LMFDB - Number field 20.0.79836369265591620072268522842253.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225)

gp: K = bnfinit(y^20 - 4*y^19 + 19*y^18 - 57*y^17 + 133*y^16 - 190*y^15 + 171*y^14 + 532*y^13 - 1330*y^12 + 3648*y^11 - 3325*y^10 + 2812*y^9 + 3534*y^8 - 9880*y^7 + 14212*y^6 - 15447*y^5 + 10906*y^4 - 6574*y^3 + 4579*y^2 - 840*y + 225, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225)

$ x^{20} - 4 x^{19} + 19 x^{18} - 57 x^{17} + 133 x^{16} - 190 x^{15} + 171 x^{14} + 532 x^{13} + \cdots + 225 $ x^20 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225

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:	$79836369265591620072268522842253$ 79836369265591620072268522842253 $\medspace = 7^{9}\cdot 19^{19}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$39.36$	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:	$7^{1/2}19^{19/20}\approx 43.38753139294459$
Ramified primes:	$7$, $19$ 7, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{133}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{8}-\frac{2}{9}a^{7}+\frac{1}{9}a^{6}+\frac{1}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{2}-\frac{2}{9}a+\frac{1}{3}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{2}{9}a^{6}-\frac{4}{9}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{45}a^{13}-\frac{1}{45}a^{12}-\frac{1}{45}a^{11}-\frac{4}{45}a^{10}+\frac{1}{45}a^{9}+\frac{1}{15}a^{8}-\frac{4}{15}a^{7}-\frac{1}{15}a^{6}+\frac{11}{45}a^{5}-\frac{11}{45}a^{4}+\frac{19}{45}a^{3}-\frac{14}{45}a^{2}+\frac{11}{45}a+\frac{1}{3}$, $\frac{1}{405}a^{14}+\frac{2}{405}a^{13}+\frac{16}{405}a^{12}+\frac{1}{135}a^{11}-\frac{56}{405}a^{10}-\frac{4}{405}a^{9}+\frac{32}{405}a^{8}+\frac{56}{405}a^{7}-\frac{83}{405}a^{6}-\frac{31}{135}a^{5}-\frac{29}{405}a^{4}+\frac{113}{405}a^{3}+\frac{4}{405}a^{2}-\frac{19}{135}a+\frac{2}{9}$, $\frac{1}{405}a^{15}+\frac{1}{135}a^{13}-\frac{4}{81}a^{12}-\frac{8}{405}a^{11}-\frac{4}{45}a^{10}+\frac{31}{405}a^{9}+\frac{2}{81}a^{8}-\frac{59}{135}a^{7}+\frac{29}{81}a^{6}+\frac{103}{405}a^{5}+\frac{1}{9}a^{4}+\frac{4}{135}a^{3}+\frac{151}{405}a^{2}+\frac{1}{27}a-\frac{4}{9}$, $\frac{1}{3645}a^{16}-\frac{4}{3645}a^{15}-\frac{4}{3645}a^{14}-\frac{19}{3645}a^{13}+\frac{113}{3645}a^{12}-\frac{7}{3645}a^{11}+\frac{26}{405}a^{10}-\frac{149}{3645}a^{9}-\frac{5}{81}a^{8}+\frac{1352}{3645}a^{7}+\frac{293}{3645}a^{6}-\frac{634}{3645}a^{5}-\frac{712}{3645}a^{4}-\frac{260}{729}a^{3}-\frac{343}{729}a^{2}-\frac{448}{1215}a+\frac{35}{81}$, $\frac{1}{18225}a^{17}+\frac{1}{18225}a^{16}-\frac{1}{1215}a^{15}-\frac{7}{6075}a^{14}-\frac{2}{225}a^{13}+\frac{11}{2025}a^{12}+\frac{829}{18225}a^{11}+\frac{2281}{18225}a^{10}+\frac{1829}{18225}a^{9}+\frac{2594}{18225}a^{8}-\frac{2542}{6075}a^{7}-\frac{758}{6075}a^{6}+\frac{1616}{6075}a^{5}+\frac{104}{2025}a^{4}-\frac{1004}{3645}a^{3}-\frac{3061}{18225}a^{2}+\frac{1822}{6075}a-\frac{14}{405}$, $\frac{1}{3444525}a^{18}+\frac{2}{137781}a^{17}-\frac{32}{1148175}a^{16}-\frac{1406}{3444525}a^{15}-\frac{1166}{3444525}a^{14}-\frac{26069}{3444525}a^{13}+\frac{29644}{688905}a^{12}+\frac{11818}{382725}a^{11}-\frac{12248}{382725}a^{10}-\frac{9832}{229635}a^{9}-\frac{112247}{688905}a^{8}-\frac{15938}{3444525}a^{7}-\frac{225044}{492075}a^{6}+\frac{1208308}{3444525}a^{5}+\frac{328028}{1148175}a^{4}+\frac{208262}{492075}a^{3}-\frac{233873}{3444525}a^{2}-\frac{352012}{1148175}a+\frac{1559}{76545}$, $\frac{1}{4837680358875}a^{19}+\frac{97891}{691097194125}a^{18}+\frac{114807841}{4837680358875}a^{17}-\frac{568818356}{4837680358875}a^{16}-\frac{146154166}{1612560119625}a^{15}+\frac{15889834}{19908149625}a^{14}+\frac{2357983139}{284569432875}a^{13}+\frac{7665323546}{967536071775}a^{12}-\frac{564659}{131262525}a^{11}+\frac{158815591006}{1612560119625}a^{10}+\frac{49001051021}{372129258375}a^{9}-\frac{37062663983}{967536071775}a^{8}-\frac{25517744992}{1612560119625}a^{7}-\frac{223976202127}{1612560119625}a^{6}-\frac{160168128922}{691097194125}a^{5}+\frac{2386723134464}{4837680358875}a^{4}-\frac{5024402507}{74425851675}a^{3}+\frac{66038207548}{284569432875}a^{2}+\frac{23062021768}{64502404785}a+\frac{1922768231}{21500801595}$ 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, 1/3*a^10 - 1/3*a^2, 1/9*a^11 - 1/9*a^10 + 1/9*a^8 - 2/9*a^7 + 1/9*a^6 + 1/9*a^5 + 4/9*a^4 - 1/9*a^2 - 2/9*a + 1/3, 1/9*a^12 - 1/9*a^10 + 1/9*a^9 - 1/9*a^8 - 1/9*a^7 + 2/9*a^6 - 4/9*a^5 + 4/9*a^4 - 1/9*a^3 - 1/3*a^2 + 1/9*a + 1/3, 1/45*a^13 - 1/45*a^12 - 1/45*a^11 - 4/45*a^10 + 1/45*a^9 + 1/15*a^8 - 4/15*a^7 - 1/15*a^6 + 11/45*a^5 - 11/45*a^4 + 19/45*a^3 - 14/45*a^2 + 11/45*a + 1/3, 1/405*a^14 + 2/405*a^13 + 16/405*a^12 + 1/135*a^11 - 56/405*a^10 - 4/405*a^9 + 32/405*a^8 + 56/405*a^7 - 83/405*a^6 - 31/135*a^5 - 29/405*a^4 + 113/405*a^3 + 4/405*a^2 - 19/135*a + 2/9, 1/405*a^15 + 1/135*a^13 - 4/81*a^12 - 8/405*a^11 - 4/45*a^10 + 31/405*a^9 + 2/81*a^8 - 59/135*a^7 + 29/81*a^6 + 103/405*a^5 + 1/9*a^4 + 4/135*a^3 + 151/405*a^2 + 1/27*a - 4/9, 1/3645*a^16 - 4/3645*a^15 - 4/3645*a^14 - 19/3645*a^13 + 113/3645*a^12 - 7/3645*a^11 + 26/405*a^10 - 149/3645*a^9 - 5/81*a^8 + 1352/3645*a^7 + 293/3645*a^6 - 634/3645*a^5 - 712/3645*a^4 - 260/729*a^3 - 343/729*a^2 - 448/1215*a + 35/81, 1/18225*a^17 + 1/18225*a^16 - 1/1215*a^15 - 7/6075*a^14 - 2/225*a^13 + 11/2025*a^12 + 829/18225*a^11 + 2281/18225*a^10 + 1829/18225*a^9 + 2594/18225*a^8 - 2542/6075*a^7 - 758/6075*a^6 + 1616/6075*a^5 + 104/2025*a^4 - 1004/3645*a^3 - 3061/18225*a^2 + 1822/6075*a - 14/405, 1/3444525*a^18 + 2/137781*a^17 - 32/1148175*a^16 - 1406/3444525*a^15 - 1166/3444525*a^14 - 26069/3444525*a^13 + 29644/688905*a^12 + 11818/382725*a^11 - 12248/382725*a^10 - 9832/229635*a^9 - 112247/688905*a^8 - 15938/3444525*a^7 - 225044/492075*a^6 + 1208308/3444525*a^5 + 328028/1148175*a^4 + 208262/492075*a^3 - 233873/3444525*a^2 - 352012/1148175*a + 1559/76545, 1/4837680358875*a^19 + 97891/691097194125*a^18 + 114807841/4837680358875*a^17 - 568818356/4837680358875*a^16 - 146154166/1612560119625*a^15 + 15889834/19908149625*a^14 + 2357983139/284569432875*a^13 + 7665323546/967536071775*a^12 - 564659/131262525*a^11 + 158815591006/1612560119625*a^10 + 49001051021/372129258375*a^9 - 37062663983/967536071775*a^8 - 25517744992/1612560119625*a^7 - 223976202127/1612560119625*a^6 - 160168128922/691097194125*a^5 + 2386723134464/4837680358875*a^4 - 5024402507/74425851675*a^3 + 66038207548/284569432875*a^2 + 23062021768/64502404785*a + 1922768231/21500801595

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:	$\frac{1752618533}{4837680358875}a^{19}-\frac{5702126539}{4837680358875}a^{18}+\frac{20899938308}{4837680358875}a^{17}-\frac{55628022508}{4837680358875}a^{16}+\frac{25272220292}{1612560119625}a^{15}+\frac{11402979592}{1612560119625}a^{14}-\frac{1697812814}{40652776125}a^{13}+\frac{177008910277}{967536071775}a^{12}-\frac{334554002}{2756513025}a^{11}-\frac{110484184777}{1612560119625}a^{10}+\frac{209543849008}{372129258375}a^{9}-\frac{247500669818}{193507214355}a^{8}+\frac{178759969664}{1612560119625}a^{7}+\frac{580823227979}{1612560119625}a^{6}-\frac{15357768298997}{4837680358875}a^{5}+\frac{13856257866352}{4837680358875}a^{4}-\frac{157665996682}{74425851675}a^{3}+\frac{226486799009}{284569432875}a^{2}-\frac{125679286799}{322512023925}a+\frac{1447247251}{21500801595}$, $\frac{102265876}{967536071775}a^{19}-\frac{12779027}{38701442871}a^{18}+\frac{40469986}{46073146275}a^{17}-\frac{7874948644}{967536071775}a^{16}+\frac{27902615651}{967536071775}a^{15}-\frac{11683594543}{138219438825}a^{14}+\frac{1734869476}{11382777315}a^{13}-\frac{13563874162}{46073146275}a^{12}+\frac{244616171}{8269539075}a^{11}-\frac{2916595778}{64502404785}a^{10}-\frac{21305289706}{14885170335}a^{9}+\frac{158724490304}{138219438825}a^{8}-\frac{2678945932462}{967536071775}a^{7}+\frac{1106970416072}{967536071775}a^{6}-\frac{328318354393}{322512023925}a^{5}+\frac{941024539246}{967536071775}a^{4}-\frac{13084568074}{74425851675}a^{3}+\frac{26825366521}{18971295525}a^{2}-\frac{14172065134}{21500801595}a-\frac{5707901}{1433386773}$, $\frac{25378298453}{4837680358875}a^{19}-\frac{58315625309}{4837680358875}a^{18}+\frac{4983577811}{76788577125}a^{17}-\frac{644606575303}{4837680358875}a^{16}+\frac{1006766286026}{4837680358875}a^{15}+\frac{90253118873}{691097194125}a^{14}-\frac{185177222563}{284569432875}a^{13}+\frac{187739824634}{46073146275}a^{12}-\frac{16119143072}{8269539075}a^{11}+\frac{12386491367738}{1612560119625}a^{10}+\frac{5099944220323}{372129258375}a^{9}-\frac{1460695468064}{138219438825}a^{8}+\frac{187154094347912}{4837680358875}a^{7}-\frac{74143935290938}{4837680358875}a^{6}-\frac{5888057591618}{537520039875}a^{5}+\frac{163240334163757}{4837680358875}a^{4}-\frac{4515060199313}{74425851675}a^{3}+\frac{1285724247421}{31618825875}a^{2}-\frac{218104962032}{11944889775}a+\frac{190869728177}{7166933865}$, $\frac{13292134187}{4837680358875}a^{19}-\frac{51030322606}{4837680358875}a^{18}+\frac{246002403422}{4837680358875}a^{17}-\frac{731599325692}{4837680358875}a^{16}+\frac{189768160261}{537520039875}a^{15}-\frac{822662421862}{1612560119625}a^{14}+\frac{145824814963}{284569432875}a^{13}+\frac{1246878876391}{967536071775}a^{12}-\frac{2817626618}{918837675}a^{11}+\frac{14830653207107}{1612560119625}a^{10}-\frac{2986361353883}{372129258375}a^{9}+\frac{7555357655498}{967536071775}a^{8}+\frac{582182888491}{76788577125}a^{7}-\frac{11866315951538}{537520039875}a^{6}+\frac{151702028008792}{4837680358875}a^{5}-\frac{24022510170536}{691097194125}a^{4}+\frac{1763475359981}{74425851675}a^{3}-\frac{3875745754939}{284569432875}a^{2}+\frac{3486515234737}{322512023925}a-\frac{4222467281}{3071543085}$, $\frac{125131642}{94856477625}a^{19}+\frac{600666151}{94856477625}a^{18}-\frac{901768004}{31618825875}a^{17}+\frac{8674543382}{94856477625}a^{16}-\frac{20768550289}{94856477625}a^{15}+\frac{31983836681}{94856477625}a^{14}-\frac{28273780736}{94856477625}a^{13}-\frac{4522369463}{6323765175}a^{12}+\frac{134107328}{54049275}a^{11}-\frac{183480293002}{31618825875}a^{10}+\frac{49489091998}{7296652125}a^{9}-\frac{76018852046}{18971295525}a^{8}-\frac{74447252419}{13550925375}a^{7}+\frac{1795517568722}{94856477625}a^{6}-\frac{825127045504}{31618825875}a^{5}+\frac{349818660931}{13550925375}a^{4}-\frac{23620397918}{1459330425}a^{3}+\frac{206997046526}{31618825875}a^{2}-\frac{3538990471}{2107921725}a+\frac{893239}{20075445}$, $\frac{59368864}{46073146275}a^{19}+\frac{1062569168}{193507214355}a^{18}-\frac{24925994869}{967536071775}a^{17}+\frac{8543268854}{107504007975}a^{16}-\frac{182920351676}{967536071775}a^{15}+\frac{275181493429}{967536071775}a^{14}-\frac{15496949347}{56913886575}a^{13}-\frac{125086471112}{193507214355}a^{12}+\frac{15767848108}{8269539075}a^{11}-\frac{183239889109}{35834669325}a^{10}+\frac{132871221482}{24808617225}a^{9}-\frac{172549605304}{38701442871}a^{8}-\frac{786703315423}{193507214355}a^{7}+\frac{387726074303}{27643887765}a^{6}-\frac{4184744269753}{193507214355}a^{5}+\frac{841241493908}{35834669325}a^{4}-\frac{191532743143}{10632264525}a^{3}+\frac{595440123011}{56913886575}a^{2}-\frac{2196772371076}{322512023925}a+\frac{44483853377}{21500801595}$, $\frac{19148254}{13110244875}a^{19}+\frac{77661697}{13110244875}a^{18}-\frac{124388338}{4370081625}a^{17}+\frac{1125974879}{13110244875}a^{16}-\frac{2663985328}{13110244875}a^{15}+\frac{3910191032}{13110244875}a^{14}-\frac{214245721}{771190875}a^{13}-\frac{678120383}{874016325}a^{12}+\frac{135916616}{67232025}a^{11}-\frac{8148778843}{1456693875}a^{10}+\frac{5122250941}{1008480375}a^{9}-\frac{11143869083}{2622048975}a^{8}-\frac{10598010133}{1872892125}a^{7}+\frac{197164494614}{13110244875}a^{6}-\frac{91513491733}{4370081625}a^{5}+\frac{41335171132}{1872892125}a^{4}-\frac{117109709}{8067843}a^{3}+\frac{2267178101}{257063625}a^{2}-\frac{1976713286}{291338775}a+\frac{759709}{924885}$, $\frac{15431854}{15357715425}a^{19}+\frac{4979513054}{967536071775}a^{18}-\frac{21871074122}{967536071775}a^{17}+\frac{24383785973}{322512023925}a^{16}-\frac{175427448772}{967536071775}a^{15}+\frac{289091489408}{967536071775}a^{14}-\frac{16462857464}{56913886575}a^{13}-\frac{17126534636}{38701442871}a^{12}+\frac{16493105948}{8269539075}a^{11}-\frac{483007032778}{107504007975}a^{10}+\frac{163360002206}{24808617225}a^{9}-\frac{126813783623}{38701442871}a^{8}-\frac{772920542977}{967536071775}a^{7}+\frac{2331437952134}{138219438825}a^{6}-\frac{17492690531098}{967536071775}a^{5}+\frac{8781180840598}{322512023925}a^{4}-\frac{158576830973}{10632264525}a^{3}+\frac{630868411672}{56913886575}a^{2}-\frac{1633582828466}{322512023925}a+\frac{27463497457}{21500801595}$, $\frac{5344863577}{1612560119625}a^{19}+\frac{10811078509}{691097194125}a^{18}-\frac{347834098666}{4837680358875}a^{17}+\frac{124074094409}{537520039875}a^{16}-\frac{2712770218322}{4837680358875}a^{15}+\frac{4316849366038}{4837680358875}a^{14}-\frac{249433382434}{284569432875}a^{13}-\frac{1563386938052}{967536071775}a^{12}+\frac{7078089518}{1181362725}a^{11}-\frac{2705194351949}{179173346625}a^{10}+\frac{2272147070813}{124043086125}a^{9}-\frac{12823626364658}{967536071775}a^{8}-\frac{53317228960004}{4837680358875}a^{7}+\frac{228094372725211}{4837680358875}a^{6}-\frac{48621116430158}{691097194125}a^{5}+\frac{13252812570668}{179173346625}a^{4}-\frac{3712552880942}{74425851675}a^{3}+\frac{5820458531807}{284569432875}a^{2}-\frac{2710919941157}{322512023925}a+\frac{43840818208}{21500801595}$ 1752618533/4837680358875a^19 - 5702126539/4837680358875a^18 + 20899938308/4837680358875a^17 - 55628022508/4837680358875a^16 + 25272220292/1612560119625a^15 + 11402979592/1612560119625a^14 - 1697812814/40652776125a^13 + 177008910277/967536071775a^12 - 334554002/2756513025a^11 - 110484184777/1612560119625a^10 + 209543849008/372129258375a^9 - 247500669818/193507214355a^8 + 178759969664/1612560119625a^7 + 580823227979/1612560119625a^6 - 15357768298997/4837680358875a^5 + 13856257866352/4837680358875a^4 - 157665996682/74425851675a^3 + 226486799009/284569432875a^2 - 125679286799/322512023925a + 1447247251/21500801595, -102265876/967536071775a^19 - 12779027/38701442871a^18 + 40469986/46073146275a^17 - 7874948644/967536071775a^16 + 27902615651/967536071775a^15 - 11683594543/138219438825a^14 + 1734869476/11382777315a^13 - 13563874162/46073146275a^12 + 244616171/8269539075a^11 - 2916595778/64502404785a^10 - 21305289706/14885170335a^9 + 158724490304/138219438825a^8 - 2678945932462/967536071775a^7 + 1106970416072/967536071775a^6 - 328318354393/322512023925a^5 + 941024539246/967536071775a^4 - 13084568074/74425851675a^3 + 26825366521/18971295525a^2 - 14172065134/21500801595a - 5707901/1433386773, 25378298453/4837680358875a^19 - 58315625309/4837680358875a^18 + 4983577811/76788577125a^17 - 644606575303/4837680358875a^16 + 1006766286026/4837680358875a^15 + 90253118873/691097194125a^14 - 185177222563/284569432875a^13 + 187739824634/46073146275a^12 - 16119143072/8269539075a^11 + 12386491367738/1612560119625a^10 + 5099944220323/372129258375a^9 - 1460695468064/138219438825a^8 + 187154094347912/4837680358875a^7 - 74143935290938/4837680358875a^6 - 5888057591618/537520039875a^5 + 163240334163757/4837680358875a^4 - 4515060199313/74425851675a^3 + 1285724247421/31618825875a^2 - 218104962032/11944889775a + 190869728177/7166933865, 13292134187/4837680358875a^19 - 51030322606/4837680358875a^18 + 246002403422/4837680358875a^17 - 731599325692/4837680358875a^16 + 189768160261/537520039875a^15 - 822662421862/1612560119625a^14 + 145824814963/284569432875a^13 + 1246878876391/967536071775a^12 - 2817626618/918837675a^11 + 14830653207107/1612560119625a^10 - 2986361353883/372129258375a^9 + 7555357655498/967536071775a^8 + 582182888491/76788577125a^7 - 11866315951538/537520039875a^6 + 151702028008792/4837680358875a^5 - 24022510170536/691097194125a^4 + 1763475359981/74425851675a^3 - 3875745754939/284569432875a^2 + 3486515234737/322512023925a - 4222467281/3071543085, -125131642/94856477625a^19 + 600666151/94856477625a^18 - 901768004/31618825875a^17 + 8674543382/94856477625a^16 - 20768550289/94856477625a^15 + 31983836681/94856477625a^14 - 28273780736/94856477625a^13 - 4522369463/6323765175a^12 + 134107328/54049275a^11 - 183480293002/31618825875a^10 + 49489091998/7296652125a^9 - 76018852046/18971295525a^8 - 74447252419/13550925375a^7 + 1795517568722/94856477625a^6 - 825127045504/31618825875a^5 + 349818660931/13550925375a^4 - 23620397918/1459330425a^3 + 206997046526/31618825875a^2 - 3538990471/2107921725a + 893239/20075445, -59368864/46073146275a^19 + 1062569168/193507214355a^18 - 24925994869/967536071775a^17 + 8543268854/107504007975a^16 - 182920351676/967536071775a^15 + 275181493429/967536071775a^14 - 15496949347/56913886575a^13 - 125086471112/193507214355a^12 + 15767848108/8269539075a^11 - 183239889109/35834669325a^10 + 132871221482/24808617225a^9 - 172549605304/38701442871a^8 - 786703315423/193507214355a^7 + 387726074303/27643887765a^6 - 4184744269753/193507214355a^5 + 841241493908/35834669325a^4 - 191532743143/10632264525a^3 + 595440123011/56913886575a^2 - 2196772371076/322512023925a + 44483853377/21500801595, -19148254/13110244875a^19 + 77661697/13110244875a^18 - 124388338/4370081625a^17 + 1125974879/13110244875a^16 - 2663985328/13110244875a^15 + 3910191032/13110244875a^14 - 214245721/771190875a^13 - 678120383/874016325a^12 + 135916616/67232025a^11 - 8148778843/1456693875a^10 + 5122250941/1008480375a^9 - 11143869083/2622048975a^8 - 10598010133/1872892125a^7 + 197164494614/13110244875a^6 - 91513491733/4370081625a^5 + 41335171132/1872892125a^4 - 117109709/8067843a^3 + 2267178101/257063625a^2 - 1976713286/291338775a + 759709/924885, -15431854/15357715425a^19 + 4979513054/967536071775a^18 - 21871074122/967536071775a^17 + 24383785973/322512023925a^16 - 175427448772/967536071775a^15 + 289091489408/967536071775a^14 - 16462857464/56913886575a^13 - 17126534636/38701442871a^12 + 16493105948/8269539075a^11 - 483007032778/107504007975a^10 + 163360002206/24808617225a^9 - 126813783623/38701442871a^8 - 772920542977/967536071775a^7 + 2331437952134/138219438825a^6 - 17492690531098/967536071775a^5 + 8781180840598/322512023925a^4 - 158576830973/10632264525a^3 + 630868411672/56913886575a^2 - 1633582828466/322512023925a + 27463497457/21500801595, -5344863577/1612560119625a^19 + 10811078509/691097194125a^18 - 347834098666/4837680358875a^17 + 124074094409/537520039875a^16 - 2712770218322/4837680358875a^15 + 4316849366038/4837680358875a^14 - 249433382434/284569432875a^13 - 1563386938052/967536071775a^12 + 7078089518/1181362725a^11 - 2705194351949/179173346625a^10 + 2272147070813/124043086125a^9 - 12823626364658/967536071775a^8 - 53317228960004/4837680358875a^7 + 228094372725211/4837680358875a^6 - 48621116430158/691097194125a^5 + 13252812570668/179173346625a^4 - 3712552880942/74425851675a^3 + 5820458531807/284569432875a^2 - 2710919941157/322512023925*a + 43840818208/21500801595 (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:	$ 259320806.1491311 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 259320806.1491311 \cdot 1}{2\cdot\sqrt{79836369265591620072268522842253}}\cr\approx \mathstrut & 1.39157193641186 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225)

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 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225, 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 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225);

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 - 4*x^19 + 19*x^18 - 57*x^17 + 133*x^16 - 190*x^15 + 171*x^14 + 532*x^13 - 1330*x^12 + 3648*x^11 - 3325*x^10 + 2812*x^9 + 3534*x^8 - 9880*x^7 + 14212*x^6 - 15447*x^5 + 10906*x^4 - 6574*x^3 + 4579*x^2 - 840*x + 225);

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{-19}) $, 4.0.48013.1, 5.1.6385729.1, 10.0.774773162367379.1

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:	data not computed
Degree 20 sibling:	data not computed
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	$20$	${\href{/padicField/3.2.0.1}{2} }^{10}$	${\href{/padicField/5.2.0.1}{2} }^{9}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$	R	${\href{/padicField/11.10.0.1}{10} }^{2}$	${\href{/padicField/13.2.0.1}{2} }^{10}$	${\href{/padicField/17.2.0.1}{2} }^{9}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	R	${\href{/padicField/23.5.0.1}{5} }^{4}$	$20$	${\href{/padicField/31.2.0.1}{2} }^{10}$	${\href{/padicField/37.4.0.1}{4} }^{5}$	${\href{/padicField/41.2.0.1}{2} }^{10}$	${\href{/padicField/43.5.0.1}{5} }^{4}$	${\href{/padicField/47.2.0.1}{2} }^{9}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	$20$	${\href{/padicField/59.2.0.1}{2} }^{10}$

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
$7$ 7	7.2.0.1	$x^{2} + 6 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	7.2.1.1	$x^{2} + 21$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$19$ 19	19.20.19.1	$x^{20} + 19$	$20$	$1$	$19$	$D_{20}$	$[\ ]_{20}^{2}$

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