LMFDB - Number field 20.0.7729954898655414581298828125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81)

gp: K = bnfinit(y^20 - 5*y^17 + 25*y^16 - 79*y^15 + 120*y^14 - 130*y^13 + 160*y^12 + 45*y^11 - 39*y^10 - 315*y^9 + 810*y^8 - 1330*y^7 + 1810*y^6 - 1714*y^5 + 1445*y^4 - 1055*y^3 + 550*y^2 - 255*y + 81, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81)

$ x^{20} - 5 x^{17} + 25 x^{16} - 79 x^{15} + 120 x^{14} - 130 x^{13} + 160 x^{12} + 45 x^{11} - 39 x^{10} + \cdots + 81 $ x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81

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:	$7729954898655414581298828125$ 7729954898655414581298828125 $\medspace = 5^{25}\cdot 11^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$24.80$	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:	$5^{13/10}11^{1/2}\approx 26.875549226221146$
Ramified primes:	$5$, $11$ 5, 11	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}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{15}a^{10}+\frac{1}{5}a^{5}+\frac{1}{3}a^{2}+\frac{2}{5}$, $\frac{1}{15}a^{11}+\frac{1}{5}a^{6}+\frac{1}{3}a^{3}+\frac{2}{5}a$, $\frac{1}{15}a^{12}+\frac{1}{5}a^{7}+\frac{1}{3}a^{4}+\frac{2}{5}a^{2}$, $\frac{1}{45}a^{13}-\frac{1}{45}a^{10}-\frac{1}{9}a^{9}-\frac{2}{45}a^{8}-\frac{1}{9}a^{7}-\frac{4}{9}a^{6}-\frac{2}{5}a^{5}+\frac{2}{9}a^{4}-\frac{14}{45}a^{3}+\frac{4}{9}a^{2}-\frac{1}{3}a+\frac{1}{5}$, $\frac{1}{45}a^{14}-\frac{1}{45}a^{11}+\frac{1}{45}a^{10}-\frac{2}{45}a^{9}-\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{2}{5}a^{6}-\frac{17}{45}a^{5}-\frac{14}{45}a^{4}+\frac{4}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{225}a^{15}+\frac{1}{45}a^{12}-\frac{1}{45}a^{11}+\frac{7}{225}a^{10}-\frac{1}{45}a^{9}-\frac{1}{45}a^{8}+\frac{4}{15}a^{7}-\frac{4}{45}a^{6}-\frac{62}{225}a^{5}-\frac{14}{45}a^{4}-\frac{1}{5}a^{3}+\frac{4}{15}a^{2}-\frac{1}{3}a+\frac{6}{25}$, $\frac{1}{2475}a^{16}+\frac{1}{495}a^{15}+\frac{1}{99}a^{14}-\frac{4}{495}a^{13}-\frac{8}{495}a^{12}+\frac{17}{2475}a^{11}-\frac{14}{495}a^{10}+\frac{1}{55}a^{9}-\frac{53}{495}a^{8}-\frac{2}{99}a^{7}-\frac{157}{2475}a^{6}+\frac{109}{495}a^{5}+\frac{101}{495}a^{4}-\frac{163}{495}a^{3}+\frac{203}{495}a^{2}+\frac{21}{275}a-\frac{9}{55}$, $\frac{1}{2475}a^{17}+\frac{4}{495}a^{14}+\frac{1}{495}a^{13}+\frac{52}{2475}a^{12}+\frac{2}{495}a^{11}-\frac{1}{55}a^{10}+\frac{56}{495}a^{9}-\frac{53}{495}a^{8}+\frac{698}{2475}a^{7}-\frac{5}{33}a^{6}+\frac{13}{55}a^{5}-\frac{17}{99}a^{4}+\frac{17}{495}a^{3}-\frac{1201}{2475}a^{2}-\frac{79}{165}a+\frac{23}{55}$, $\frac{1}{423225}a^{18}-\frac{17}{141075}a^{17}+\frac{23}{423225}a^{16}-\frac{173}{423225}a^{15}-\frac{28}{7695}a^{14}+\frac{1537}{423225}a^{13}-\frac{6092}{423225}a^{12}+\frac{12706}{423225}a^{11}-\frac{1222}{141075}a^{10}-\frac{524}{84645}a^{9}-\frac{524}{12825}a^{8}+\frac{127327}{423225}a^{7}+\frac{62108}{141075}a^{6}+\frac{157846}{423225}a^{5}+\frac{16817}{84645}a^{4}-\frac{99521}{423225}a^{3}-\frac{157709}{423225}a^{2}-\frac{54676}{141075}a+\frac{953}{5225}$, $\frac{1}{25816725}a^{19}-\frac{1}{8605575}a^{18}+\frac{2876}{25816725}a^{17}+\frac{17}{271755}a^{16}-\frac{2978}{2346975}a^{15}+\frac{50737}{25816725}a^{14}+\frac{20999}{2346975}a^{13}+\frac{62632}{25816725}a^{12}+\frac{12191}{573705}a^{11}+\frac{328769}{25816725}a^{10}-\frac{11434}{782325}a^{9}+\frac{504571}{25816725}a^{8}-\frac{252443}{2868525}a^{7}+\frac{1344797}{5163345}a^{6}-\frac{10679399}{25816725}a^{5}+\frac{7352284}{25816725}a^{4}+\frac{7579918}{25816725}a^{3}+\frac{458252}{956175}a^{2}+\frac{32374}{573705}a+\frac{46523}{318725}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 + 1/3*a^7 + 1/3*a^6 + 1/3*a^5 + 1/3*a^4 + 1/3*a^3 + 1/3*a^2 + 1/3*a, 1/3*a^9 - 1/3*a, 1/15*a^10 + 1/5*a^5 + 1/3*a^2 + 2/5, 1/15*a^11 + 1/5*a^6 + 1/3*a^3 + 2/5*a, 1/15*a^12 + 1/5*a^7 + 1/3*a^4 + 2/5*a^2, 1/45*a^13 - 1/45*a^10 - 1/9*a^9 - 2/45*a^8 - 1/9*a^7 - 4/9*a^6 - 2/5*a^5 + 2/9*a^4 - 14/45*a^3 + 4/9*a^2 - 1/3*a + 1/5, 1/45*a^14 - 1/45*a^11 + 1/45*a^10 - 2/45*a^9 - 1/9*a^8 - 4/9*a^7 - 2/5*a^6 - 17/45*a^5 - 14/45*a^4 + 4/9*a^3 + 1/3*a^2 + 1/5*a - 1/5, 1/225*a^15 + 1/45*a^12 - 1/45*a^11 + 7/225*a^10 - 1/45*a^9 - 1/45*a^8 + 4/15*a^7 - 4/45*a^6 - 62/225*a^5 - 14/45*a^4 - 1/5*a^3 + 4/15*a^2 - 1/3*a + 6/25, 1/2475*a^16 + 1/495*a^15 + 1/99*a^14 - 4/495*a^13 - 8/495*a^12 + 17/2475*a^11 - 14/495*a^10 + 1/55*a^9 - 53/495*a^8 - 2/99*a^7 - 157/2475*a^6 + 109/495*a^5 + 101/495*a^4 - 163/495*a^3 + 203/495*a^2 + 21/275*a - 9/55, 1/2475*a^17 + 4/495*a^14 + 1/495*a^13 + 52/2475*a^12 + 2/495*a^11 - 1/55*a^10 + 56/495*a^9 - 53/495*a^8 + 698/2475*a^7 - 5/33*a^6 + 13/55*a^5 - 17/99*a^4 + 17/495*a^3 - 1201/2475*a^2 - 79/165*a + 23/55, 1/423225*a^18 - 17/141075*a^17 + 23/423225*a^16 - 173/423225*a^15 - 28/7695*a^14 + 1537/423225*a^13 - 6092/423225*a^12 + 12706/423225*a^11 - 1222/141075*a^10 - 524/84645*a^9 - 524/12825*a^8 + 127327/423225*a^7 + 62108/141075*a^6 + 157846/423225*a^5 + 16817/84645*a^4 - 99521/423225*a^3 - 157709/423225*a^2 - 54676/141075*a + 953/5225, 1/25816725*a^19 - 1/8605575*a^18 + 2876/25816725*a^17 + 17/271755*a^16 - 2978/2346975*a^15 + 50737/25816725*a^14 + 20999/2346975*a^13 + 62632/25816725*a^12 + 12191/573705*a^11 + 328769/25816725*a^10 - 11434/782325*a^9 + 504571/25816725*a^8 - 252443/2868525*a^7 + 1344797/5163345*a^6 - 10679399/25816725*a^5 + 7352284/25816725*a^4 + 7579918/25816725*a^3 + 458252/956175*a^2 + 32374/573705*a + 46523/318725

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

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

Class group and class number

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:	$ -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:	$\frac{85726}{25816725}a^{19}-\frac{5228}{956175}a^{18}-\frac{533866}{25816725}a^{17}-\frac{797759}{25816725}a^{16}+\frac{2824889}{25816725}a^{15}-\frac{7695953}{25816725}a^{14}+\frac{9678553}{25816725}a^{13}+\frac{4838998}{25816725}a^{12}-\frac{92401}{782325}a^{11}+\frac{8040518}{25816725}a^{10}-\frac{5638723}{2868525}a^{9}-\frac{92737643}{25816725}a^{8}+\frac{34784224}{8605575}a^{7}-\frac{44017142}{25816725}a^{6}+\frac{4541242}{25816725}a^{5}-\frac{15279941}{25816725}a^{4}-\frac{104409554}{25816725}a^{3}+\frac{12763882}{8605575}a^{2}-\frac{7490999}{2868525}a+\frac{515396}{318725}$, $\frac{33422}{25816725}a^{19}+\frac{94822}{5163345}a^{18}+\frac{184837}{25816725}a^{17}-\frac{13867}{956175}a^{16}-\frac{190238}{2868525}a^{15}+\frac{2790353}{8605575}a^{14}-\frac{1105079}{1032669}a^{13}+\frac{33634819}{25816725}a^{12}-\frac{22705003}{25816725}a^{11}+\frac{43736821}{25816725}a^{10}+\frac{52967071}{25816725}a^{9}-\frac{6654599}{5163345}a^{8}-\frac{161180719}{25816725}a^{7}+\frac{282537203}{25816725}a^{6}-\frac{330257801}{25816725}a^{5}+\frac{450361873}{25816725}a^{4}-\frac{68391403}{5163345}a^{3}+\frac{17485483}{2346975}a^{2}-\frac{47342522}{8605575}a+\frac{532162}{318725}$, $\frac{888736}{25816725}a^{19}+\frac{79886}{2346975}a^{18}-\frac{72652}{25816725}a^{17}-\frac{146099}{782325}a^{16}+\frac{1198331}{1721115}a^{15}-\frac{160204}{86925}a^{14}+\frac{37153577}{25816725}a^{13}-\frac{14879464}{25816725}a^{12}+\frac{2696944}{1358775}a^{11}+\frac{28200577}{5163345}a^{10}+\frac{34564163}{25816725}a^{9}-\frac{18896963}{1358775}a^{8}+\frac{420497974}{25816725}a^{7}-\frac{392306306}{25816725}a^{6}+\frac{102156589}{5163345}a^{5}-\frac{238413886}{25816725}a^{4}+\frac{154850249}{25816725}a^{3}-\frac{113551133}{25816725}a^{2}-\frac{6077971}{8605575}a-\frac{2289}{5795}$, $\frac{4298}{573705}a^{19}+\frac{20962}{2346975}a^{18}+\frac{566}{344223}a^{17}-\frac{950318}{25816725}a^{16}+\frac{3835823}{25816725}a^{15}-\frac{20051}{54351}a^{14}+\frac{6051029}{25816725}a^{13}-\frac{2422}{54351}a^{12}+\frac{698164}{2346975}a^{11}+\frac{610393}{452925}a^{10}+\frac{216817}{469395}a^{9}-\frac{22327073}{8605575}a^{8}+\frac{18204748}{5163345}a^{7}-\frac{19732148}{8605575}a^{6}+\frac{90253124}{25816725}a^{5}-\frac{8221}{24705}a^{4}+\frac{890578}{2346975}a^{3}+\frac{4729807}{5163345}a^{2}-\frac{7362164}{8605575}a+\frac{363492}{318725}$, $\frac{2897}{271755}a^{19}+\frac{55283}{8605575}a^{18}+\frac{17354}{25816725}a^{17}-\frac{1294364}{25816725}a^{16}+\frac{6298694}{25816725}a^{15}-\frac{3513244}{5163345}a^{14}+\frac{19892833}{25816725}a^{13}-\frac{16386602}{25816725}a^{12}+\frac{885121}{956175}a^{11}+\frac{30099038}{25816725}a^{10}+\frac{495122}{1721115}a^{9}-\frac{92676638}{25816725}a^{8}+\frac{21370738}{2868525}a^{7}-\frac{19062802}{2346975}a^{6}+\frac{261277942}{25816725}a^{5}-\frac{8292338}{1032669}a^{4}+\frac{215389831}{25816725}a^{3}-\frac{15520091}{2868525}a^{2}+\frac{8366741}{2868525}a-\frac{419279}{318725}$, $\frac{202319}{25816725}a^{19}+\frac{45146}{2868525}a^{18}+\frac{235237}{25816725}a^{17}-\frac{1025749}{25816725}a^{16}+\frac{3042454}{25816725}a^{15}-\frac{620717}{2346975}a^{14}-\frac{1800457}{25816725}a^{13}+\frac{3339959}{25816725}a^{12}+\frac{2779394}{8605575}a^{11}+\frac{44735338}{25816725}a^{10}+\frac{1496741}{956175}a^{9}-\frac{5433158}{2346975}a^{8}+\frac{6411047}{8605575}a^{7}+\frac{5841743}{25816725}a^{6}+\frac{31991027}{25816725}a^{5}+\frac{65980136}{25816725}a^{4}-\frac{34438594}{25816725}a^{3}+\frac{954884}{452925}a^{2}-\frac{4056739}{2868525}a+\frac{232526}{318725}$, $\frac{55741}{8605575}a^{19}+\frac{45256}{2346975}a^{18}+\frac{25316}{1721115}a^{17}-\frac{589781}{25816725}a^{16}+\frac{1923857}{25816725}a^{15}-\frac{141791}{1358775}a^{14}-\frac{11488648}{25816725}a^{13}+\frac{129766}{271755}a^{12}-\frac{584702}{2346975}a^{11}+\frac{294089}{150975}a^{10}+\frac{65708569}{25816725}a^{9}-\frac{8165614}{8605575}a^{8}-\frac{85292}{5163345}a^{7}+\frac{25111784}{8605575}a^{6}-\frac{169319989}{25816725}a^{5}+\frac{7947473}{1358775}a^{4}-\frac{130067926}{25816725}a^{3}+\frac{4369283}{1032669}a^{2}-\frac{18918233}{8605575}a+\frac{111373}{318725}$, $\frac{1610569}{25816725}a^{19}+\frac{1930997}{25816725}a^{18}-\frac{11203}{469395}a^{17}-\frac{850571}{2346975}a^{16}+\frac{6242267}{5163345}a^{15}-\frac{6781667}{2346975}a^{14}+\frac{10540213}{8605575}a^{13}+\frac{2298616}{1721115}a^{12}+\frac{4044028}{2346975}a^{11}+\frac{58299601}{5163345}a^{10}+\frac{65482432}{25816725}a^{9}-\frac{751848374}{25816725}a^{8}+\frac{127964173}{5163345}a^{7}-\frac{177079808}{25816725}a^{6}+\frac{2032684}{156465}a^{5}+\frac{4929692}{782325}a^{4}-\frac{10646789}{782325}a^{3}+\frac{41119747}{5163345}a^{2}-\frac{78039098}{8605575}a+\frac{50385}{12749}$, $\frac{58}{18117}a^{19}-\frac{366229}{25816725}a^{18}-\frac{207659}{8605575}a^{17}-\frac{400061}{25816725}a^{16}+\frac{4211567}{25816725}a^{15}-\frac{2541367}{5163345}a^{14}+\frac{22920962}{25816725}a^{13}-\frac{6615619}{25816725}a^{12}-\frac{6181972}{25816725}a^{11}+\frac{92386}{2868525}a^{10}-\frac{16139221}{5163345}a^{9}-\frac{23733784}{8605575}a^{8}+\frac{238924949}{25816725}a^{7}-\frac{62279461}{8605575}a^{6}+\frac{80024471}{25816725}a^{5}-\frac{15746372}{5163345}a^{4}-\frac{63042631}{25816725}a^{3}+\frac{45381422}{25816725}a^{2}-\frac{15244028}{8605575}a+\frac{530373}{318725}$ 85726/25816725a^19 - 5228/956175a^18 - 533866/25816725a^17 - 797759/25816725a^16 + 2824889/25816725a^15 - 7695953/25816725a^14 + 9678553/25816725a^13 + 4838998/25816725a^12 - 92401/782325a^11 + 8040518/25816725a^10 - 5638723/2868525a^9 - 92737643/25816725a^8 + 34784224/8605575a^7 - 44017142/25816725a^6 + 4541242/25816725a^5 - 15279941/25816725a^4 - 104409554/25816725a^3 + 12763882/8605575a^2 - 7490999/2868525a + 515396/318725, 33422/25816725a^19 + 94822/5163345a^18 + 184837/25816725a^17 - 13867/956175a^16 - 190238/2868525a^15 + 2790353/8605575a^14 - 1105079/1032669a^13 + 33634819/25816725a^12 - 22705003/25816725a^11 + 43736821/25816725a^10 + 52967071/25816725a^9 - 6654599/5163345a^8 - 161180719/25816725a^7 + 282537203/25816725a^6 - 330257801/25816725a^5 + 450361873/25816725a^4 - 68391403/5163345a^3 + 17485483/2346975a^2 - 47342522/8605575a + 532162/318725, 888736/25816725a^19 + 79886/2346975a^18 - 72652/25816725a^17 - 146099/782325a^16 + 1198331/1721115a^15 - 160204/86925a^14 + 37153577/25816725a^13 - 14879464/25816725a^12 + 2696944/1358775a^11 + 28200577/5163345a^10 + 34564163/25816725a^9 - 18896963/1358775a^8 + 420497974/25816725a^7 - 392306306/25816725a^6 + 102156589/5163345a^5 - 238413886/25816725a^4 + 154850249/25816725a^3 - 113551133/25816725a^2 - 6077971/8605575a - 2289/5795, 4298/573705a^19 + 20962/2346975a^18 + 566/344223a^17 - 950318/25816725a^16 + 3835823/25816725a^15 - 20051/54351a^14 + 6051029/25816725a^13 - 2422/54351a^12 + 698164/2346975a^11 + 610393/452925a^10 + 216817/469395a^9 - 22327073/8605575a^8 + 18204748/5163345a^7 - 19732148/8605575a^6 + 90253124/25816725a^5 - 8221/24705a^4 + 890578/2346975a^3 + 4729807/5163345a^2 - 7362164/8605575a + 363492/318725, 2897/271755a^19 + 55283/8605575a^18 + 17354/25816725a^17 - 1294364/25816725a^16 + 6298694/25816725a^15 - 3513244/5163345a^14 + 19892833/25816725a^13 - 16386602/25816725a^12 + 885121/956175a^11 + 30099038/25816725a^10 + 495122/1721115a^9 - 92676638/25816725a^8 + 21370738/2868525a^7 - 19062802/2346975a^6 + 261277942/25816725a^5 - 8292338/1032669a^4 + 215389831/25816725a^3 - 15520091/2868525a^2 + 8366741/2868525a - 419279/318725, 202319/25816725a^19 + 45146/2868525a^18 + 235237/25816725a^17 - 1025749/25816725a^16 + 3042454/25816725a^15 - 620717/2346975a^14 - 1800457/25816725a^13 + 3339959/25816725a^12 + 2779394/8605575a^11 + 44735338/25816725a^10 + 1496741/956175a^9 - 5433158/2346975a^8 + 6411047/8605575a^7 + 5841743/25816725a^6 + 31991027/25816725a^5 + 65980136/25816725a^4 - 34438594/25816725a^3 + 954884/452925a^2 - 4056739/2868525a + 232526/318725, 55741/8605575a^19 + 45256/2346975a^18 + 25316/1721115a^17 - 589781/25816725a^16 + 1923857/25816725a^15 - 141791/1358775a^14 - 11488648/25816725a^13 + 129766/271755a^12 - 584702/2346975a^11 + 294089/150975a^10 + 65708569/25816725a^9 - 8165614/8605575a^8 - 85292/5163345a^7 + 25111784/8605575a^6 - 169319989/25816725a^5 + 7947473/1358775a^4 - 130067926/25816725a^3 + 4369283/1032669a^2 - 18918233/8605575a + 111373/318725, 1610569/25816725a^19 + 1930997/25816725a^18 - 11203/469395a^17 - 850571/2346975a^16 + 6242267/5163345a^15 - 6781667/2346975a^14 + 10540213/8605575a^13 + 2298616/1721115a^12 + 4044028/2346975a^11 + 58299601/5163345a^10 + 65482432/25816725a^9 - 751848374/25816725a^8 + 127964173/5163345a^7 - 177079808/25816725a^6 + 2032684/156465a^5 + 4929692/782325a^4 - 10646789/782325a^3 + 41119747/5163345a^2 - 78039098/8605575a + 50385/12749, 58/18117a^19 - 366229/25816725a^18 - 207659/8605575a^17 - 400061/25816725a^16 + 4211567/25816725a^15 - 2541367/5163345a^14 + 22920962/25816725a^13 - 6615619/25816725a^12 - 6181972/25816725a^11 + 92386/2868525a^10 - 16139221/5163345a^9 - 23733784/8605575a^8 + 238924949/25816725a^7 - 62279461/8605575a^6 + 80024471/25816725a^5 - 15746372/5163345a^4 - 63042631/25816725a^3 + 45381422/25816725a^2 - 15244028/8605575*a + 530373/318725 (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:	$ 2264798.97636 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2264798.97636 \cdot 1}{2\cdot\sqrt{7729954898655414581298828125}}\cr\approx \mathstrut & 1.23512203000 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81)

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 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81, 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 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81);

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 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81);

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

$D_{20}$ (as 20T10):

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 13 conjugacy class representatives for $D_{20}$

Character table for $D_{20}$

Intermediate fields

$\Q(\sqrt{-11}) $, 4.0.605.1, 5.1.1890625.1, 10.0.39319091796875.1

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:	20.2.3513615863025188446044921875.1
Minimal sibling:	20.2.3513615863025188446044921875.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$20$	${\href{/padicField/3.2.0.1}{2} }^{9}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$	R	$20$	R	${\href{/padicField/13.4.0.1}{4} }^{5}$	$20$	${\href{/padicField/19.2.0.1}{2} }^{10}$	${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.2.0.1}{2} }^{10}$	${\href{/padicField/31.10.0.1}{10} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{10}$	$20$	${\href{/padicField/47.2.0.1}{2} }^{9}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.5.0.1}{5} }^{4}$

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

# 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	5.10.13.8	$x^{10} + 20 x^{4} + 10$	$10$	$1$	$13$	$D_{10}$	$[3/2]_{2}^{2}$
$5$ 5	5.10.12.7	$x^{10} - 50 x^{8} + 30 x^{7} + 175 x^{6} - 740 x^{5} + 225 x^{4} - 250 x^{3} + 150 x^{2} + 25$	$5$	$2$	$12$	$D_{10}$	$[3/2]_{2}^{2}$
$11$ 11	11.4.2.1	$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	11.4.2.1	$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	11.4.2.1	$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	11.4.2.1	$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	11.4.2.1	$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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