LMFDB - Number field 20.0.401221017379430122673511505126953125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101)

gp: K = bnfinit(y^20 + 60*y^18 + 1530*y^16 - 61*y^15 + 21600*y^14 - 2745*y^13 + 184275*y^12 - 49410*y^11 + 976936*y^10 - 452925*y^9 + 3246330*y^8 - 2223450*y^7 + 7022340*y^6 - 5859721*y^5 + 10764225*y^4 - 10075065*y^3 + 9995400*y^2 - 13549320*y + 18662101, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101)

$ x^{20} + 60 x^{18} + 1530 x^{16} - 61 x^{15} + 21600 x^{14} - 2745 x^{13} + 184275 x^{12} + \cdots + 18662101 $ x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$20$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 10]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$401221017379430122673511505126953125$ 401221017379430122673511505126953125 $\medspace = 5^{35}\cdot 13^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$60.28$	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:	$5^{7/4}13^{1/2}\approx 60.27943648904789$
Ramified primes:	$5$, $13$ 5, 13	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{5}) $
$\card{ \Gal(K/\Q) }$:	$20$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$325=5^{2}\cdot 13$
Dirichlet character group:	$\lbrace$$\chi_{325}(1,·)$, $\chi_{325}(66,·)$, $\chi_{325}(131,·)$, $\chi_{325}(196,·)$, $\chi_{325}(261,·)$, $\chi_{325}(12,·)$, $\chi_{325}(77,·)$, $\chi_{325}(14,·)$, $\chi_{325}(79,·)$, $\chi_{325}(144,·)$, $\chi_{325}(209,·)$, $\chi_{325}(274,·)$, $\chi_{325}(142,·)$, $\chi_{325}(207,·)$, $\chi_{325}(272,·)$, $\chi_{325}(38,·)$, $\chi_{325}(103,·)$, $\chi_{325}(168,·)$, $\chi_{325}(233,·)$, $\chi_{325}(298,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}-\frac{8}{19}a^{8}-\frac{8}{19}a^{6}-\frac{2}{19}a^{5}+\frac{1}{19}a^{4}+\frac{8}{19}a^{3}-\frac{8}{19}a^{2}+\frac{5}{19}a-\frac{4}{19}$, $\frac{1}{19}a^{11}-\frac{8}{19}a^{9}-\frac{8}{19}a^{7}-\frac{2}{19}a^{6}+\frac{1}{19}a^{5}+\frac{8}{19}a^{4}-\frac{8}{19}a^{3}+\frac{5}{19}a^{2}-\frac{4}{19}a$, $\frac{1}{19}a^{12}+\frac{4}{19}a^{8}-\frac{2}{19}a^{7}-\frac{6}{19}a^{6}-\frac{8}{19}a^{5}-\frac{7}{19}a^{3}+\frac{8}{19}a^{2}+\frac{2}{19}a+\frac{6}{19}$, $\frac{1}{315732481}a^{13}-\frac{7415831}{315732481}a^{12}+\frac{39}{315732481}a^{11}-\frac{1089932}{315732481}a^{10}+\frac{585}{315732481}a^{9}+\frac{18520916}{315732481}a^{8}-\frac{49848285}{315732481}a^{7}+\frac{91238201}{315732481}a^{6}+\frac{14742}{315732481}a^{5}-\frac{8233950}{315732481}a^{4}+\frac{132962105}{315732481}a^{3}+\frac{17644007}{315732481}a^{2}-\frac{132930515}{315732481}a+\frac{6422776}{16617499}$, $\frac{1}{315732481}a^{14}+\frac{42}{315732481}a^{12}+\frac{5629994}{315732481}a^{11}+\frac{693}{315732481}a^{10}-\frac{13620186}{315732481}a^{9}+\frac{5670}{315732481}a^{8}+\frac{2732758}{315732481}a^{7}-\frac{132916178}{315732481}a^{6}-\frac{126901762}{315732481}a^{5}-\frac{99657366}{315732481}a^{4}-\frac{44035218}{315732481}a^{3}+\frac{16653220}{315732481}a^{2}+\frac{26754866}{315732481}a+\frac{4374}{315732481}$, $\frac{1}{315732481}a^{15}+\frac{1362415}{315732481}a^{12}-\frac{945}{315732481}a^{11}-\frac{1078040}{315732481}a^{10}-\frac{18900}{315732481}a^{9}+\frac{122199232}{315732481}a^{8}+\frac{66316906}{315732481}a^{7}+\frac{95763552}{315732481}a^{6}-\frac{33806534}{315732481}a^{5}-\frac{47176797}{315732481}a^{4}-\frac{150450516}{315732481}a^{3}-\frac{132680963}{315732481}a^{2}+\frac{49458837}{315732481}a+\frac{59284440}{315732481}$, $\frac{1}{315732481}a^{16}-\frac{1080}{315732481}a^{12}-\frac{4359728}{315732481}a^{11}-\frac{23760}{315732481}a^{10}-\frac{126436076}{315732481}a^{9}-\frac{218700}{315732481}a^{8}+\frac{7221232}{315732481}a^{7}-\frac{150537267}{315732481}a^{6}-\frac{91194933}{315732481}a^{5}-\frac{1856642}{16617499}a^{4}+\frac{67171817}{315732481}a^{3}-\frac{101279634}{315732481}a^{2}+\frac{92709206}{315732481}a-\frac{196830}{315732481}$, $\frac{1}{315732481}a^{17}-\frac{3822690}{315732481}a^{12}+\frac{18360}{315732481}a^{11}-\frac{7397714}{315732481}a^{10}+\frac{413100}{315732481}a^{9}+\frac{2341716}{315732481}a^{8}+\frac{86656679}{315732481}a^{7}-\frac{79089424}{315732481}a^{6}-\frac{69207335}{315732481}a^{5}+\frac{48250283}{315732481}a^{4}-\frac{77397594}{315732481}a^{3}-\frac{78209577}{315732481}a^{2}-\frac{139519161}{315732481}a+\frac{68448947}{315732481}$, $\frac{1}{315732481}a^{18}+\frac{22032}{315732481}a^{12}-\frac{7870295}{315732481}a^{11}+\frac{545292}{315732481}a^{10}-\frac{37981997}{315732481}a^{9}+\frac{5353776}{315732481}a^{8}+\frac{152369311}{315732481}a^{7}+\frac{7437199}{16617499}a^{6}+\frac{19172156}{315732481}a^{5}-\frac{96019731}{315732481}a^{4}+\frac{52656972}{315732481}a^{3}-\frac{90779006}{315732481}a^{2}-\frac{112858232}{315732481}a+\frac{5353776}{315732481}$, $\frac{1}{315732481}a^{19}-\frac{5531871}{315732481}a^{12}-\frac{16524}{16617499}a^{11}-\frac{192170}{16617499}a^{10}-\frac{396576}{16617499}a^{9}-\frac{73791543}{315732481}a^{8}-\frac{51196997}{315732481}a^{7}-\frac{54958238}{315732481}a^{6}+\frac{110944493}{315732481}a^{5}-\frac{66515704}{315732481}a^{4}+\frac{86727338}{315732481}a^{3}+\frac{70326140}{315732481}a^{2}+\frac{95671494}{315732481}a+\frac{135774889}{315732481}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/19*a^10 - 8/19*a^8 - 8/19*a^6 - 2/19*a^5 + 1/19*a^4 + 8/19*a^3 - 8/19*a^2 + 5/19*a - 4/19, 1/19*a^11 - 8/19*a^9 - 8/19*a^7 - 2/19*a^6 + 1/19*a^5 + 8/19*a^4 - 8/19*a^3 + 5/19*a^2 - 4/19*a, 1/19*a^12 + 4/19*a^8 - 2/19*a^7 - 6/19*a^6 - 8/19*a^5 - 7/19*a^3 + 8/19*a^2 + 2/19*a + 6/19, 1/315732481*a^13 - 7415831/315732481*a^12 + 39/315732481*a^11 - 1089932/315732481*a^10 + 585/315732481*a^9 + 18520916/315732481*a^8 - 49848285/315732481*a^7 + 91238201/315732481*a^6 + 14742/315732481*a^5 - 8233950/315732481*a^4 + 132962105/315732481*a^3 + 17644007/315732481*a^2 - 132930515/315732481*a + 6422776/16617499, 1/315732481*a^14 + 42/315732481*a^12 + 5629994/315732481*a^11 + 693/315732481*a^10 - 13620186/315732481*a^9 + 5670/315732481*a^8 + 2732758/315732481*a^7 - 132916178/315732481*a^6 - 126901762/315732481*a^5 - 99657366/315732481*a^4 - 44035218/315732481*a^3 + 16653220/315732481*a^2 + 26754866/315732481*a + 4374/315732481, 1/315732481*a^15 + 1362415/315732481*a^12 - 945/315732481*a^11 - 1078040/315732481*a^10 - 18900/315732481*a^9 + 122199232/315732481*a^8 + 66316906/315732481*a^7 + 95763552/315732481*a^6 - 33806534/315732481*a^5 - 47176797/315732481*a^4 - 150450516/315732481*a^3 - 132680963/315732481*a^2 + 49458837/315732481*a + 59284440/315732481, 1/315732481*a^16 - 1080/315732481*a^12 - 4359728/315732481*a^11 - 23760/315732481*a^10 - 126436076/315732481*a^9 - 218700/315732481*a^8 + 7221232/315732481*a^7 - 150537267/315732481*a^6 - 91194933/315732481*a^5 - 1856642/16617499*a^4 + 67171817/315732481*a^3 - 101279634/315732481*a^2 + 92709206/315732481*a - 196830/315732481, 1/315732481*a^17 - 3822690/315732481*a^12 + 18360/315732481*a^11 - 7397714/315732481*a^10 + 413100/315732481*a^9 + 2341716/315732481*a^8 + 86656679/315732481*a^7 - 79089424/315732481*a^6 - 69207335/315732481*a^5 + 48250283/315732481*a^4 - 77397594/315732481*a^3 - 78209577/315732481*a^2 - 139519161/315732481*a + 68448947/315732481, 1/315732481*a^18 + 22032/315732481*a^12 - 7870295/315732481*a^11 + 545292/315732481*a^10 - 37981997/315732481*a^9 + 5353776/315732481*a^8 + 152369311/315732481*a^7 + 7437199/16617499*a^6 + 19172156/315732481*a^5 - 96019731/315732481*a^4 + 52656972/315732481*a^3 - 90779006/315732481*a^2 - 112858232/315732481*a + 5353776/315732481, 1/315732481*a^19 - 5531871/315732481*a^12 - 16524/16617499*a^11 - 192170/16617499*a^10 - 396576/16617499*a^9 - 73791543/315732481*a^8 - 51196997/315732481*a^7 - 54958238/315732481*a^6 + 110944493/315732481*a^5 - 66515704/315732481*a^4 + 86727338/315732481*a^3 + 70326140/315732481*a^2 + 95671494/315732481*a + 135774889/315732481

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{15362}$, which has order $15362$ (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:	$9$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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:	$\frac{61}{16617499}a^{15}+\frac{2745}{16617499}a^{13}+\frac{49410}{16617499}a^{11}-\frac{3964}{16617499}a^{10}+\frac{452925}{16617499}a^{9}-\frac{118920}{16617499}a^{8}+\frac{2223450}{16617499}a^{7}-\frac{1248660}{16617499}a^{6}+\frac{5603094}{16617499}a^{5}-\frac{5351400}{16617499}a^{4}+\frac{6225660}{16617499}a^{3}-\frac{8027100}{16617499}a^{2}+\frac{2001105}{16617499}a-\frac{1926504}{16617499}$, $\frac{97}{315732481}a^{18}+\frac{5238}{315732481}a^{16}+\frac{117855}{315732481}a^{14}+\frac{1429974}{315732481}a^{12}+\frac{10111959}{315732481}a^{10}+\frac{42003522}{315732481}a^{8}-\frac{919480}{315732481}a^{7}+\frac{98008218}{315732481}a^{6}-\frac{19309080}{315732481}a^{5}+\frac{114555060}{315732481}a^{4}-\frac{115854480}{315732481}a^{3}+\frac{51549777}{315732481}a^{2}-\frac{173781720}{315732481}a+\frac{3818502}{315732481}$, $\frac{61}{315732481}a^{19}-\frac{97}{315732481}a^{18}+\frac{3694}{315732481}a^{17}-\frac{5319}{315732481}a^{16}+\frac{93356}{315732481}a^{15}-\frac{123024}{315732481}a^{14}+\frac{1275507}{315732481}a^{13}-\frac{1559592}{315732481}a^{12}+\frac{10166129}{315732481}a^{11}-\frac{11694200}{315732481}a^{10}+\frac{47443954}{315732481}a^{9}-\frac{51736903}{315732481}a^{8}+\frac{125307085}{315732481}a^{7}-\frac{127921416}{315732481}a^{6}+\frac{197764743}{315732481}a^{5}-\frac{188006983}{315732481}a^{4}+\frac{292694571}{315732481}a^{3}-\frac{354059670}{315732481}a^{2}+\frac{179365037}{315732481}a-\frac{39293561}{315732481}$, $\frac{40}{315732481}a^{19}+\frac{2063}{315732481}a^{17}+\frac{44812}{315732481}a^{15}-\frac{2683}{315732481}a^{14}+\frac{531788}{315732481}a^{13}-\frac{5183}{16617499}a^{12}+\frac{3743521}{315732481}a^{11}-\frac{1423111}{315732481}a^{10}+\frac{16055457}{315732481}a^{9}-\frac{535115}{16617499}a^{8}+\frac{43213572}{315732481}a^{7}-\frac{37183015}{315732481}a^{6}+\frac{80358615}{315732481}a^{5}-\frac{74849913}{315732481}a^{4}+\frac{105682239}{315732481}a^{3}-\frac{123058332}{315732481}a^{2}+\frac{57775923}{315732481}a-\frac{4067064}{16617499}$, $\frac{21}{315732481}a^{19}+\frac{63}{16617499}a^{17}+\frac{427}{315732481}a^{16}+\frac{1512}{16617499}a^{15}+\frac{19215}{315732481}a^{14}+\frac{19845}{16617499}a^{13}+\frac{345870}{315732481}a^{12}+\frac{2913281}{315732481}a^{11}+\frac{3170475}{315732481}a^{10}+\frac{13032411}{315732481}a^{9}+\frac{15564150}{315732481}a^{8}+\frac{29654352}{315732481}a^{7}+\frac{41018047}{315732481}a^{6}+\frac{20132658}{315732481}a^{5}+\frac{59297123}{315732481}a^{4}-\frac{16922115}{315732481}a^{3}-\frac{39894744}{315732481}a^{2}-\frac{121954504}{315732481}a-\frac{202109976}{315732481}$, $\frac{508}{315732481}a^{16}+\frac{24384}{315732481}a^{14}+\frac{475488}{315732481}a^{12}+\frac{4828032}{315732481}a^{10}-\frac{173383}{315732481}a^{9}+\frac{27157680}{315732481}a^{8}-\frac{4681341}{315732481}a^{7}+\frac{82954368}{315732481}a^{6}-\frac{42132069}{315732481}a^{5}+\frac{124431552}{315732481}a^{4}-\frac{140440230}{315732481}a^{3}+\frac{71103744}{315732481}a^{2}-\frac{126396207}{315732481}a+\frac{6665976}{315732481}$, $\frac{217}{315732481}a^{17}+\frac{11067}{315732481}a^{15}+\frac{232407}{315732481}a^{13}+\frac{2589678}{315732481}a^{11}+\frac{16434495}{315732481}a^{9}-\frac{399331}{315732481}a^{8}+\frac{59164182}{315732481}a^{7}-\frac{9583944}{315732481}a^{6}+\frac{112949802}{315732481}a^{5}-\frac{71879580}{315732481}a^{4}+\frac{96814116}{315732481}a^{3}-\frac{172510992}{315732481}a^{2}+\frac{24203529}{315732481}a-\frac{64691622}{315732481}$, $\frac{6160}{315732481}a^{13}-\frac{14209}{315732481}a^{12}+\frac{240240}{315732481}a^{11}-\frac{511524}{315732481}a^{10}+\frac{3603600}{315732481}a^{9}-\frac{6905574}{315732481}a^{8}+\frac{25945920}{315732481}a^{7}-\frac{42968016}{315732481}a^{6}+\frac{90810720}{315732481}a^{5}-\frac{120847545}{315732481}a^{4}+\frac{136216080}{315732481}a^{3}-\frac{124300332}{315732481}a^{2}+\frac{58378320}{315732481}a-\frac{20716722}{315732481}$, $\frac{40}{315732481}a^{19}+\frac{120}{16617499}a^{17}+\frac{2880}{16617499}a^{15}+\frac{37800}{16617499}a^{13}+\frac{294840}{16617499}a^{11}+\frac{1389960}{16617499}a^{9}+\frac{3849120}{16617499}a^{7}-\frac{2117473}{315732481}a^{6}+\frac{5773680}{16617499}a^{5}-\frac{38114514}{315732481}a^{4}+\frac{3936600}{16617499}a^{3}-\frac{171515313}{315732481}a^{2}+\frac{787320}{16617499}a-\frac{114343542}{315732481}$ 61/16617499a^15 + 2745/16617499a^13 + 49410/16617499a^11 - 3964/16617499a^10 + 452925/16617499a^9 - 118920/16617499a^8 + 2223450/16617499a^7 - 1248660/16617499a^6 + 5603094/16617499a^5 - 5351400/16617499a^4 + 6225660/16617499a^3 - 8027100/16617499a^2 + 2001105/16617499a - 1926504/16617499, 97/315732481a^18 + 5238/315732481a^16 + 117855/315732481a^14 + 1429974/315732481a^12 + 10111959/315732481a^10 + 42003522/315732481a^8 - 919480/315732481a^7 + 98008218/315732481a^6 - 19309080/315732481a^5 + 114555060/315732481a^4 - 115854480/315732481a^3 + 51549777/315732481a^2 - 173781720/315732481a + 3818502/315732481, 61/315732481a^19 - 97/315732481a^18 + 3694/315732481a^17 - 5319/315732481a^16 + 93356/315732481a^15 - 123024/315732481a^14 + 1275507/315732481a^13 - 1559592/315732481a^12 + 10166129/315732481a^11 - 11694200/315732481a^10 + 47443954/315732481a^9 - 51736903/315732481a^8 + 125307085/315732481a^7 - 127921416/315732481a^6 + 197764743/315732481a^5 - 188006983/315732481a^4 + 292694571/315732481a^3 - 354059670/315732481a^2 + 179365037/315732481a - 39293561/315732481, 40/315732481a^19 + 2063/315732481a^17 + 44812/315732481a^15 - 2683/315732481a^14 + 531788/315732481a^13 - 5183/16617499a^12 + 3743521/315732481a^11 - 1423111/315732481a^10 + 16055457/315732481a^9 - 535115/16617499a^8 + 43213572/315732481a^7 - 37183015/315732481a^6 + 80358615/315732481a^5 - 74849913/315732481a^4 + 105682239/315732481a^3 - 123058332/315732481a^2 + 57775923/315732481a - 4067064/16617499, 21/315732481a^19 + 63/16617499a^17 + 427/315732481a^16 + 1512/16617499a^15 + 19215/315732481a^14 + 19845/16617499a^13 + 345870/315732481a^12 + 2913281/315732481a^11 + 3170475/315732481a^10 + 13032411/315732481a^9 + 15564150/315732481a^8 + 29654352/315732481a^7 + 41018047/315732481a^6 + 20132658/315732481a^5 + 59297123/315732481a^4 - 16922115/315732481a^3 - 39894744/315732481a^2 - 121954504/315732481a - 202109976/315732481, 508/315732481a^16 + 24384/315732481a^14 + 475488/315732481a^12 + 4828032/315732481a^10 - 173383/315732481a^9 + 27157680/315732481a^8 - 4681341/315732481a^7 + 82954368/315732481a^6 - 42132069/315732481a^5 + 124431552/315732481a^4 - 140440230/315732481a^3 + 71103744/315732481a^2 - 126396207/315732481a + 6665976/315732481, 217/315732481a^17 + 11067/315732481a^15 + 232407/315732481a^13 + 2589678/315732481a^11 + 16434495/315732481a^9 - 399331/315732481a^8 + 59164182/315732481a^7 - 9583944/315732481a^6 + 112949802/315732481a^5 - 71879580/315732481a^4 + 96814116/315732481a^3 - 172510992/315732481a^2 + 24203529/315732481a - 64691622/315732481, 6160/315732481a^13 - 14209/315732481a^12 + 240240/315732481a^11 - 511524/315732481a^10 + 3603600/315732481a^9 - 6905574/315732481a^8 + 25945920/315732481a^7 - 42968016/315732481a^6 + 90810720/315732481a^5 - 120847545/315732481a^4 + 136216080/315732481a^3 - 124300332/315732481a^2 + 58378320/315732481a - 20716722/315732481, 40/315732481a^19 + 120/16617499a^17 + 2880/16617499a^15 + 37800/16617499a^13 + 294840/16617499a^11 + 1389960/16617499a^9 + 3849120/16617499a^7 - 2117473/315732481a^6 + 5773680/16617499a^5 - 38114514/315732481a^4 + 3936600/16617499a^3 - 171515313/315732481a^2 + 787320/16617499a - 114343542/315732481 (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:	$ 161406.837641 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 161406.837641 \cdot 15362}{2\cdot\sqrt{401221017379430122673511505126953125}}\cr\approx \mathstrut & 0.187692336190 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101)

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^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101, 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^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101);

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^20 + 60*x^18 + 1530*x^16 - 61*x^15 + 21600*x^14 - 2745*x^13 + 184275*x^12 - 49410*x^11 + 976936*x^10 - 452925*x^9 + 3246330*x^8 - 2223450*x^7 + 7022340*x^6 - 5859721*x^5 + 10764225*x^4 - 10075065*x^3 + 9995400*x^2 - 13549320*x + 18662101);

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

$C_{20}$ (as 20T1):

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 cyclic group of order 20

The 20 conjugacy class representatives for $C_{20}$

Character table for $C_{20}$

Intermediate fields

$\Q(\sqrt{5}) $, 4.0.21125.1, 5.5.390625.1, $\Q(\zeta_{25})^+$

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$20$	$20$	R	${\href{/padicField/7.4.0.1}{4} }^{5}$	${\href{/padicField/11.10.0.1}{10} }^{2}$	R	$20$	${\href{/padicField/19.5.0.1}{5} }^{4}$	$20$	${\href{/padicField/29.10.0.1}{10} }^{2}$	${\href{/padicField/31.10.0.1}{10} }^{2}$	$20$	${\href{/padicField/41.10.0.1}{10} }^{2}$	${\href{/padicField/43.4.0.1}{4} }^{5}$	$20$	$20$	${\href{/padicField/59.5.0.1}{5} }^{4}$

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

# 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
$5$ 5		Deg $20$	$20$	$1$	$35$
$13$ 13	13.20.10.2	$x^{20} - 2599051 x^{10} + 24134045 x^{8} - 501988136 x^{6} + 815730721 x^{4} - 10604499373 x^{2} + 275716983698$	$2$	$10$	$10$	20T1	$[\ ]_{2}^{10}$

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