LMFDB - Number field 20.0.1591368644495819328669.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1)

gp: K = bnfinit(y^20 - 6*y^19 + 11*y^18 - 13*y^16 - 19*y^15 + 50*y^14 + 8*y^13 - 42*y^12 - 38*y^11 + 42*y^10 + 31*y^9 + 17*y^8 - 59*y^7 + 6*y^6 + 2*y^5 - y^4 + 15*y^3 - 2*y^2 - 3*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*x^2 - 3*x + 1)

$ x^{20} - 6 x^{19} + 11 x^{18} - 13 x^{16} - 19 x^{15} + 50 x^{14} + 8 x^{13} - 42 x^{12} - 38 x^{11} + \cdots + 1 $ x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*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:	$1591368644495819328669$ 1591368644495819328669 $\medspace = 3^{10}\cdot 1609^{4}\cdot 4021$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$11.48$	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:	$3^{1/2}1609^{1/2}4021^{1/2}\approx 4405.606314685869$
Ramified primes:	$3$, $1609$, $4021$ 3, 1609, 4021	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{4021}) $
$\card{ \Aut(K/\Q) }$:	$2$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{73}a^{18}-\frac{22}{73}a^{17}-\frac{23}{73}a^{16}+\frac{27}{73}a^{15}-\frac{35}{73}a^{14}-\frac{26}{73}a^{13}+\frac{33}{73}a^{12}+\frac{26}{73}a^{11}+\frac{17}{73}a^{10}+\frac{20}{73}a^{9}+\frac{22}{73}a^{8}-\frac{11}{73}a^{7}+\frac{23}{73}a^{6}+\frac{23}{73}a^{5}+\frac{31}{73}a^{4}-\frac{28}{73}a^{3}+\frac{15}{73}a^{2}-\frac{2}{73}a+\frac{7}{73}$, $\frac{1}{14026147}a^{19}+\frac{46492}{14026147}a^{18}+\frac{2911223}{14026147}a^{17}+\frac{2392303}{14026147}a^{16}-\frac{2529864}{14026147}a^{15}+\frac{6752822}{14026147}a^{14}+\frac{5119915}{14026147}a^{13}-\frac{1906743}{14026147}a^{12}-\frac{4495788}{14026147}a^{11}+\frac{2465816}{14026147}a^{10}-\frac{2667284}{14026147}a^{9}+\frac{4850906}{14026147}a^{8}+\frac{267352}{14026147}a^{7}+\frac{5511968}{14026147}a^{6}-\frac{5833173}{14026147}a^{5}+\frac{5215422}{14026147}a^{4}+\frac{2296085}{14026147}a^{3}+\frac{6128470}{14026147}a^{2}-\frac{4364178}{14026147}a+\frac{2440481}{14026147}$ 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, a^17, 1/73*a^18 - 22/73*a^17 - 23/73*a^16 + 27/73*a^15 - 35/73*a^14 - 26/73*a^13 + 33/73*a^12 + 26/73*a^11 + 17/73*a^10 + 20/73*a^9 + 22/73*a^8 - 11/73*a^7 + 23/73*a^6 + 23/73*a^5 + 31/73*a^4 - 28/73*a^3 + 15/73*a^2 - 2/73*a + 7/73, 1/14026147*a^19 + 46492/14026147*a^18 + 2911223/14026147*a^17 + 2392303/14026147*a^16 - 2529864/14026147*a^15 + 6752822/14026147*a^14 + 5119915/14026147*a^13 - 1906743/14026147*a^12 - 4495788/14026147*a^11 + 2465816/14026147*a^10 - 2667284/14026147*a^9 + 4850906/14026147*a^8 + 267352/14026147*a^7 + 5511968/14026147*a^6 - 5833173/14026147*a^5 + 5215422/14026147*a^4 + 2296085/14026147*a^3 + 6128470/14026147*a^2 - 4364178/14026147*a + 2440481/14026147

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

Trivial group, which has order $1$ (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{82}{73} a^{19} - \frac{435}{73} a^{18} + \frac{627}{73} a^{17} + 4 a^{16} - \frac{655}{73} a^{15} - \frac{1940}{73} a^{14} + \frac{2590}{73} a^{13} + \frac{1830}{73} a^{12} - \frac{1410}{73} a^{11} - \frac{3630}{73} a^{10} + \frac{714}{73} a^{9} + \frac{2060}{73} a^{8} + \frac{2887}{73} a^{7} - \frac{2616}{73} a^{6} - \frac{354}{73} a^{5} - \frac{591}{73} a^{4} - \frac{237}{73} a^{3} + \frac{661}{73} a^{2} + \frac{99}{73} a - \frac{53}{73} $ (82)/(73)a^(19) - (435)/(73)a^(18) + (627)/(73)a^(17) + 4a^(16) - (655)/(73)a^(15) - (1940)/(73)a^(14) + (2590)/(73)a^(13) + (1830)/(73)a^(12) - (1410)/(73)a^(11) - (3630)/(73)a^(10) + (714)/(73)a^(9) + (2060)/(73)a^(8) + (2887)/(73)a^(7) - (2616)/(73)a^(6) - (354)/(73)a^(5) - (591)/(73)a^(4) - (237)/(73)a^(3) + (661)/(73)a^(2) + (99)/(73)*a - (53)/(73) (order $6$)	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:	$\frac{6091276}{14026147}a^{19}-\frac{25159907}{14026147}a^{18}+\frac{8997927}{14026147}a^{17}+\frac{75373549}{14026147}a^{16}-\frac{24452770}{14026147}a^{15}-\frac{197087877}{14026147}a^{14}+\frac{26815742}{14026147}a^{13}+\frac{353863516}{14026147}a^{12}+\frac{42091283}{14026147}a^{11}-\frac{380032631}{14026147}a^{10}-\frac{240020198}{14026147}a^{9}+\frac{213591143}{14026147}a^{8}+\frac{356411029}{14026147}a^{7}+\frac{41131374}{14026147}a^{6}-\frac{218996892}{14026147}a^{5}-\frac{39592315}{14026147}a^{4}-\frac{76764613}{14026147}a^{3}+\frac{2797946}{14026147}a^{2}+\frac{44729688}{14026147}a-\frac{2178519}{14026147}$, $\frac{15993462}{14026147}a^{19}-\frac{86558157}{14026147}a^{18}+\frac{1793881}{192139}a^{17}+\frac{49085547}{14026147}a^{16}-\frac{147803598}{14026147}a^{15}-\frac{361361032}{14026147}a^{14}+\frac{565730951}{14026147}a^{13}+\frac{319319854}{14026147}a^{12}-\frac{377890415}{14026147}a^{11}-\frac{688718057}{14026147}a^{10}+\frac{266624319}{14026147}a^{9}+\frac{436144362}{14026147}a^{8}+\frac{487338726}{14026147}a^{7}-\frac{605283641}{14026147}a^{6}-\frac{53236751}{14026147}a^{5}-\frac{72544924}{14026147}a^{4}-\frac{45830047}{14026147}a^{3}+\frac{169248534}{14026147}a^{2}+\frac{32975494}{14026147}a-\frac{27483849}{14026147}$, $\frac{10117353}{14026147}a^{19}-\frac{59841870}{14026147}a^{18}+\frac{100860572}{14026147}a^{17}+\frac{31789519}{14026147}a^{16}-\frac{148147317}{14026147}a^{15}-\frac{241577026}{14026147}a^{14}+\frac{498866903}{14026147}a^{13}+\frac{254328426}{14026147}a^{12}-\frac{458887443}{14026147}a^{11}-\frac{588851910}{14026147}a^{10}+\frac{349870105}{14026147}a^{9}+\frac{542779785}{14026147}a^{8}+\frac{318135101}{14026147}a^{7}-\frac{636538029}{14026147}a^{6}-\frac{192398509}{14026147}a^{5}+\frac{20298702}{14026147}a^{4}+\frac{25260783}{14026147}a^{3}+\frac{178674866}{14026147}a^{2}+\frac{42441970}{14026147}a-\frac{34636200}{14026147}$, $\frac{2390396}{14026147}a^{19}-\frac{3902382}{14026147}a^{18}-\frac{24583339}{14026147}a^{17}+\frac{60647848}{14026147}a^{16}+\frac{19165308}{14026147}a^{15}-\frac{93136311}{14026147}a^{14}-\frac{132323650}{14026147}a^{13}+\frac{230744154}{14026147}a^{12}+\frac{151882287}{14026147}a^{11}-\frac{112622029}{14026147}a^{10}-\frac{292182510}{14026147}a^{9}-\frac{45556}{14026147}a^{8}+\frac{145229935}{14026147}a^{7}+\frac{243607942}{14026147}a^{6}-\frac{99800833}{14026147}a^{5}+\frac{22147078}{14026147}a^{4}-\frac{116981661}{14026147}a^{3}-\frac{64125820}{14026147}a^{2}+\frac{268104}{192139}a-\frac{4070466}{14026147}$, $\frac{7480162}{14026147}a^{19}-\frac{31950590}{14026147}a^{18}+\frac{19264687}{14026147}a^{17}+\frac{72926297}{14026147}a^{16}-\frac{24369358}{14026147}a^{15}-\frac{213704300}{14026147}a^{14}+\frac{51938913}{14026147}a^{13}+\frac{4488287}{192139}a^{12}+\frac{56362181}{14026147}a^{11}-\frac{351304335}{14026147}a^{10}-\frac{229384028}{14026147}a^{9}+\frac{124489390}{14026147}a^{8}+\frac{363094953}{14026147}a^{7}+\frac{63941449}{14026147}a^{6}-\frac{133783885}{14026147}a^{5}-\frac{69624238}{14026147}a^{4}-\frac{103505803}{14026147}a^{3}+\frac{312899}{14026147}a^{2}+\frac{47321056}{14026147}a-\frac{400728}{14026147}$, $\frac{9878730}{14026147}a^{19}-\frac{50381715}{14026147}a^{18}+\frac{62946208}{14026147}a^{17}+\frac{58588940}{14026147}a^{16}-\frac{79454051}{14026147}a^{15}-\frac{255143887}{14026147}a^{14}+\frac{259073755}{14026147}a^{13}+\frac{320129432}{14026147}a^{12}-\frac{136365688}{14026147}a^{11}-\frac{484222371}{14026147}a^{10}-\frac{37596529}{14026147}a^{9}+\frac{286478780}{14026147}a^{8}+\frac{411859085}{14026147}a^{7}-\frac{203267901}{14026147}a^{6}-\frac{140938271}{14026147}a^{5}-\frac{64037649}{14026147}a^{4}-\frac{91289620}{14026147}a^{3}+\frac{87074128}{14026147}a^{2}+\frac{18675767}{14026147}a-\frac{4120106}{14026147}$, $\frac{955036}{14026147}a^{19}-\frac{9474748}{14026147}a^{18}+\frac{26670641}{14026147}a^{17}-\frac{12609911}{14026147}a^{16}-\frac{39165009}{14026147}a^{15}-\frac{12567301}{14026147}a^{14}+\frac{143913631}{14026147}a^{13}-\frac{20826476}{14026147}a^{12}-\frac{163355265}{14026147}a^{11}-\frac{76813469}{14026147}a^{10}+\frac{188400814}{14026147}a^{9}+\frac{127010033}{14026147}a^{8}-\frac{24828413}{14026147}a^{7}-\frac{220765085}{14026147}a^{6}+\frac{1790780}{14026147}a^{5}+\frac{29795959}{14026147}a^{4}+\frac{44238969}{14026147}a^{3}+\frac{43501184}{14026147}a^{2}-\frac{12960654}{14026147}a-\frac{98960}{192139}$, $\frac{21634329}{14026147}a^{19}-\frac{104755234}{14026147}a^{18}+\frac{120005737}{14026147}a^{17}+\frac{120749972}{14026147}a^{16}-\frac{109911452}{14026147}a^{15}-\frac{545830644}{14026147}a^{14}+\frac{438776760}{14026147}a^{13}+\frac{623179172}{14026147}a^{12}-\frac{84225939}{14026147}a^{11}-\frac{940255405}{14026147}a^{10}-\frac{2493635}{192139}a^{9}+\frac{399772380}{14026147}a^{8}+\frac{868911354}{14026147}a^{7}-\frac{318085502}{14026147}a^{6}-\frac{134533837}{14026147}a^{5}-\frac{198062314}{14026147}a^{4}-\frac{147321640}{14026147}a^{3}+\frac{70422303}{14026147}a^{2}+\frac{35701495}{14026147}a-\frac{12823039}{14026147}$, $\frac{6804340}{14026147}a^{19}-\frac{37732653}{14026147}a^{18}+\frac{59069995}{14026147}a^{17}+\frac{20382294}{14026147}a^{16}-\frac{70674087}{14026147}a^{15}-\frac{160833164}{14026147}a^{14}+\frac{270142724}{14026147}a^{13}+\frac{147078598}{14026147}a^{12}-\frac{194358734}{14026147}a^{11}-\frac{341916758}{14026147}a^{10}+\frac{149627745}{14026147}a^{9}+\frac{242483102}{14026147}a^{8}+\frac{227601461}{14026147}a^{7}-\frac{312423623}{14026147}a^{6}-\frac{60505872}{14026147}a^{5}-\frac{36735629}{14026147}a^{4}+\frac{5409963}{14026147}a^{3}+\frac{78093580}{14026147}a^{2}+\frac{30677903}{14026147}a-\frac{24769813}{14026147}$ 6091276/14026147a^19 - 25159907/14026147a^18 + 8997927/14026147a^17 + 75373549/14026147a^16 - 24452770/14026147a^15 - 197087877/14026147a^14 + 26815742/14026147a^13 + 353863516/14026147a^12 + 42091283/14026147a^11 - 380032631/14026147a^10 - 240020198/14026147a^9 + 213591143/14026147a^8 + 356411029/14026147a^7 + 41131374/14026147a^6 - 218996892/14026147a^5 - 39592315/14026147a^4 - 76764613/14026147a^3 + 2797946/14026147a^2 + 44729688/14026147a - 2178519/14026147, 15993462/14026147a^19 - 86558157/14026147a^18 + 1793881/192139a^17 + 49085547/14026147a^16 - 147803598/14026147a^15 - 361361032/14026147a^14 + 565730951/14026147a^13 + 319319854/14026147a^12 - 377890415/14026147a^11 - 688718057/14026147a^10 + 266624319/14026147a^9 + 436144362/14026147a^8 + 487338726/14026147a^7 - 605283641/14026147a^6 - 53236751/14026147a^5 - 72544924/14026147a^4 - 45830047/14026147a^3 + 169248534/14026147a^2 + 32975494/14026147a - 27483849/14026147, 10117353/14026147a^19 - 59841870/14026147a^18 + 100860572/14026147a^17 + 31789519/14026147a^16 - 148147317/14026147a^15 - 241577026/14026147a^14 + 498866903/14026147a^13 + 254328426/14026147a^12 - 458887443/14026147a^11 - 588851910/14026147a^10 + 349870105/14026147a^9 + 542779785/14026147a^8 + 318135101/14026147a^7 - 636538029/14026147a^6 - 192398509/14026147a^5 + 20298702/14026147a^4 + 25260783/14026147a^3 + 178674866/14026147a^2 + 42441970/14026147a - 34636200/14026147, 2390396/14026147a^19 - 3902382/14026147a^18 - 24583339/14026147a^17 + 60647848/14026147a^16 + 19165308/14026147a^15 - 93136311/14026147a^14 - 132323650/14026147a^13 + 230744154/14026147a^12 + 151882287/14026147a^11 - 112622029/14026147a^10 - 292182510/14026147a^9 - 45556/14026147a^8 + 145229935/14026147a^7 + 243607942/14026147a^6 - 99800833/14026147a^5 + 22147078/14026147a^4 - 116981661/14026147a^3 - 64125820/14026147a^2 + 268104/192139a - 4070466/14026147, 7480162/14026147a^19 - 31950590/14026147a^18 + 19264687/14026147a^17 + 72926297/14026147a^16 - 24369358/14026147a^15 - 213704300/14026147a^14 + 51938913/14026147a^13 + 4488287/192139a^12 + 56362181/14026147a^11 - 351304335/14026147a^10 - 229384028/14026147a^9 + 124489390/14026147a^8 + 363094953/14026147a^7 + 63941449/14026147a^6 - 133783885/14026147a^5 - 69624238/14026147a^4 - 103505803/14026147a^3 + 312899/14026147a^2 + 47321056/14026147a - 400728/14026147, 9878730/14026147a^19 - 50381715/14026147a^18 + 62946208/14026147a^17 + 58588940/14026147a^16 - 79454051/14026147a^15 - 255143887/14026147a^14 + 259073755/14026147a^13 + 320129432/14026147a^12 - 136365688/14026147a^11 - 484222371/14026147a^10 - 37596529/14026147a^9 + 286478780/14026147a^8 + 411859085/14026147a^7 - 203267901/14026147a^6 - 140938271/14026147a^5 - 64037649/14026147a^4 - 91289620/14026147a^3 + 87074128/14026147a^2 + 18675767/14026147a - 4120106/14026147, 955036/14026147a^19 - 9474748/14026147a^18 + 26670641/14026147a^17 - 12609911/14026147a^16 - 39165009/14026147a^15 - 12567301/14026147a^14 + 143913631/14026147a^13 - 20826476/14026147a^12 - 163355265/14026147a^11 - 76813469/14026147a^10 + 188400814/14026147a^9 + 127010033/14026147a^8 - 24828413/14026147a^7 - 220765085/14026147a^6 + 1790780/14026147a^5 + 29795959/14026147a^4 + 44238969/14026147a^3 + 43501184/14026147a^2 - 12960654/14026147a - 98960/192139, 21634329/14026147a^19 - 104755234/14026147a^18 + 120005737/14026147a^17 + 120749972/14026147a^16 - 109911452/14026147a^15 - 545830644/14026147a^14 + 438776760/14026147a^13 + 623179172/14026147a^12 - 84225939/14026147a^11 - 940255405/14026147a^10 - 2493635/192139a^9 + 399772380/14026147a^8 + 868911354/14026147a^7 - 318085502/14026147a^6 - 134533837/14026147a^5 - 198062314/14026147a^4 - 147321640/14026147a^3 + 70422303/14026147a^2 + 35701495/14026147a - 12823039/14026147, 6804340/14026147a^19 - 37732653/14026147a^18 + 59069995/14026147a^17 + 20382294/14026147a^16 - 70674087/14026147a^15 - 160833164/14026147a^14 + 270142724/14026147a^13 + 147078598/14026147a^12 - 194358734/14026147a^11 - 341916758/14026147a^10 + 149627745/14026147a^9 + 242483102/14026147a^8 + 227601461/14026147a^7 - 312423623/14026147a^6 - 60505872/14026147a^5 - 36735629/14026147a^4 + 5409963/14026147a^3 + 78093580/14026147a^2 + 30677903/14026147*a - 24769813/14026147 (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:	$ 427.42684139 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 427.42684139 \cdot 1}{6\cdot\sqrt{1591368644495819328669}}\cr\approx \mathstrut & 0.17124733681 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*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 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*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 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*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 - 6*x^19 + 11*x^18 - 13*x^16 - 19*x^15 + 50*x^14 + 8*x^13 - 42*x^12 - 38*x^11 + 42*x^10 + 31*x^9 + 17*x^8 - 59*x^7 + 6*x^6 + 2*x^5 - x^4 + 15*x^3 - 2*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

$C_2^8.(D_4\times S_5)$ (as 20T887):

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 non-solvable group of order 245760

The 201 conjugacy class representatives for $C_2^8.(D_4\times S_5)$

Character table for $C_2^8.(D_4\times S_5)$

Intermediate fields

$\Q(\sqrt{-3}) $, 5.1.1609.1, 10.0.629098083.1

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

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

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

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

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

Sibling fields

Degree 20 siblings:	data not computed
Degree 40 siblings:	data not computed
Minimal sibling:	20.2.26332894319002837533243.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$20$	R	${\href{/padicField/5.10.0.1}{10} }^{2}$	${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$	${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$	${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$	${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$	${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$	${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	$20$	${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.4.0.1}{4} }^{5}$	${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$	${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
$3$ 3	3.20.10.1	$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$1609$ 1609	$\Q_{1609}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{1609}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{1609}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{1609}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $4$	$2$	$2$	$2$
$4021$ 4021		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$

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