LMFDB - Number field 20.4.163840000000000000000000000.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25)

gp:K = bnfinit(y^20 - 10*y^19 + 45*y^18 - 120*y^17 + 215*y^16 - 292*y^15 + 350*y^14 - 400*y^13 + 385*y^12 - 230*y^11 + 11*y^10 + 60*y^9 + 55*y^8 - 140*y^7 + 90*y^6 - 20*y^5 + 25*y^4 - 50*y^3 - 75*y^2 + 100*y - 25, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25)

$ x^{20} - 10 x^{19} + 45 x^{18} - 120 x^{17} + 215 x^{16} - 292 x^{15} + 350 x^{14} - 400 x^{13} + \cdots - 25 $ x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25

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:	$163840000000000000000000000$ 163840000000000000000000000 $\medspace = 2^{36}\cdot 5^{22}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$20.45$	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:	$2^{13/6}5^{23/20}\approx 28.57900880593445$
Ramified primes:	$2$, $5$ 2, 5	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:	$\Q$
$\Aut(K/\Q)$:	$C_2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{5}$, $\frac{1}{5}a^{11}-\frac{1}{5}a^{6}$, $\frac{1}{10}a^{12}-\frac{1}{10}a^{10}+\frac{2}{5}a^{7}-\frac{1}{2}a^{6}-\frac{2}{5}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{10}a^{13}-\frac{1}{10}a^{11}+\frac{2}{5}a^{8}-\frac{1}{2}a^{7}-\frac{2}{5}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{10}a^{14}-\frac{1}{10}a^{10}+\frac{2}{5}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{2}{5}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{10}a^{15}-\frac{1}{10}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{2}{5}a^{6}-\frac{1}{10}a^{5}-\frac{1}{2}a$, $\frac{1}{10}a^{16}-\frac{1}{2}a^{8}+\frac{2}{5}a^{6}-\frac{1}{2}$, $\frac{1}{10}a^{17}-\frac{1}{2}a^{9}+\frac{2}{5}a^{7}-\frac{1}{2}a$, $\frac{1}{61850}a^{18}-\frac{9}{61850}a^{17}-\frac{442}{30925}a^{16}+\frac{1091}{61850}a^{15}-\frac{917}{30925}a^{14}+\frac{2309}{61850}a^{13}+\frac{717}{30925}a^{12}+\frac{2197}{30925}a^{11}-\frac{3281}{61850}a^{10}+\frac{2592}{30925}a^{9}-\frac{15}{1237}a^{8}-\frac{1989}{6185}a^{7}-\frac{80}{1237}a^{6}-\frac{765}{2474}a^{5}-\frac{674}{6185}a^{4}+\frac{1161}{2474}a^{3}+\frac{557}{2474}a^{2}+\frac{1203}{2474}a+\frac{439}{1237}$, $\frac{1}{78611350}a^{19}+\frac{313}{39305675}a^{18}+\frac{2838501}{78611350}a^{17}-\frac{1154009}{78611350}a^{16}+\frac{1705291}{78611350}a^{15}-\frac{2807491}{78611350}a^{14}-\frac{1823673}{39305675}a^{13}-\frac{1206273}{39305675}a^{12}+\frac{3808697}{39305675}a^{11}-\frac{5480001}{78611350}a^{10}-\frac{765937}{1572227}a^{9}+\frac{3835803}{7861135}a^{8}+\frac{3946391}{15722270}a^{7}+\frac{936715}{3144454}a^{6}-\frac{6455421}{15722270}a^{5}-\frac{1038409}{3144454}a^{4}-\frac{639}{2474}a^{3}-\frac{395282}{1572227}a^{2}-\frac{202091}{1572227}a-\frac{214798}{1572227}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/5*a^10 - 1/5*a^5, 1/5*a^11 - 1/5*a^6, 1/10*a^12 - 1/10*a^10 + 2/5*a^7 - 1/2*a^6 - 2/5*a^5 - 1/2*a^2 - 1/2, 1/10*a^13 - 1/10*a^11 + 2/5*a^8 - 1/2*a^7 - 2/5*a^6 - 1/2*a^3 - 1/2*a, 1/10*a^14 - 1/10*a^10 + 2/5*a^9 - 1/2*a^8 - 1/2*a^6 - 2/5*a^5 - 1/2*a^4 - 1/2, 1/10*a^15 - 1/10*a^11 - 1/2*a^9 - 1/2*a^7 - 2/5*a^6 - 1/10*a^5 - 1/2*a, 1/10*a^16 - 1/2*a^8 + 2/5*a^6 - 1/2, 1/10*a^17 - 1/2*a^9 + 2/5*a^7 - 1/2*a, 1/61850*a^18 - 9/61850*a^17 - 442/30925*a^16 + 1091/61850*a^15 - 917/30925*a^14 + 2309/61850*a^13 + 717/30925*a^12 + 2197/30925*a^11 - 3281/61850*a^10 + 2592/30925*a^9 - 15/1237*a^8 - 1989/6185*a^7 - 80/1237*a^6 - 765/2474*a^5 - 674/6185*a^4 + 1161/2474*a^3 + 557/2474*a^2 + 1203/2474*a + 439/1237, 1/78611350*a^19 + 313/39305675*a^18 + 2838501/78611350*a^17 - 1154009/78611350*a^16 + 1705291/78611350*a^15 - 2807491/78611350*a^14 - 1823673/39305675*a^13 - 1206273/39305675*a^12 + 3808697/39305675*a^11 - 5480001/78611350*a^10 - 765937/1572227*a^9 + 3835803/7861135*a^8 + 3946391/15722270*a^7 + 936715/3144454*a^6 - 6455421/15722270*a^5 - 1038409/3144454*a^4 - 639/2474*a^3 - 395282/1572227*a^2 - 202091/1572227*a - 214798/1572227

comment:Integral basis

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

Ideal class group:		Trivial group, which has order $1$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$	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:	$\frac{576}{6355}a^{19}-\frac{5472}{6355}a^{18}+\frac{23184}{6355}a^{17}-\frac{57528}{6355}a^{16}+\frac{95076}{6355}a^{15}-\frac{120654}{6355}a^{14}+\frac{141273}{6355}a^{13}-\frac{319527}{12710}a^{12}+\frac{142196}{6355}a^{11}-\frac{126577}{12710}a^{10}-\frac{20542}{6355}a^{9}+\frac{17934}{6355}a^{8}+\frac{47002}{6355}a^{7}-\frac{130801}{12710}a^{6}+\frac{5798}{1271}a^{5}-\frac{676}{1271}a^{4}+2a^{3}-\frac{7707}{2542}a^{2}-\frac{10249}{1271}a+\frac{10249}{2542}$, $\frac{884082}{7861135}a^{19}-\frac{82873123}{78611350}a^{18}+\frac{172810176}{39305675}a^{17}-\frac{840843883}{78611350}a^{16}+\frac{678507306}{39305675}a^{15}-\frac{842863529}{39305675}a^{14}+\frac{979919774}{39305675}a^{13}-\frac{1104735491}{39305675}a^{12}+\frac{955325154}{39305675}a^{11}-\frac{734653227}{78611350}a^{10}-\frac{208739956}{39305675}a^{9}+\frac{47674209}{15722270}a^{8}+\frac{15342100}{1572227}a^{7}-\frac{94577598}{7861135}a^{6}+\frac{35246366}{7861135}a^{5}-\frac{5137373}{7861135}a^{4}+\frac{3927}{1237}a^{3}-\frac{12603995}{3144454}a^{2}-\frac{16488171}{1572227}a+\frac{15186667}{3144454}$, $\frac{5115677}{78611350}a^{19}-\frac{24291522}{39305675}a^{18}+\frac{205758881}{78611350}a^{17}-\frac{509215409}{78611350}a^{16}+\frac{416982148}{39305675}a^{15}-\frac{1036564573}{78611350}a^{14}+\frac{1181671969}{78611350}a^{13}-\frac{655585298}{39305675}a^{12}+\frac{561375002}{39305675}a^{11}-\frac{370592441}{78611350}a^{10}-\frac{415825559}{78611350}a^{9}+\frac{35075511}{7861135}a^{8}+\frac{12595085}{3144454}a^{7}-\frac{97653947}{15722270}a^{6}+\frac{14153173}{7861135}a^{5}+\frac{12956283}{15722270}a^{4}+\frac{1163}{1237}a^{3}-\frac{3816715}{1572227}a^{2}-\frac{11249425}{1572227}a+\frac{6283726}{1572227}$, $\frac{5271579}{78611350}a^{19}-\frac{24676812}{39305675}a^{18}+\frac{203848061}{78611350}a^{17}-\frac{485156659}{78611350}a^{16}+\frac{376895818}{39305675}a^{15}-\frac{449801561}{39305675}a^{14}+\frac{1053288539}{78611350}a^{13}-\frac{1248560181}{78611350}a^{12}+\frac{556070052}{39305675}a^{11}-\frac{423155721}{78611350}a^{10}-\frac{192725217}{78611350}a^{9}-\frac{14830751}{15722270}a^{8}+\frac{139570591}{15722270}a^{7}-\frac{106105657}{15722270}a^{6}-\frac{10294468}{7861135}a^{5}+\frac{21059302}{7861135}a^{4}+\frac{3774}{1237}a^{3}-\frac{16452037}{3144454}a^{2}-\frac{9279784}{1572227}a+\frac{6158737}{1572227}$, $\frac{184686}{7861135}a^{19}-\frac{18659837}{78611350}a^{18}+\frac{42279384}{39305675}a^{17}-\frac{226588157}{78611350}a^{16}+\frac{203560284}{39305675}a^{15}-\frac{276503956}{39305675}a^{14}+\frac{661480967}{78611350}a^{13}-\frac{379435664}{39305675}a^{12}+\frac{751379877}{78611350}a^{11}-\frac{498623403}{78611350}a^{10}+\frac{57467226}{39305675}a^{9}+\frac{631319}{3144454}a^{8}+\frac{41442373}{15722270}a^{7}-\frac{38957789}{7861135}a^{6}+\frac{26405714}{7861135}a^{5}-\frac{1134217}{7861135}a^{4}-\frac{1669}{2474}a^{3}-\frac{2482457}{3144454}a^{2}-\frac{1913243}{3144454}a+\frac{3044239}{3144454}$, $\frac{1774782}{39305675}a^{19}-\frac{16326609}{39305675}a^{18}+\frac{66675716}{39305675}a^{17}-\frac{158215819}{39305675}a^{16}+\frac{495298897}{78611350}a^{15}-\frac{296704448}{39305675}a^{14}+\frac{663253443}{78611350}a^{13}-\frac{711135757}{78611350}a^{12}+\frac{274955039}{39305675}a^{11}-\frac{81884127}{78611350}a^{10}-\frac{334515653}{78611350}a^{9}+\frac{21177407}{7861135}a^{8}+\frac{22258432}{7861135}a^{7}-\frac{67570169}{15722270}a^{6}+\frac{6271165}{3144454}a^{5}-\frac{1392592}{7861135}a^{4}+\frac{1899}{2474}a^{3}-\frac{4037289}{3144454}a^{2}-\frac{7699040}{1572227}a+\frac{4627033}{3144454}$, $\frac{1200288}{39305675}a^{19}-\frac{4566941}{15722270}a^{18}+\frac{9684688}{7861135}a^{17}-\frac{24109462}{7861135}a^{16}+\frac{80263019}{15722270}a^{15}-\frac{513042887}{78611350}a^{14}+\frac{59701104}{7861135}a^{13}-\frac{26504329}{3144454}a^{12}+\frac{116443801}{15722270}a^{11}-\frac{9859945}{3144454}a^{10}-\frac{127763229}{78611350}a^{9}+\frac{26949623}{15722270}a^{8}+\frac{30178969}{15722270}a^{7}-\frac{26783661}{7861135}a^{6}+\frac{33297961}{15722270}a^{5}-\frac{15969227}{15722270}a^{4}+\frac{1654}{1237}a^{3}-\frac{3046244}{1572227}a^{2}-\frac{6550633}{3144454}a+\frac{2150189}{1572227}$, $\frac{8678237}{78611350}a^{19}-\frac{41213682}{39305675}a^{18}+\frac{349151921}{78611350}a^{17}-\frac{865026089}{78611350}a^{16}+\frac{711004678}{39305675}a^{15}-\frac{1782809563}{78611350}a^{14}+\frac{1027722737}{39305675}a^{13}-\frac{2303238411}{78611350}a^{12}+\frac{2025815669}{78611350}a^{11}-\frac{410495163}{39305675}a^{10}-\frac{464266479}{78611350}a^{9}+\frac{7661311}{1572227}a^{8}+\frac{14155585}{1572227}a^{7}-\frac{101461942}{7861135}a^{6}+\frac{39944623}{7861135}a^{5}+\frac{8775223}{15722270}a^{4}+\frac{3563}{2474}a^{3}-\frac{13186323}{3144454}a^{2}-\frac{32032409}{3144454}a+\frac{14975891}{3144454}$, $\frac{484402}{39305675}a^{19}-\frac{7794099}{78611350}a^{18}+\frac{25915771}{78611350}a^{17}-\frac{42366219}{78611350}a^{16}+\frac{26258981}{78611350}a^{15}+\frac{19830353}{78611350}a^{14}-\frac{27678093}{39305675}a^{13}+\frac{89949089}{78611350}a^{12}-\frac{178211811}{78611350}a^{11}+\frac{283519329}{78611350}a^{10}-\frac{125890596}{39305675}a^{9}+\frac{5623132}{7861135}a^{8}+\frac{2104215}{3144454}a^{7}+\frac{3345109}{7861135}a^{6}-\frac{3313205}{3144454}a^{5}+\frac{7596289}{15722270}a^{4}-\frac{210}{1237}a^{3}+\frac{1583364}{1572227}a^{2}-\frac{3649645}{1572227}a-\frac{3104449}{3144454}$, $\frac{87023}{1267925}a^{19}-\frac{1651797}{2535850}a^{18}+\frac{6979583}{2535850}a^{17}-\frac{8608761}{1267925}a^{16}+\frac{14061964}{1267925}a^{15}-\frac{17523982}{1267925}a^{14}+\frac{40470827}{2535850}a^{13}-\frac{22844064}{1267925}a^{12}+\frac{39986607}{2535850}a^{11}-\frac{14883843}{2535850}a^{10}-\frac{11469197}{2535850}a^{9}+\frac{184649}{50717}a^{8}+\frac{500097}{101434}a^{7}-\frac{1649173}{253585}a^{6}+\frac{231972}{253585}a^{5}+\frac{376457}{253585}a^{4}+\frac{4127}{2474}a^{3}-\frac{302213}{101434}a^{2}-\frac{336776}{50717}a+\frac{213201}{50717}$, $\frac{181267}{2535850}a^{19}-\frac{861828}{1267925}a^{18}+\frac{7275999}{2535850}a^{17}-\frac{17900041}{2535850}a^{16}+\frac{14596877}{1267925}a^{15}-\frac{3657021}{253585}a^{14}+\frac{42607571}{2535850}a^{13}-\frac{47802249}{2535850}a^{12}+\frac{20471968}{1267925}a^{11}-\frac{15027909}{2535850}a^{10}-\frac{10911087}{2535850}a^{9}+\frac{359513}{101434}a^{8}+\frac{2100743}{507170}a^{7}-\frac{2995861}{507170}a^{6}+\frac{571012}{253585}a^{5}+\frac{22807}{253585}a^{4}+\frac{971}{1237}a^{3}-\frac{205627}{101434}a^{2}-\frac{341624}{50717}a+\frac{206892}{50717}$ 576/6355a^19 - 5472/6355a^18 + 23184/6355a^17 - 57528/6355a^16 + 95076/6355a^15 - 120654/6355a^14 + 141273/6355a^13 - 319527/12710a^12 + 142196/6355a^11 - 126577/12710a^10 - 20542/6355a^9 + 17934/6355a^8 + 47002/6355a^7 - 130801/12710a^6 + 5798/1271a^5 - 676/1271a^4 + 2a^3 - 7707/2542a^2 - 10249/1271a + 10249/2542, 884082/7861135a^19 - 82873123/78611350a^18 + 172810176/39305675a^17 - 840843883/78611350a^16 + 678507306/39305675a^15 - 842863529/39305675a^14 + 979919774/39305675a^13 - 1104735491/39305675a^12 + 955325154/39305675a^11 - 734653227/78611350a^10 - 208739956/39305675a^9 + 47674209/15722270a^8 + 15342100/1572227a^7 - 94577598/7861135a^6 + 35246366/7861135a^5 - 5137373/7861135a^4 + 3927/1237a^3 - 12603995/3144454a^2 - 16488171/1572227a + 15186667/3144454, 5115677/78611350a^19 - 24291522/39305675a^18 + 205758881/78611350a^17 - 509215409/78611350a^16 + 416982148/39305675a^15 - 1036564573/78611350a^14 + 1181671969/78611350a^13 - 655585298/39305675a^12 + 561375002/39305675a^11 - 370592441/78611350a^10 - 415825559/78611350a^9 + 35075511/7861135a^8 + 12595085/3144454a^7 - 97653947/15722270a^6 + 14153173/7861135a^5 + 12956283/15722270a^4 + 1163/1237a^3 - 3816715/1572227a^2 - 11249425/1572227a + 6283726/1572227, 5271579/78611350a^19 - 24676812/39305675a^18 + 203848061/78611350a^17 - 485156659/78611350a^16 + 376895818/39305675a^15 - 449801561/39305675a^14 + 1053288539/78611350a^13 - 1248560181/78611350a^12 + 556070052/39305675a^11 - 423155721/78611350a^10 - 192725217/78611350a^9 - 14830751/15722270a^8 + 139570591/15722270a^7 - 106105657/15722270a^6 - 10294468/7861135a^5 + 21059302/7861135a^4 + 3774/1237a^3 - 16452037/3144454a^2 - 9279784/1572227a + 6158737/1572227, 184686/7861135a^19 - 18659837/78611350a^18 + 42279384/39305675a^17 - 226588157/78611350a^16 + 203560284/39305675a^15 - 276503956/39305675a^14 + 661480967/78611350a^13 - 379435664/39305675a^12 + 751379877/78611350a^11 - 498623403/78611350a^10 + 57467226/39305675a^9 + 631319/3144454a^8 + 41442373/15722270a^7 - 38957789/7861135a^6 + 26405714/7861135a^5 - 1134217/7861135a^4 - 1669/2474a^3 - 2482457/3144454a^2 - 1913243/3144454a + 3044239/3144454, 1774782/39305675a^19 - 16326609/39305675a^18 + 66675716/39305675a^17 - 158215819/39305675a^16 + 495298897/78611350a^15 - 296704448/39305675a^14 + 663253443/78611350a^13 - 711135757/78611350a^12 + 274955039/39305675a^11 - 81884127/78611350a^10 - 334515653/78611350a^9 + 21177407/7861135a^8 + 22258432/7861135a^7 - 67570169/15722270a^6 + 6271165/3144454a^5 - 1392592/7861135a^4 + 1899/2474a^3 - 4037289/3144454a^2 - 7699040/1572227a + 4627033/3144454, 1200288/39305675a^19 - 4566941/15722270a^18 + 9684688/7861135a^17 - 24109462/7861135a^16 + 80263019/15722270a^15 - 513042887/78611350a^14 + 59701104/7861135a^13 - 26504329/3144454a^12 + 116443801/15722270a^11 - 9859945/3144454a^10 - 127763229/78611350a^9 + 26949623/15722270a^8 + 30178969/15722270a^7 - 26783661/7861135a^6 + 33297961/15722270a^5 - 15969227/15722270a^4 + 1654/1237a^3 - 3046244/1572227a^2 - 6550633/3144454a + 2150189/1572227, 8678237/78611350a^19 - 41213682/39305675a^18 + 349151921/78611350a^17 - 865026089/78611350a^16 + 711004678/39305675a^15 - 1782809563/78611350a^14 + 1027722737/39305675a^13 - 2303238411/78611350a^12 + 2025815669/78611350a^11 - 410495163/39305675a^10 - 464266479/78611350a^9 + 7661311/1572227a^8 + 14155585/1572227a^7 - 101461942/7861135a^6 + 39944623/7861135a^5 + 8775223/15722270a^4 + 3563/2474a^3 - 13186323/3144454a^2 - 32032409/3144454a + 14975891/3144454, 484402/39305675a^19 - 7794099/78611350a^18 + 25915771/78611350a^17 - 42366219/78611350a^16 + 26258981/78611350a^15 + 19830353/78611350a^14 - 27678093/39305675a^13 + 89949089/78611350a^12 - 178211811/78611350a^11 + 283519329/78611350a^10 - 125890596/39305675a^9 + 5623132/7861135a^8 + 2104215/3144454a^7 + 3345109/7861135a^6 - 3313205/3144454a^5 + 7596289/15722270a^4 - 210/1237a^3 + 1583364/1572227a^2 - 3649645/1572227a - 3104449/3144454, 87023/1267925a^19 - 1651797/2535850a^18 + 6979583/2535850a^17 - 8608761/1267925a^16 + 14061964/1267925a^15 - 17523982/1267925a^14 + 40470827/2535850a^13 - 22844064/1267925a^12 + 39986607/2535850a^11 - 14883843/2535850a^10 - 11469197/2535850a^9 + 184649/50717a^8 + 500097/101434a^7 - 1649173/253585a^6 + 231972/253585a^5 + 376457/253585a^4 + 4127/2474a^3 - 302213/101434a^2 - 336776/50717a + 213201/50717, 181267/2535850a^19 - 861828/1267925a^18 + 7275999/2535850a^17 - 17900041/2535850a^16 + 14596877/1267925a^15 - 3657021/253585a^14 + 42607571/2535850a^13 - 47802249/2535850a^12 + 20471968/1267925a^11 - 15027909/2535850a^10 - 10911087/2535850a^9 + 359513/101434a^8 + 2100743/507170a^7 - 2995861/507170a^6 + 571012/253585a^5 + 22807/253585a^4 + 971/1237a^3 - 205627/101434a^2 - 341624/50717*a + 206892/50717	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:	$ 331332.875442 $	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 4 $

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 331332.875442 \cdot 1}{2\cdot\sqrt{163840000000000000000000000}}\cr\approx \mathstrut & 0.503017962444 \end{aligned}\]

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 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25) 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 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25, 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 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25); 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 - 10*x^19 + 45*x^18 - 120*x^17 + 215*x^16 - 292*x^15 + 350*x^14 - 400*x^13 + 385*x^12 - 230*x^11 + 11*x^10 + 60*x^9 + 55*x^8 - 140*x^7 + 90*x^6 - 20*x^5 + 25*x^4 - 50*x^3 - 75*x^2 + 100*x - 25); 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

$S_5$ (as 20T32):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

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

A non-solvable group of order 120

The 7 conjugacy class representatives for $S_5$

Character table for $S_5$

Intermediate fields

$\Q(\sqrt{5}) $, 10.2.12800000000000.1

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 5 sibling:	5.1.800000.1
Degree 6 sibling:	6.2.3200000.1
Degree 10 siblings:	10.2.12800000000000.1, 10.2.3200000000000.2
Degree 12 sibling:	12.4.256000000000000.1
Degree 15 sibling:	deg 15
Degree 20 siblings:	deg 20, deg 20
Degree 24 sibling:	data not computed
Degree 30 siblings:	data not computed
Degree 40 sibling:	data not computed
Minimal sibling:	5.1.800000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }$	R	${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.5.0.1}{5} }^{4}$	${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$	${\href{/padicField/29.5.0.1}{5} }^{4}$	${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$	${\href{/padicField/37.2.0.1}{2} }^{10}$	${\href{/padicField/41.3.0.1}{3} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.5.0.1}{5} }^{4}$

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	2.2.4.16a1.1	$x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 14 x^{2} + 8 x + 7$	$4$	$2$	$16$	$S_4$	$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
$2$ 2	2.2.6.20a1.45	$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 231 x^{6} + 228 x^{5} + 180 x^{4} + 110 x^{3} + 55 x^{2} + 20 x + 9$	$6$	$2$	$20$	$S_4$	$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
$5$ 5	5.1.10.11a2.2	$x^{10} + 20 x^{2} + 5$	$10$	$1$	$11$	$F_5$	$$[\frac{5}{4}]_{4}$$
$5$ 5	5.1.10.11a2.2	$x^{10} + 20 x^{2} + 5$	$10$	$1$	$11$	$F_5$	$$[\frac{5}{4}]_{4}$$

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