7591}{3492}a+\frac{4055}{1746}$, $\frac{5753}{1787904}a^{23}-\frac{4963}{893952}a^{22}+\frac{1679}{893952}a^{21}-\frac{709}{55872}a^{20}+\frac{4411}{223488}a^{19}-\frac{2735}{148992}a^{18}+\frac{565}{24832}a^{17}-\frac{4999}{223488}a^{16}+\frac{23867}{223488}a^{15}-\frac{7781}{74496}a^{14}+\frac{1505}{24832}a^{13}-\frac{43841}{223488}a^{12}+\frac{6961}{49664}a^{11}-\frac{25385}{74496}a^{10}+\frac{62335}{223488}a^{9}+\frac{32581}{111744}a^{8}+\frac{23117}{37248}a^{7}-\frac{7609}{9312}a^{6}+\frac{2209}{6984}a^{5}+\frac{9905}{13968}a^{4}-\frac{15283}{6984}a^{3}-\frac{6823}{3492}a^{2}-\frac{14185}{3492}a+\frac{2759}{291}$, $\frac{2189}{3575808}a^{23}-\frac{323}{198656}a^{22}-\frac{2155}{893952}a^{21}-\frac{925}{893952}a^{20}+\frac{281}{99328}a^{19}+\frac{1205}{893952}a^{18}+\frac{317}{74496}a^{17}+\frac{4285}{223488}a^{16}+\frac{4261}{446976}a^{15}-\frac{7633}{446976}a^{14}-\frac{2689}{49664}a^{13}+\frac{733}{446976}a^{12}-\frac{62023}{893952}a^{11}-\frac{3067}{148992}a^{10}+\frac{9473}{223488}a^{9}+\frac{2239}{4656}a^{8}-\frac{5953}{111744}a^{7}+\frac{1651}{18624}a^{6}-\frac{7873}{27936}a^{5}+\frac{1165}{6984}a^{4}-\frac{225}{97}a^{3}-\frac{875}{1164}a^{2}-\frac{1759}{3492}a+\frac{6881}{1746}$, $\frac{6497}{3575808}a^{23}-\frac{1597}{446976}a^{22}+\frac{2521}{893952}a^{21}-\frac{6095}{893952}a^{20}+\frac{16943}{893952}a^{19}-\frac{8095}{297984}a^{18}+\frac{5125}{148992}a^{17}-\frac{5263}{111744}a^{16}+\frac{45137}{446976}a^{15}-\frac{24365}{148992}a^{14}+\frac{26083}{148992}a^{13}-\frac{107321}{446976}a^{12}+\frac{37353}{99328}a^{11}-\frac{48155}{74496}a^{10}+\frac{197435}{223488}a^{9}-\frac{87277}{111744}a^{8}+\frac{14111}{12416}a^{7}-\frac{18005}{9312}a^{6}+\frac{65465}{27936}a^{5}-\frac{29705}{13968}a^{4}+\frac{12379}{6984}a^{3}-\frac{3305}{873}a^{2}+\frac{15427}{3492}a+\frac{20}{97}$, $\frac{4025}{1191936}a^{23}-\frac{4985}{1787904}a^{22}-\frac{379}{111744}a^{21}-\frac{473}{893952}a^{20}+\frac{1819}{893952}a^{19}+\frac{8243}{893952}a^{18}-\frac{973}{37248}a^{17}+\frac{147}{3104}a^{16}-\frac{3565}{446976}a^{15}+\frac{24161}{446976}a^{14}-\frac{26537}{148992}a^{13}+\frac{30133}{148992}a^{12}-\frac{342841}{893952}a^{11}+\frac{53819}{148992}a^{10}-\frac{6425}{9312}a^{9}+\frac{168967}{111744}a^{8}-\frac{128845}{111744}a^{7}+\frac{11983}{9312}a^{6}-\frac{20615}{9312}a^{5}+\frac{29249}{6984}a^{4}-\frac{5099}{873}a^{3}+\frac{2735}{873}a^{2}-\frac{21143}{3492}a+\frac{9253}{873}$, $\frac{5069}{3575808}a^{23}-\frac{713}{223488}a^{22}+\frac{3377}{446976}a^{21}-\frac{17369}{893952}a^{20}+\frac{20867}{893952}a^{19}-\frac{4117}{99328}a^{18}+\frac{9415}{148992}a^{17}-\frac{23237}{223488}a^{16}+\frac{78461}{446976}a^{15}-\frac{31009}{148992}a^{14}+\frac{52699}{148992}a^{13}-\frac{253313}{446976}a^{12}+\frac{220375}{297984}a^{11}-\frac{79439}{74496}a^{10}+\frac{32287}{27936}a^{9}-\frac{92027}{55872}a^{8}+\frac{94693}{37248}a^{7}-\frac{6263}{2328}a^{6}+\frac{100811}{27936}a^{5}-\frac{54857}{13968}a^{4}+\frac{4829}{873}a^{3}-\frac{24191}{3492}a^{2}+\frac{7807}{3492}a-\frac{556}{291}$, $\frac{2021}{1787904}a^{23}-\frac{337}{148992}a^{22}+\frac{4543}{893952}a^{21}-\frac{275}{55872}a^{20}+\frac{2081}{148992}a^{19}-\frac{5809}{446976}a^{18}+\frac{151}{6208}a^{17}-\frac{10015}{223488}a^{16}+\frac{9277}{223488}a^{15}-\frac{23227}{223488}a^{14}+\frac{8941}{74496}a^{13}-\frac{31643}{223488}a^{12}+\frac{120953}{446976}a^{11}-\frac{8783}{37248}a^{10}+\frac{90559}{223488}a^{9}-\frac{19439}{37248}a^{8}+\frac{25331}{55872}a^{7}-\frac{6557}{6208}a^{6}+\frac{4519}{6984}a^{5}-\frac{6877}{6984}a^{4}+\frac{1615}{1164}a^{3}+\frac{659}{1164}a^{2}+\frac{2069}{1746}a+\frac{5275}{1746}$ 971/1191936a^23 + 1157/595968a^22 - 103/37248a^21 + 283/297984a^20 - 2797/297984a^19 + 1089/99328a^18 - 675/24832a^17 + 1031/37248a^16 - 4381/148992a^15 + 5247/49664a^14 - 4845/49664a^13 + 25579/148992a^12 - 26299/99328a^11 + 11877/49664a^10 - 11839/18624a^9 + 22541/37248a^8 - 7053/12416a^7 + 7529/6208a^6 - 10471/9312a^5 + 5753/2328a^4 - 3425/2328a^3 + 343/291a^2 - 4949/1164a - 157/194, 3079/1787904a^23 + 563/893952a^22 - 473/148992a^21 - 95/148992a^20 - 79/6984a^19 + 3491/446976a^18 - 973/37248a^17 + 935/27936a^16 - 419/24832a^15 + 23963/223488a^14 - 7109/74496a^13 + 39719/223488a^12 - 148189/446976a^11 + 16967/74496a^10 - 75257/111744a^9 + 10463/13968a^8 - 66377/111744a^7 + 13729/9312a^6 - 15257/13968a^5 + 13453/4656a^4 - 2189/873a^3 + 2269/1746a^2 - 6983/1164a + 329/873, 2917/1787904a^23 - 3299/1787904a^22 - 107/99328a^21 - 57/24832a^20 + 791/446976a^19 - 1709/223488a^18 + 209/148992a^17 + 563/111744a^16 + 2323/74496a^15 - 1141/55872a^14 + 65/9312a^13 - 53/111744a^12 - 22465/446976a^11 - 17537/148992a^10 + 10487/223488a^9 + 17881/111744a^8 + 3019/111744a^7 - 65/9312a^6 + 2879/6984a^5 + 749/1164a^4 - 9829/6984a^3 - 2035/3492a^2 - 1141/1164a + 98/873, 459/198656a^23 - 3661/595968a^22 + 2929/297984a^21 - 2771/148992a^20 + 6221/148992a^19 - 4417/74496a^18 + 12649/148992a^17 - 1523/12416a^16 + 15895/74496a^15 - 13081/37248a^14 + 5625/12416a^13 - 12287/18624a^12 + 47981/49664a^11 - 199477/148992a^10 + 134555/74496a^9 - 6107/3104a^8 + 99707/37248a^7 - 77647/18624a^6 + 5771/1164a^5 - 1555/291a^4 + 3013/582a^3 - 1904/291a^2 + 2723/388a + 145/194, 2513/893952a^23 - 69/198656a^22 + 379/446976a^21 - 1189/223488a^20 + 437/74496a^19 - 4001/446976a^18 - 385/148992a^17 - 2549/223488a^16 + 4897/111744a^15 - 3881/223488a^14 + 1559/74496a^13 - 1903/223488a^12 + 2005/111744a^11 - 8373/49664a^10 - 7483/55872a^9 + 14/97a^8 + 13175/111744a^7 - 2063/18624a^6 + 1895/6984a^5 + 2737/1746a^4 - 901/2328a^3 + 13/582a^2 - 7591/3492a + 4055/1746, 5753/1787904a^23 - 4963/893952a^22 + 1679/893952a^21 - 709/55872a^20 + 4411/223488a^19 - 2735/148992a^18 + 565/24832a^17 - 4999/223488a^16 + 23867/223488a^15 - 7781/74496a^14 + 1505/24832a^13 - 43841/223488a^12 + 6961/49664a^11 - 25385/74496a^10 + 62335/223488a^9 + 32581/111744a^8 + 23117/37248a^7 - 7609/9312a^6 + 2209/6984a^5 + 9905/13968a^4 - 15283/6984a^3 - 6823/3492a^2 - 14185/3492a + 2759/291, 2189/3575808a^23 - 323/198656a^22 - 2155/893952a^21 - 925/893952a^20 + 281/99328a^19 + 1205/893952a^18 + 317/74496a^17 + 4285/223488a^16 + 4261/446976a^15 - 7633/446976a^14 - 2689/49664a^13 + 733/446976a^12 - 62023/893952a^11 - 3067/148992a^10 + 9473/223488a^9 + 2239/4656a^8 - 5953/111744a^7 + 1651/18624a^6 - 7873/27936a^5 + 1165/6984a^4 - 225/97a^3 - 875/1164a^2 - 1759/3492a + 6881/1746, 6497/3575808a^23 - 1597/446976a^22 + 2521/893952a^21 - 6095/893952a^20 + 16943/893952a^19 - 8095/297984a^18 + 5125/148992a^17 - 5263/111744a^16 + 45137/446976a^15 - 24365/148992a^14 + 26083/148992a^13 - 107321/446976a^12 + 37353/99328a^11 - 48155/74496a^10 + 197435/223488a^9 - 87277/111744a^8 + 14111/12416a^7 - 18005/9312a^6 + 65465/27936a^5 - 29705/13968a^4 + 12379/6984a^3 - 3305/873a^2 + 15427/3492a + 20/97, 4025/1191936a^23 - 4985/1787904a^22 - 379/111744a^21 - 473/893952a^20 + 1819/893952a^19 + 8243/893952a^18 - 973/37248a^17 + 147/3104a^16 - 3565/446976a^15 + 24161/446976a^14 - 26537/148992a^13 + 30133/148992a^12 - 342841/893952a^11 + 53819/148992a^10 - 6425/9312a^9 + 168967/111744a^8 - 128845/111744a^7 + 11983/9312a^6 - 20615/9312a^5 + 29249/6984a^4 - 5099/873a^3 + 2735/873a^2 - 21143/3492a + 9253/873, 5069/3575808a^23 - 713/223488a^22 + 3377/446976a^21 - 17369/893952a^20 + 20867/893952a^19 - 4117/99328a^18 + 9415/148992a^17 - 23237/223488a^16 + 78461/446976a^15 - 31009/148992a^14 + 52699/148992a^13 - 253313/446976a^12 + 220375/297984a^11 - 79439/74496a^10 + 32287/27936a^9 - 92027/55872a^8 + 94693/37248a^7 - 6263/2328a^6 + 100811/27936a^5 - 54857/13968a^4 + 4829/873a^3 - 24191/3492a^2 + 7807/3492a - 556/291, 2021/1787904a^23 - 337/148992a^22 + 4543/893952a^21 - 275/55872a^20 + 2081/148992a^19 - 5809/446976a^18 + 151/6208a^17 - 10015/223488a^16 + 9277/223488a^15 - 23227/223488a^14 + 8941/74496a^13 - 31643/223488a^12 + 120953/446976a^11 - 8783/37248a^10 + 90559/223488a^9 - 19439/37248a^8 + 25331/55872a^7 - 6557/6208a^6 + 4519/6984a^5 - 6877/6984a^4 + 1615/1164a^3 + 659/1164a^2 + 2069/1746*a + 5275/1746 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 75194114.41726962 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 75194114.41726962 \cdot 4}{12\cdot\sqrt{59344171108947314029391991313268736}}\cr\approx \mathstrut & 0.389521990077421 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 4*x^21 + 4*x^20 - 4*x^19 + 8*x^18 - 16*x^17 + 40*x^16 - 24*x^15 + 40*x^14 - 104*x^13 + 100*x^12 - 208*x^11 + 160*x^10 - 192*x^9 + 640*x^8 - 512*x^7 + 512*x^6 - 512*x^5 + 1024*x^4 - 2048*x^3 + 4096);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^3\times S_4$ (as 24T400):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 192

The 40 conjugacy class representatives for $C_2^3\times S_4$

Character table for $C_2^3\times S_4$ is not computed

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{-3}) $, 3.3.404.1, $\Q(\zeta_{12})$, 6.6.18280192.1, 6.0.1142512.1, 6.6.30847824.1, 6.0.493565184.1, 6.0.2611456.1, 6.6.70509312.1, 6.0.4406832.1, 12.0.4971563078713344.1, 12.0.334165419556864.1, 12.0.243606590856953856.1, 12.12.243606590856953856.1, 12.0.243606590856953856.3, 12.0.243606590856953856.2, 12.0.951588245534976.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Sibling fields

Degree 24 siblings:	data not computed
Degree 32 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.6.0.1}{6} }^{4}$	R	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.6.0.1}{6} }^{4}$	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.2.0.1}{2} }^{12}$	${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.12.16.13	$x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$	$6$	$2$	$16$	$D_6$	$[2]_{3}^{2}$
$2$ 2	2.12.16.13	$x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$	$6$	$2$	$16$	$D_6$	$[2]_{3}^{2}$
$3$ 3	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$7$ 7	7.4.0.1	$x^{4} + 5 x^{2} + 4 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	7.4.0.1	$x^{4} + 5 x^{2} + 4 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	7.4.0.1	$x^{4} + 5 x^{2} + 4 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.0.1	$x^{4} + 5 x^{2} + 4 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$101$ 101	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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