LMFDB - Number field 20.4.4800653894273660658369343520768.1

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Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32)

gp:K = bnfinit(y^20 + 6*y^18 + 20*y^16 + 40*y^14 + 6*y^12 - 236*y^10 - 524*y^8 + 72*y^6 + 737*y^4 - 298*y^2 + 32, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32)

$ x^{20} + 6x^{18} + 20x^{16} + 40x^{14} + 6x^{12} - 236x^{10} - 524x^{8} + 72x^{6} + 737x^{4} - 298x^{2} + 32 $ x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$20$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(4, 8)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$4800653894273660658369343520768$ 4800653894273660658369343520768 $\medspace = 2^{61}\cdot 113^{6}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$34.20$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	not computed
Ramified primes:	$2$, $113$ 2, 113	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q(\sqrt{2}) $
$\Aut(K/\Q)$:	$C_2$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{11}+\frac{1}{4}a^{5}-\frac{1}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{2}a^{5}+\frac{5}{16}a^{4}+\frac{3}{16}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{11}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{3}{16}a^{5}+\frac{3}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{3}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{7}{16}a^{3}-\frac{1}{2}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{11}-\frac{1}{8}a^{9}-\frac{1}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{7}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{32}a^{16}-\frac{1}{32}a^{12}-\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{3}{32}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{5}{32}a^{4}+\frac{3}{16}a^{3}-\frac{1}{16}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{64}a^{17}-\frac{1}{64}a^{16}+\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{11}+\frac{3}{32}a^{10}+\frac{7}{64}a^{9}+\frac{1}{64}a^{8}-\frac{3}{16}a^{7}+\frac{1}{16}a^{6}-\frac{17}{64}a^{5}-\frac{15}{64}a^{4}+\frac{3}{32}a^{3}-\frac{9}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{5137344}a^{18}+\frac{52255}{5137344}a^{16}-\frac{22093}{1712448}a^{14}-\frac{120491}{5137344}a^{12}-\frac{1}{16}a^{11}-\frac{521891}{5137344}a^{10}-\frac{1}{8}a^{9}-\frac{224395}{5137344}a^{8}+\frac{1}{8}a^{7}-\frac{171415}{1712448}a^{6}-\frac{1}{2}a^{5}+\frac{161813}{1712448}a^{4}+\frac{3}{16}a^{3}-\frac{244037}{2568672}a^{2}+\frac{1}{8}a-\frac{38951}{80271}$, $\frac{1}{5137344}a^{19}-\frac{1751}{321084}a^{17}-\frac{1}{64}a^{16}-\frac{22093}{1712448}a^{15}+\frac{60161}{2568672}a^{13}-\frac{1}{64}a^{12}+\frac{280819}{5137344}a^{11}+\frac{3}{32}a^{10}-\frac{36031}{1284336}a^{9}+\frac{1}{64}a^{8}-\frac{278443}{1712448}a^{7}+\frac{1}{16}a^{6}-\frac{66257}{856224}a^{5}-\frac{15}{64}a^{4}-\frac{81883}{1284336}a^{3}-\frac{9}{32}a^{2}+\frac{89747}{642168}a$ 1, a, a^2, a^3, a^4, a^5, 1/2*a^6 - 1/2*a^2, 1/2*a^7 - 1/2*a^3, 1/4*a^8 - 1/4*a^7 - 1/4*a^4 + 1/4*a^3 - 1/2*a^2 - 1/2*a, 1/4*a^9 - 1/4*a^7 - 1/4*a^5 - 1/4*a^3 - 1/2*a, 1/4*a^10 - 1/4*a^7 - 1/4*a^6 - 1/2*a^4 + 1/4*a^3 - 1/2*a, 1/8*a^11 + 1/4*a^5 - 1/8*a^3 + 1/4*a, 1/16*a^12 - 1/16*a^11 - 1/8*a^10 - 1/8*a^9 - 1/8*a^8 + 1/8*a^7 - 1/2*a^5 + 5/16*a^4 + 3/16*a^3 - 3/8*a^2 + 1/8*a - 1/2, 1/16*a^13 - 1/16*a^11 + 1/8*a^7 - 1/4*a^6 - 3/16*a^5 + 3/16*a^3 + 1/4*a^2 - 1/8*a - 1/2, 1/16*a^14 - 1/16*a^11 - 1/8*a^10 - 1/8*a^9 - 1/8*a^7 - 3/16*a^6 - 1/2*a^5 - 1/2*a^4 + 7/16*a^3 - 1/2*a^2 - 3/8*a - 1/2, 1/16*a^15 - 1/16*a^11 - 1/8*a^9 - 1/16*a^7 - 1/4*a^6 + 1/4*a^5 - 7/16*a^3 + 1/4*a^2 - 1/8*a - 1/2, 1/32*a^16 - 1/32*a^12 - 1/16*a^11 - 1/16*a^10 - 1/8*a^9 + 3/32*a^8 + 1/8*a^7 - 1/8*a^6 - 1/2*a^5 + 5/32*a^4 + 3/16*a^3 - 1/16*a^2 + 1/8*a - 1/2, 1/64*a^17 - 1/64*a^16 + 1/64*a^13 - 1/64*a^12 - 1/32*a^11 + 3/32*a^10 + 7/64*a^9 + 1/64*a^8 - 3/16*a^7 + 1/16*a^6 - 17/64*a^5 - 15/64*a^4 + 3/32*a^3 - 9/32*a^2 - 1/8*a, 1/5137344*a^18 + 52255/5137344*a^16 - 22093/1712448*a^14 - 120491/5137344*a^12 - 1/16*a^11 - 521891/5137344*a^10 - 1/8*a^9 - 224395/5137344*a^8 + 1/8*a^7 - 171415/1712448*a^6 - 1/2*a^5 + 161813/1712448*a^4 + 3/16*a^3 - 244037/2568672*a^2 + 1/8*a - 38951/80271, 1/5137344*a^19 - 1751/321084*a^17 - 1/64*a^16 - 22093/1712448*a^15 + 60161/2568672*a^13 - 1/64*a^12 + 280819/5137344*a^11 + 3/32*a^10 - 36031/1284336*a^9 + 1/64*a^8 - 278443/1712448*a^7 + 1/16*a^6 - 66257/856224*a^5 - 15/64*a^4 - 81883/1284336*a^3 - 9/32*a^2 + 89747/642168*a

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{2}$, which has order $2$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

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

oscar:UK, fUK = unit_group(OK)

Rank:	$11$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{21739}{5137344}a^{19}+\frac{9527}{1712448}a^{18}+\frac{136795}{5137344}a^{17}+\frac{46157}{1712448}a^{16}+\frac{168965}{1712448}a^{15}+\frac{44065}{570816}a^{14}+\frac{1173217}{5137344}a^{13}+\frac{171599}{1712448}a^{12}+\frac{1077943}{5137344}a^{11}-\frac{390901}{1712448}a^{10}-\frac{3586159}{5137344}a^{9}-\frac{2495537}{1712448}a^{8}-\frac{3849817}{1712448}a^{7}-\frac{1072085}{570816}a^{6}-\frac{2230543}{1712448}a^{5}+\frac{1709263}{570816}a^{4}+\frac{1113841}{2568672}a^{3}+\frac{4523921}{856224}a^{2}-\frac{1501771}{642168}a-\frac{20101}{26757}$, $\frac{1}{16}a^{19}-\frac{1051}{142704}a^{18}+\frac{3}{8}a^{17}-\frac{2861}{71352}a^{16}+\frac{5}{4}a^{15}-\frac{5333}{47568}a^{14}+\frac{5}{2}a^{13}-\frac{11339}{71352}a^{12}+\frac{3}{8}a^{11}+\frac{42455}{142704}a^{10}-\frac{59}{4}a^{9}+\frac{148637}{71352}a^{8}-\frac{131}{4}a^{7}+\frac{136069}{47568}a^{6}+\frac{9}{2}a^{5}-\frac{116419}{23784}a^{4}+\frac{737}{16}a^{3}-\frac{312563}{35676}a^{2}-\frac{149}{8}a+\frac{110275}{17838}$, $\frac{6061}{1712448}a^{19}+\frac{281}{5137344}a^{18}+\frac{12115}{428112}a^{17}-\frac{2969}{2568672}a^{16}+\frac{57707}{570816}a^{15}-\frac{509}{1712448}a^{14}+\frac{205481}{856224}a^{13}-\frac{7985}{642168}a^{12}+\frac{298759}{1712448}a^{11}-\frac{76525}{5137344}a^{10}-\frac{98653}{107028}a^{9}-\frac{101401}{2568672}a^{8}-\frac{1773643}{570816}a^{7}-\frac{112043}{1712448}a^{6}-\frac{371381}{285408}a^{5}-\frac{171593}{428112}a^{4}+\frac{1975991}{428112}a^{3}+\frac{911635}{1284336}a^{2}-\frac{227299}{214056}a+\frac{23521}{160542}$, $\frac{6061}{1712448}a^{19}-\frac{281}{5137344}a^{18}+\frac{12115}{428112}a^{17}+\frac{2969}{2568672}a^{16}+\frac{57707}{570816}a^{15}+\frac{509}{1712448}a^{14}+\frac{205481}{856224}a^{13}+\frac{7985}{642168}a^{12}+\frac{298759}{1712448}a^{11}+\frac{76525}{5137344}a^{10}-\frac{98653}{107028}a^{9}+\frac{101401}{2568672}a^{8}-\frac{1773643}{570816}a^{7}+\frac{112043}{1712448}a^{6}-\frac{371381}{285408}a^{5}+\frac{171593}{428112}a^{4}+\frac{1975991}{428112}a^{3}-\frac{911635}{1284336}a^{2}-\frac{227299}{214056}a-\frac{23521}{160542}$, $\frac{116381}{1712448}a^{19}+\frac{21143}{642168}a^{18}+\frac{180023}{428112}a^{17}+\frac{1023433}{5137344}a^{16}+\frac{820111}{570816}a^{15}+\frac{18149}{26757}a^{14}+\frac{2554795}{856224}a^{13}+\frac{7139689}{5137344}a^{12}+\frac{1638407}{1712448}a^{11}+\frac{1008305}{2568672}a^{10}-\frac{3394903}{214056}a^{9}-\frac{38820217}{5137344}a^{8}-\frac{22012823}{570816}a^{7}-\frac{7609135}{428112}a^{6}-\frac{598255}{285408}a^{5}-\frac{331555}{1712448}a^{4}+\frac{21232213}{428112}a^{3}+\frac{58237393}{2568672}a^{2}-\frac{2318015}{214056}a-\frac{406703}{80271}$, $\frac{116381}{1712448}a^{19}-\frac{21143}{642168}a^{18}+\frac{180023}{428112}a^{17}-\frac{1023433}{5137344}a^{16}+\frac{820111}{570816}a^{15}-\frac{18149}{26757}a^{14}+\frac{2554795}{856224}a^{13}-\frac{7139689}{5137344}a^{12}+\frac{1638407}{1712448}a^{11}-\frac{1008305}{2568672}a^{10}-\frac{3394903}{214056}a^{9}+\frac{38820217}{5137344}a^{8}-\frac{22012823}{570816}a^{7}+\frac{7609135}{428112}a^{6}-\frac{598255}{285408}a^{5}+\frac{331555}{1712448}a^{4}+\frac{21232213}{428112}a^{3}-\frac{58237393}{2568672}a^{2}-\frac{2318015}{214056}a+\frac{406703}{80271}$, $\frac{397115}{5137344}a^{18}+\frac{2475161}{5137344}a^{16}+\frac{2834305}{1712448}a^{14}+\frac{17777087}{5137344}a^{12}+\frac{6408863}{5137344}a^{10}-\frac{92329013}{5137344}a^{8}-\frac{76151213}{1712448}a^{6}-\frac{6379049}{1712448}a^{4}+\frac{143307425}{2568672}a^{2}-\frac{2097731}{160542}$, $\frac{490741}{1284336}a^{19}-\frac{895}{47568}a^{18}+\frac{6133289}{2568672}a^{17}-\frac{1477}{11892}a^{16}+\frac{3527411}{428112}a^{15}-\frac{7125}{15856}a^{14}+\frac{44535119}{2568672}a^{13}-\frac{5969}{5946}a^{12}+\frac{8492263}{1284336}a^{11}-\frac{31261}{47568}a^{10}-\frac{227518433}{2568672}a^{9}+\frac{50155}{11892}a^{8}-\frac{95254387}{428112}a^{7}+\frac{199897}{15856}a^{6}-\frac{24109289}{856224}a^{5}+\frac{5644}{991}a^{4}+\frac{88454855}{321084}a^{3}-\frac{34724}{2973}a^{2}-\frac{28166545}{642168}a-\frac{16889}{5946}$, $\frac{63167}{321084}a^{19}+\frac{64423}{642168}a^{18}+\frac{818569}{642168}a^{17}+\frac{210011}{321084}a^{16}+\frac{969131}{214056}a^{15}+\frac{496145}{214056}a^{14}+\frac{12764729}{1284336}a^{13}+\frac{816415}{160542}a^{12}+\frac{7174181}{1284336}a^{11}+\frac{1811503}{642168}a^{10}-\frac{28547233}{642168}a^{9}-\frac{7361873}{321084}a^{8}-\frac{13341101}{107028}a^{7}-\frac{13726117}{214056}a^{6}-\frac{17843405}{428112}a^{5}-\frac{1102423}{53514}a^{4}+\frac{171676621}{1284336}a^{3}+\frac{22918831}{321084}a^{2}+\frac{4570309}{642168}a+\frac{103057}{160542}$, $\frac{41575}{5137344}a^{19}-\frac{22367}{856224}a^{18}+\frac{144047}{2568672}a^{17}-\frac{270949}{1712448}a^{16}+\frac{318905}{1712448}a^{15}-\frac{154687}{285408}a^{14}+\frac{496369}{1284336}a^{13}-\frac{1892461}{1712448}a^{12}+\frac{435685}{5137344}a^{11}-\frac{137089}{428112}a^{10}-\frac{5802629}{2568672}a^{9}+\frac{10391797}{1712448}a^{8}-\frac{9430681}{1712448}a^{7}+\frac{4078001}{285408}a^{6}+\frac{43277}{107028}a^{5}+\frac{287119}{570816}a^{4}+\frac{14135081}{1284336}a^{3}-\frac{16667425}{856224}a^{2}-\frac{1688375}{321084}a+\frac{277201}{53514}$, $\frac{10523}{2568672}a^{19}-\frac{5663}{642168}a^{18}+\frac{23015}{2568672}a^{17}-\frac{41159}{642168}a^{16}-\frac{19823}{856224}a^{15}-\frac{56587}{214056}a^{14}-\frac{608245}{2568672}a^{13}-\frac{444593}{642168}a^{12}-\frac{2446387}{2568672}a^{11}-\frac{591113}{642168}a^{10}-\frac{5033651}{2568672}a^{9}+\frac{701123}{642168}a^{8}+\frac{370951}{856224}a^{7}+\frac{1424063}{214056}a^{6}+\frac{8933071}{856224}a^{5}+\frac{1899707}{214056}a^{4}+\frac{1859827}{160542}a^{3}+\frac{379217}{160542}a^{2}-\frac{1968043}{642168}a-\frac{163391}{160542}$ 21739/5137344a^19 + 9527/1712448a^18 + 136795/5137344a^17 + 46157/1712448a^16 + 168965/1712448a^15 + 44065/570816a^14 + 1173217/5137344a^13 + 171599/1712448a^12 + 1077943/5137344a^11 - 390901/1712448a^10 - 3586159/5137344a^9 - 2495537/1712448a^8 - 3849817/1712448a^7 - 1072085/570816a^6 - 2230543/1712448a^5 + 1709263/570816a^4 + 1113841/2568672a^3 + 4523921/856224a^2 - 1501771/642168a - 20101/26757, 1/16a^19 - 1051/142704a^18 + 3/8a^17 - 2861/71352a^16 + 5/4a^15 - 5333/47568a^14 + 5/2a^13 - 11339/71352a^12 + 3/8a^11 + 42455/142704a^10 - 59/4a^9 + 148637/71352a^8 - 131/4a^7 + 136069/47568a^6 + 9/2a^5 - 116419/23784a^4 + 737/16a^3 - 312563/35676a^2 - 149/8a + 110275/17838, 6061/1712448a^19 + 281/5137344a^18 + 12115/428112a^17 - 2969/2568672a^16 + 57707/570816a^15 - 509/1712448a^14 + 205481/856224a^13 - 7985/642168a^12 + 298759/1712448a^11 - 76525/5137344a^10 - 98653/107028a^9 - 101401/2568672a^8 - 1773643/570816a^7 - 112043/1712448a^6 - 371381/285408a^5 - 171593/428112a^4 + 1975991/428112a^3 + 911635/1284336a^2 - 227299/214056a + 23521/160542, 6061/1712448a^19 - 281/5137344a^18 + 12115/428112a^17 + 2969/2568672a^16 + 57707/570816a^15 + 509/1712448a^14 + 205481/856224a^13 + 7985/642168a^12 + 298759/1712448a^11 + 76525/5137344a^10 - 98653/107028a^9 + 101401/2568672a^8 - 1773643/570816a^7 + 112043/1712448a^6 - 371381/285408a^5 + 171593/428112a^4 + 1975991/428112a^3 - 911635/1284336a^2 - 227299/214056a - 23521/160542, 116381/1712448a^19 + 21143/642168a^18 + 180023/428112a^17 + 1023433/5137344a^16 + 820111/570816a^15 + 18149/26757a^14 + 2554795/856224a^13 + 7139689/5137344a^12 + 1638407/1712448a^11 + 1008305/2568672a^10 - 3394903/214056a^9 - 38820217/5137344a^8 - 22012823/570816a^7 - 7609135/428112a^6 - 598255/285408a^5 - 331555/1712448a^4 + 21232213/428112a^3 + 58237393/2568672a^2 - 2318015/214056a - 406703/80271, 116381/1712448a^19 - 21143/642168a^18 + 180023/428112a^17 - 1023433/5137344a^16 + 820111/570816a^15 - 18149/26757a^14 + 2554795/856224a^13 - 7139689/5137344a^12 + 1638407/1712448a^11 - 1008305/2568672a^10 - 3394903/214056a^9 + 38820217/5137344a^8 - 22012823/570816a^7 + 7609135/428112a^6 - 598255/285408a^5 + 331555/1712448a^4 + 21232213/428112a^3 - 58237393/2568672a^2 - 2318015/214056a + 406703/80271, 397115/5137344a^18 + 2475161/5137344a^16 + 2834305/1712448a^14 + 17777087/5137344a^12 + 6408863/5137344a^10 - 92329013/5137344a^8 - 76151213/1712448a^6 - 6379049/1712448a^4 + 143307425/2568672a^2 - 2097731/160542, 490741/1284336a^19 - 895/47568a^18 + 6133289/2568672a^17 - 1477/11892a^16 + 3527411/428112a^15 - 7125/15856a^14 + 44535119/2568672a^13 - 5969/5946a^12 + 8492263/1284336a^11 - 31261/47568a^10 - 227518433/2568672a^9 + 50155/11892a^8 - 95254387/428112a^7 + 199897/15856a^6 - 24109289/856224a^5 + 5644/991a^4 + 88454855/321084a^3 - 34724/2973a^2 - 28166545/642168a - 16889/5946, 63167/321084a^19 + 64423/642168a^18 + 818569/642168a^17 + 210011/321084a^16 + 969131/214056a^15 + 496145/214056a^14 + 12764729/1284336a^13 + 816415/160542a^12 + 7174181/1284336a^11 + 1811503/642168a^10 - 28547233/642168a^9 - 7361873/321084a^8 - 13341101/107028a^7 - 13726117/214056a^6 - 17843405/428112a^5 - 1102423/53514a^4 + 171676621/1284336a^3 + 22918831/321084a^2 + 4570309/642168a + 103057/160542, 41575/5137344a^19 - 22367/856224a^18 + 144047/2568672a^17 - 270949/1712448a^16 + 318905/1712448a^15 - 154687/285408a^14 + 496369/1284336a^13 - 1892461/1712448a^12 + 435685/5137344a^11 - 137089/428112a^10 - 5802629/2568672a^9 + 10391797/1712448a^8 - 9430681/1712448a^7 + 4078001/285408a^6 + 43277/107028a^5 + 287119/570816a^4 + 14135081/1284336a^3 - 16667425/856224a^2 - 1688375/321084a + 277201/53514, 10523/2568672a^19 - 5663/642168a^18 + 23015/2568672a^17 - 41159/642168a^16 - 19823/856224a^15 - 56587/214056a^14 - 608245/2568672a^13 - 444593/642168a^12 - 2446387/2568672a^11 - 591113/642168a^10 - 5033651/2568672a^9 + 701123/642168a^8 + 370951/856224a^7 + 1424063/214056a^6 + 8933071/856224a^5 + 1899707/214056a^4 + 1859827/160542a^3 + 379217/160542a^2 - 1968043/642168*a - 163391/160542 (assuming GRH)	comment:Fundamental units 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:	$ 572116182.073 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 3 $ (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 572116182.073 \cdot 1}{2\cdot\sqrt{4800653894273660658369343520768}}\cr\approx \mathstrut & 5.0741460967 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 6*x^18 + 20*x^16 + 40*x^14 + 6*x^12 - 236*x^10 - 524*x^8 + 72*x^6 + 737*x^4 - 298*x^2 + 32); 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

$C_2^9.S_6:C_2$ (as 20T951):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A non-solvable group of order 737280

The 65 conjugacy class representatives for $C_2^9.S_6:C_2$

Character table for $C_2^9.S_6:C_2$

Intermediate fields

10.2.24207794634752.1

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

comment:Intermediate fields

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

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

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

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

Sibling fields

Degree 20 siblings:	data not computed
Degree 40 siblings:	data not computed
Minimal sibling:	20.4.4800653894273660658369343520768.2

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$16{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$	$20$	${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$	$16{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$	$16{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	$20$	${\href{/padicField/31.5.0.1}{5} }^{4}$	${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.10.0.1}{10} }^{2}$	${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$	$16{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	$20$

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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	2.2.1.0a1.1	$x^{2} + x + 1$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$$[3]$$
	2.1.16.58n1.779	$x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$	$16$	$1$	$58$	16T333	$$[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$$
$113$ 113	113.1.2.1a1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	113.1.2.1a1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	113.4.1.0a1.1	$x^{4} + 62 x + 3$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	113.4.1.0a1.1	$x^{4} + 62 x + 3$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	113.4.2.4a1.2	$x^{8} + 124 x^{5} + 6 x^{4} + 3844 x^{2} + 372 x + 122$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$

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