LMFDB - Number field 20.0.20403517554797011816436767578125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1)

gp: K = bnfinit(y^20 - y^19 + 6*y^18 - 7*y^17 + 30*y^16 - 18*y^15 + 123*y^14 - 78*y^13 + 538*y^12 - 412*y^11 + 704*y^10 - 501*y^9 + 837*y^8 + 172*y^7 + 309*y^6 + 49*y^5 + 174*y^4 - 46*y^3 + 13*y^2 - 3*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1)

$ x^{20} - x^{19} + 6 x^{18} - 7 x^{17} + 30 x^{16} - 18 x^{15} + 123 x^{14} - 78 x^{13} + 538 x^{12} + \cdots + 1 $ x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1

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:	$20403517554797011816436767578125$ 20403517554797011816436767578125 $\medspace = 5^{15}\cdot 401^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$36.77$	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:	$5^{3/4}401^{1/2}\approx 66.95757085563274$
Ramified primes:	$5$, $401$ 5, 401	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}) $
$\card{ \Aut(K/\Q) }$:	$4$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1417708231}a^{17}+\frac{384167857}{1417708231}a^{16}+\frac{637500389}{1417708231}a^{15}-\frac{134369331}{1417708231}a^{14}+\frac{605418008}{1417708231}a^{13}-\frac{311749414}{1417708231}a^{12}+\frac{208049519}{1417708231}a^{11}+\frac{78541616}{1417708231}a^{10}-\frac{549468139}{1417708231}a^{9}+\frac{686496298}{1417708231}a^{8}-\frac{644161275}{1417708231}a^{7}-\frac{489623395}{1417708231}a^{6}+\frac{232383097}{1417708231}a^{5}+\frac{321375228}{1417708231}a^{4}+\frac{128034204}{1417708231}a^{3}+\frac{158908518}{1417708231}a^{2}-\frac{15990677}{1417708231}a-\frac{149797153}{1417708231}$, $\frac{1}{1417708231}a^{18}+\frac{662726879}{1417708231}a^{16}+\frac{484932519}{1417708231}a^{15}-\frac{6714589}{1417708231}a^{14}+\frac{397177271}{1417708231}a^{13}+\frac{506524719}{1417708231}a^{12}+\frac{674365493}{1417708231}a^{11}+\frac{236237116}{1417708231}a^{10}-\frac{276252692}{1417708231}a^{9}-\frac{644775711}{1417708231}a^{8}+\frac{466747914}{1417708231}a^{7}-\frac{638809667}{1417708231}a^{6}+\frac{506107805}{1417708231}a^{5}+\frac{377415341}{1417708231}a^{4}+\frac{150752643}{1417708231}a^{3}-\frac{569769128}{1417708231}a^{2}-\frac{177763377}{1417708231}a+\frac{255558026}{1417708231}$, $\frac{1}{1417708231}a^{19}-\frac{130521986}{1417708231}a^{16}-\frac{567182095}{1417708231}a^{15}-\frac{517410571}{1417708231}a^{14}-\frac{437154121}{1417708231}a^{13}-\frac{449439250}{1417708231}a^{12}-\frac{245569725}{1417708231}a^{11}-\frac{597585616}{1417708231}a^{10}-\frac{646655140}{1417708231}a^{9}-\frac{167879803}{1417708231}a^{8}+\frac{221075082}{1417708231}a^{7}-\frac{405494546}{1417708231}a^{6}-\frac{117115702}{1417708231}a^{5}+\frac{14865276}{1417708231}a^{4}+\frac{8276476}{1417708231}a^{3}-\frac{277849714}{1417708231}a^{2}+\frac{463062711}{1417708231}a+\frac{456636863}{1417708231}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, 1/1417708231*a^17 + 384167857/1417708231*a^16 + 637500389/1417708231*a^15 - 134369331/1417708231*a^14 + 605418008/1417708231*a^13 - 311749414/1417708231*a^12 + 208049519/1417708231*a^11 + 78541616/1417708231*a^10 - 549468139/1417708231*a^9 + 686496298/1417708231*a^8 - 644161275/1417708231*a^7 - 489623395/1417708231*a^6 + 232383097/1417708231*a^5 + 321375228/1417708231*a^4 + 128034204/1417708231*a^3 + 158908518/1417708231*a^2 - 15990677/1417708231*a - 149797153/1417708231, 1/1417708231*a^18 + 662726879/1417708231*a^16 + 484932519/1417708231*a^15 - 6714589/1417708231*a^14 + 397177271/1417708231*a^13 + 506524719/1417708231*a^12 + 674365493/1417708231*a^11 + 236237116/1417708231*a^10 - 276252692/1417708231*a^9 - 644775711/1417708231*a^8 + 466747914/1417708231*a^7 - 638809667/1417708231*a^6 + 506107805/1417708231*a^5 + 377415341/1417708231*a^4 + 150752643/1417708231*a^3 - 569769128/1417708231*a^2 - 177763377/1417708231*a + 255558026/1417708231, 1/1417708231*a^19 - 130521986/1417708231*a^16 - 567182095/1417708231*a^15 - 517410571/1417708231*a^14 - 437154121/1417708231*a^13 - 449439250/1417708231*a^12 - 245569725/1417708231*a^11 - 597585616/1417708231*a^10 - 646655140/1417708231*a^9 - 167879803/1417708231*a^8 + 221075082/1417708231*a^7 - 405494546/1417708231*a^6 - 117115702/1417708231*a^5 + 14865276/1417708231*a^4 + 8276476/1417708231*a^3 - 277849714/1417708231*a^2 + 463062711/1417708231*a + 456636863/1417708231

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

$C_{29}$, which has order $29$ (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:	$ \frac{325947825}{1417708231} a^{19} - \frac{216420225}{1417708231} a^{18} + \frac{1847037675}{1417708231} a^{17} - \frac{1629739125}{1417708231} a^{16} + \frac{9017889825}{1417708231} a^{15} - \frac{2607582600}{1417708231} a^{14} + \frac{38154039180}{1417708231} a^{13} - \frac{12060069525}{1417708231} a^{12} + \frac{166885286400}{1417708231} a^{11} - \frac{75837193950}{1417708231} a^{10} + \frac{184703767500}{1417708231} a^{9} - \frac{88343307058}{1417708231} a^{8} + \frac{218385042750}{1417708231} a^{7} + \frac{147002469075}{1417708231} a^{6} + \frac{119405553225}{1417708231} a^{5} + \frac{49544069400}{1417708231} a^{4} + \frac{58447871898}{1417708231} a^{3} + \frac{3911373900}{1417708231} a^{2} - \frac{760544925}{1417708231} a + \frac{434597100}{1417708231} $ (325947825)/(1417708231)a^(19) - (216420225)/(1417708231)a^(18) + (1847037675)/(1417708231)a^(17) - (1629739125)/(1417708231)a^(16) + (9017889825)/(1417708231)a^(15) - (2607582600)/(1417708231)a^(14) + (38154039180)/(1417708231)a^(13) - (12060069525)/(1417708231)a^(12) + (166885286400)/(1417708231)a^(11) - (75837193950)/(1417708231)a^(10) + (184703767500)/(1417708231)a^(9) - (88343307058)/(1417708231)a^(8) + (218385042750)/(1417708231)a^(7) + (147002469075)/(1417708231)a^(6) + (119405553225)/(1417708231)a^(5) + (49544069400)/(1417708231)a^(4) + (58447871898)/(1417708231)a^(3) + (3911373900)/(1417708231)a^(2) - (760544925)/(1417708231)*a + (434597100)/(1417708231) (order $10$)	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:	$\frac{51219150}{1417708231}a^{19}-\frac{10243830}{1417708231}a^{18}+\frac{235608090}{1417708231}a^{17}-\frac{92797916}{1417708231}a^{16}+\frac{1075602150}{1417708231}a^{15}+\frac{460972350}{1417708231}a^{14}+\frac{4712161800}{1417708231}a^{13}+\frac{1290722580}{1417708231}a^{12}+\frac{20752189557}{1417708231}a^{11}+\frac{2079497490}{1417708231}a^{10}+\frac{3441926880}{1417708231}a^{9}+\frac{10336024470}{1417708231}a^{8}+\frac{4927282230}{1417708231}a^{7}+\frac{52364509749}{1417708231}a^{6}+\frac{2284374090}{1417708231}a^{5}+\frac{1311210240}{1417708231}a^{4}-\frac{338046390}{1417708231}a^{3}+\frac{102438300}{1417708231}a^{2}-\frac{5197833885}{1417708231}a+\frac{1427952061}{1417708231}$, $\frac{1727553}{1417708231}a^{19}-\frac{10365318}{1417708231}a^{18}+\frac{12092871}{1417708231}a^{17}-\frac{51826590}{1417708231}a^{16}+\frac{67847513}{1417708231}a^{15}-\frac{212489019}{1417708231}a^{14}+\frac{134749134}{1417708231}a^{13}-\frac{929423514}{1417708231}a^{12}+\frac{711751836}{1417708231}a^{11}-\frac{4204627898}{1417708231}a^{10}+\frac{865504053}{1417708231}a^{9}-\frac{1445961861}{1417708231}a^{8}-\frac{297139116}{1417708231}a^{7}-\frac{533813877}{1417708231}a^{6}-\frac{9025308596}{1417708231}a^{5}-\frac{300594222}{1417708231}a^{4}+\frac{79467438}{1417708231}a^{3}-\frac{22458189}{1417708231}a^{2}+\frac{5182659}{1417708231}a+\frac{368435119}{1417708231}$, $\frac{51219150}{1417708231}a^{19}-\frac{10243830}{1417708231}a^{18}+\frac{235608090}{1417708231}a^{17}-\frac{92797916}{1417708231}a^{16}+\frac{1075602150}{1417708231}a^{15}+\frac{460972350}{1417708231}a^{14}+\frac{4712161800}{1417708231}a^{13}+\frac{1290722580}{1417708231}a^{12}+\frac{20752189557}{1417708231}a^{11}+\frac{2079497490}{1417708231}a^{10}+\frac{3441926880}{1417708231}a^{9}+\frac{10336024470}{1417708231}a^{8}+\frac{4927282230}{1417708231}a^{7}+\frac{52364509749}{1417708231}a^{6}+\frac{2284374090}{1417708231}a^{5}+\frac{1311210240}{1417708231}a^{4}-\frac{338046390}{1417708231}a^{3}+\frac{102438300}{1417708231}a^{2}-\frac{3780125654}{1417708231}a+\frac{10243830}{1417708231}$, $\frac{79105780}{1417708231}a^{19}-\frac{15821156}{1417708231}a^{18}+\frac{363886588}{1417708231}a^{17}-\frac{144785803}{1417708231}a^{16}+\frac{1661221380}{1417708231}a^{15}+\frac{711952020}{1417708231}a^{14}+\frac{7277731760}{1417708231}a^{13}+\frac{1993465656}{1417708231}a^{12}+\frac{32020194887}{1417708231}a^{11}+\frac{3211694668}{1417708231}a^{10}+\frac{5315908416}{1417708231}a^{9}+\frac{15963546404}{1417708231}a^{8}+\frac{7609976036}{1417708231}a^{7}+\frac{83383635119}{1417708231}a^{6}+\frac{3528117788}{1417708231}a^{5}+\frac{2025107968}{1417708231}a^{4}-\frac{522098148}{1417708231}a^{3}+\frac{158211560}{1417708231}a^{2}-\frac{1104028965}{1417708231}a+\frac{15821156}{1417708231}$, $\frac{93706444}{1417708231}a^{19}-\frac{163986277}{1417708231}a^{18}+\frac{609546447}{1417708231}a^{17}-\frac{1054197495}{1417708231}a^{16}+\frac{3162592485}{1417708231}a^{15}-\frac{3631124705}{1417708231}a^{14}+\frac{12088131276}{1417708231}a^{13}-\frac{15521896311}{1417708231}a^{12}+\frac{53014420693}{1417708231}a^{11}-\frac{74590329424}{1417708231}a^{10}+\frac{82321111054}{1417708231}a^{9}-\frac{86772167144}{1417708231}a^{8}+\frac{96368395944}{1417708231}a^{7}-\frac{30969979742}{1417708231}a^{6}-\frac{2740913487}{1417708231}a^{5}-\frac{21154229733}{1417708231}a^{4}+\frac{5622386640}{1417708231}a^{3}-\frac{20836418512}{1417708231}a^{2}+\frac{1792534007}{1417708231}a-\frac{1534841286}{1417708231}$, $\frac{27886630}{1417708231}a^{19}-\frac{5577326}{1417708231}a^{18}+\frac{128278498}{1417708231}a^{17}-\frac{51987887}{1417708231}a^{16}+\frac{585619230}{1417708231}a^{15}+\frac{250979670}{1417708231}a^{14}+\frac{2565569960}{1417708231}a^{13}+\frac{702743076}{1417708231}a^{12}+\frac{11268005330}{1417708231}a^{11}+\frac{1132197178}{1417708231}a^{10}+\frac{1873981536}{1417708231}a^{9}+\frac{5627521934}{1417708231}a^{8}+\frac{2682693806}{1417708231}a^{7}+\frac{31019125370}{1417708231}a^{6}+\frac{1243743698}{1417708231}a^{5}+\frac{713897728}{1417708231}a^{4}-\frac{184051758}{1417708231}a^{3}+\frac{55773260}{1417708231}a^{2}+\frac{2676096689}{1417708231}a+\frac{5577326}{1417708231}$, $\frac{478906038}{1417708231}a^{19}-\frac{656500560}{1417708231}a^{18}+\frac{2930135213}{1417708231}a^{17}-\frac{4342331624}{1417708231}a^{16}+\frac{14922281007}{1417708231}a^{15}-\frac{13361137089}{1417708231}a^{14}+\frac{58742589234}{1417708231}a^{13}-\frac{58303369329}{1417708231}a^{12}+\frac{257194318963}{1417708231}a^{11}-\frac{288746615540}{1417708231}a^{10}+\frac{347833474905}{1417708231}a^{9}-\frac{338436696468}{1417708231}a^{8}+\frac{419034674920}{1417708231}a^{7}-\frac{32290571672}{1417708231}a^{6}+\frac{32843924479}{1417708231}a^{5}-\frac{84976990941}{1417708231}a^{4}+\frac{22606474527}{1417708231}a^{3}-\frac{72277342992}{1417708231}a^{2}-\frac{8473111703}{1417708231}a-\frac{5271934394}{1417708231}$, $\frac{9821766}{1417708231}a^{19}-\frac{14215091}{1417708231}a^{18}+\frac{76536393}{1417708231}a^{17}-\frac{119465557}{1417708231}a^{16}+\frac{420085791}{1417708231}a^{15}-\frac{468703336}{1417708231}a^{14}+\frac{1782943542}{1417708231}a^{13}-\frac{1881495414}{1417708231}a^{12}+\frac{7542506200}{1417708231}a^{11}-\frac{8747663499}{1417708231}a^{10}+\frac{17089529483}{1417708231}a^{9}-\frac{19179256480}{1417708231}a^{8}+\frac{26053832973}{1417708231}a^{7}-\frac{15742767588}{1417708231}a^{6}+\frac{19232631579}{1417708231}a^{5}-\frac{4587591660}{1417708231}a^{4}+\frac{1218477931}{1417708231}a^{3}+\frac{2346342018}{1417708231}a^{2}-\frac{2384110787}{1417708231}a+\frac{196233141}{1417708231}$, $\frac{57946280}{1417708231}a^{19}+\frac{22643483}{1417708231}a^{18}+\frac{276775783}{1417708231}a^{17}+\frac{73882357}{1417708231}a^{16}+\frac{1233427261}{1417708231}a^{15}+\frac{1334551247}{1417708231}a^{14}+\frac{5968077411}{1417708231}a^{13}+\frac{5342097875}{1417708231}a^{12}+\frac{26181458574}{1417708231}a^{11}+\frac{19296954324}{1417708231}a^{10}+\frac{13212827382}{1417708231}a^{9}+\frac{26217700865}{1417708231}a^{8}+\frac{16256294677}{1417708231}a^{7}+\frac{73736641270}{1417708231}a^{6}+\frac{42064844705}{1417708231}a^{5}+\frac{29473488063}{1417708231}a^{4}+\frac{24259611146}{1417708231}a^{3}+\frac{12581751599}{1417708231}a^{2}+\frac{1874128171}{1417708231}a+\frac{217446841}{1417708231}$ 51219150/1417708231a^19 - 10243830/1417708231a^18 + 235608090/1417708231a^17 - 92797916/1417708231a^16 + 1075602150/1417708231a^15 + 460972350/1417708231a^14 + 4712161800/1417708231a^13 + 1290722580/1417708231a^12 + 20752189557/1417708231a^11 + 2079497490/1417708231a^10 + 3441926880/1417708231a^9 + 10336024470/1417708231a^8 + 4927282230/1417708231a^7 + 52364509749/1417708231a^6 + 2284374090/1417708231a^5 + 1311210240/1417708231a^4 - 338046390/1417708231a^3 + 102438300/1417708231a^2 - 5197833885/1417708231a + 1427952061/1417708231, 1727553/1417708231a^19 - 10365318/1417708231a^18 + 12092871/1417708231a^17 - 51826590/1417708231a^16 + 67847513/1417708231a^15 - 212489019/1417708231a^14 + 134749134/1417708231a^13 - 929423514/1417708231a^12 + 711751836/1417708231a^11 - 4204627898/1417708231a^10 + 865504053/1417708231a^9 - 1445961861/1417708231a^8 - 297139116/1417708231a^7 - 533813877/1417708231a^6 - 9025308596/1417708231a^5 - 300594222/1417708231a^4 + 79467438/1417708231a^3 - 22458189/1417708231a^2 + 5182659/1417708231a + 368435119/1417708231, 51219150/1417708231a^19 - 10243830/1417708231a^18 + 235608090/1417708231a^17 - 92797916/1417708231a^16 + 1075602150/1417708231a^15 + 460972350/1417708231a^14 + 4712161800/1417708231a^13 + 1290722580/1417708231a^12 + 20752189557/1417708231a^11 + 2079497490/1417708231a^10 + 3441926880/1417708231a^9 + 10336024470/1417708231a^8 + 4927282230/1417708231a^7 + 52364509749/1417708231a^6 + 2284374090/1417708231a^5 + 1311210240/1417708231a^4 - 338046390/1417708231a^3 + 102438300/1417708231a^2 - 3780125654/1417708231a + 10243830/1417708231, 79105780/1417708231a^19 - 15821156/1417708231a^18 + 363886588/1417708231a^17 - 144785803/1417708231a^16 + 1661221380/1417708231a^15 + 711952020/1417708231a^14 + 7277731760/1417708231a^13 + 1993465656/1417708231a^12 + 32020194887/1417708231a^11 + 3211694668/1417708231a^10 + 5315908416/1417708231a^9 + 15963546404/1417708231a^8 + 7609976036/1417708231a^7 + 83383635119/1417708231a^6 + 3528117788/1417708231a^5 + 2025107968/1417708231a^4 - 522098148/1417708231a^3 + 158211560/1417708231a^2 - 1104028965/1417708231a + 15821156/1417708231, 93706444/1417708231a^19 - 163986277/1417708231a^18 + 609546447/1417708231a^17 - 1054197495/1417708231a^16 + 3162592485/1417708231a^15 - 3631124705/1417708231a^14 + 12088131276/1417708231a^13 - 15521896311/1417708231a^12 + 53014420693/1417708231a^11 - 74590329424/1417708231a^10 + 82321111054/1417708231a^9 - 86772167144/1417708231a^8 + 96368395944/1417708231a^7 - 30969979742/1417708231a^6 - 2740913487/1417708231a^5 - 21154229733/1417708231a^4 + 5622386640/1417708231a^3 - 20836418512/1417708231a^2 + 1792534007/1417708231a - 1534841286/1417708231, 27886630/1417708231a^19 - 5577326/1417708231a^18 + 128278498/1417708231a^17 - 51987887/1417708231a^16 + 585619230/1417708231a^15 + 250979670/1417708231a^14 + 2565569960/1417708231a^13 + 702743076/1417708231a^12 + 11268005330/1417708231a^11 + 1132197178/1417708231a^10 + 1873981536/1417708231a^9 + 5627521934/1417708231a^8 + 2682693806/1417708231a^7 + 31019125370/1417708231a^6 + 1243743698/1417708231a^5 + 713897728/1417708231a^4 - 184051758/1417708231a^3 + 55773260/1417708231a^2 + 2676096689/1417708231a + 5577326/1417708231, 478906038/1417708231a^19 - 656500560/1417708231a^18 + 2930135213/1417708231a^17 - 4342331624/1417708231a^16 + 14922281007/1417708231a^15 - 13361137089/1417708231a^14 + 58742589234/1417708231a^13 - 58303369329/1417708231a^12 + 257194318963/1417708231a^11 - 288746615540/1417708231a^10 + 347833474905/1417708231a^9 - 338436696468/1417708231a^8 + 419034674920/1417708231a^7 - 32290571672/1417708231a^6 + 32843924479/1417708231a^5 - 84976990941/1417708231a^4 + 22606474527/1417708231a^3 - 72277342992/1417708231a^2 - 8473111703/1417708231a - 5271934394/1417708231, 9821766/1417708231a^19 - 14215091/1417708231a^18 + 76536393/1417708231a^17 - 119465557/1417708231a^16 + 420085791/1417708231a^15 - 468703336/1417708231a^14 + 1782943542/1417708231a^13 - 1881495414/1417708231a^12 + 7542506200/1417708231a^11 - 8747663499/1417708231a^10 + 17089529483/1417708231a^9 - 19179256480/1417708231a^8 + 26053832973/1417708231a^7 - 15742767588/1417708231a^6 + 19232631579/1417708231a^5 - 4587591660/1417708231a^4 + 1218477931/1417708231a^3 + 2346342018/1417708231a^2 - 2384110787/1417708231a + 196233141/1417708231, 57946280/1417708231a^19 + 22643483/1417708231a^18 + 276775783/1417708231a^17 + 73882357/1417708231a^16 + 1233427261/1417708231a^15 + 1334551247/1417708231a^14 + 5968077411/1417708231a^13 + 5342097875/1417708231a^12 + 26181458574/1417708231a^11 + 19296954324/1417708231a^10 + 13212827382/1417708231a^9 + 26217700865/1417708231a^8 + 16256294677/1417708231a^7 + 73736641270/1417708231a^6 + 42064844705/1417708231a^5 + 29473488063/1417708231a^4 + 24259611146/1417708231a^3 + 12581751599/1417708231a^2 + 1874128171/1417708231*a + 217446841/1417708231 (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:	$ 2526424.45141 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2526424.45141 \cdot 29}{10\cdot\sqrt{20403517554797011816436767578125}}\cr\approx \mathstrut & 0.155543001191 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1)

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 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1, 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 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1);

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 - x^19 + 6*x^18 - 7*x^17 + 30*x^16 - 18*x^15 + 123*x^14 - 78*x^13 + 538*x^12 - 412*x^11 + 704*x^10 - 501*x^9 + 837*x^8 + 172*x^7 + 309*x^6 + 49*x^5 + 174*x^4 - 46*x^3 + 13*x^2 - 3*x + 1);

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

$C_4\times D_5$ (as 20T6):

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 40

The 16 conjugacy class representatives for $C_4\times D_5$

Character table for $C_4\times D_5$

Intermediate fields

$\Q(\sqrt{5}) $, $\Q(\zeta_{5})$, 5.5.160801.1, 10.10.80803005003125.1

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

Galois closure:	deg 40
Degree 20 sibling:	deg 20
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	$20$	${\href{/padicField/3.4.0.1}{4} }^{5}$	R	$20$	${\href{/padicField/11.5.0.1}{5} }^{4}$	${\href{/padicField/13.4.0.1}{4} }^{5}$	${\href{/padicField/17.4.0.1}{4} }^{5}$	${\href{/padicField/19.2.0.1}{2} }^{10}$	${\href{/padicField/23.4.0.1}{4} }^{5}$	${\href{/padicField/29.10.0.1}{10} }^{2}$	${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{5}$	${\href{/padicField/41.5.0.1}{5} }^{4}$	$20$	$20$	${\href{/padicField/53.4.0.1}{4} }^{5}$	${\href{/padicField/59.2.0.1}{2} }^{10}$

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 $20$	$4$	$5$	$15$
$401$ 401	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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