LMFDB - Number field 20.4.5597618628270744384765625.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 1)

gp: K = bnfinit(y^20 - 11*y^18 + 63*y^16 - 218*y^14 + 493*y^12 - 827*y^10 + 1137*y^8 - 1157*y^6 + 180*y^4 - 21*y^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 1)

$ x^{20} - 11 x^{18} + 63 x^{16} - 218 x^{14} + 493 x^{12} - 827 x^{10} + 1137 x^{8} - 1157 x^{6} + \cdots + 1 $ x^20 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 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:	$[4, 8]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$5597618628270744384765625$ 5597618628270744384765625 $\medspace = 5^{12}\cdot 13^{6}\cdot 41^{6}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.27$	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:	$5^{3/4}13^{3/4}41^{1/2}\approx 146.58075214439185$
Ramified primes:	$5$, $13$, $41$ 5, 13, 41	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$4$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{82}a^{16}-\frac{7}{82}a^{14}-\frac{5}{41}a^{12}+\frac{8}{41}a^{10}+\frac{23}{82}a^{8}+\frac{21}{82}a^{6}-\frac{1}{2}a^{5}+\frac{11}{41}a^{4}-\frac{1}{2}a^{3}+\frac{18}{41}a^{2}-\frac{1}{2}a+\frac{31}{82}$, $\frac{1}{82}a^{17}-\frac{7}{82}a^{15}-\frac{5}{41}a^{13}+\frac{8}{41}a^{11}+\frac{23}{82}a^{9}+\frac{21}{82}a^{7}-\frac{1}{2}a^{6}+\frac{11}{41}a^{5}-\frac{1}{2}a^{4}+\frac{18}{41}a^{3}-\frac{1}{2}a^{2}+\frac{31}{82}a$, $\frac{1}{3728603222}a^{18}-\frac{18208461}{3728603222}a^{16}+\frac{907902983}{3728603222}a^{14}+\frac{715922473}{3728603222}a^{12}+\frac{338098780}{1864301611}a^{10}-\frac{1}{2}a^{9}-\frac{1841451617}{3728603222}a^{8}-\frac{53047185}{3728603222}a^{6}-\frac{471704429}{3728603222}a^{4}+\frac{9481597}{1864301611}a^{2}-\frac{1}{2}a-\frac{1613351339}{3728603222}$, $\frac{1}{3728603222}a^{19}-\frac{18208461}{3728603222}a^{17}+\frac{907902983}{3728603222}a^{15}+\frac{715922473}{3728603222}a^{13}+\frac{338098780}{1864301611}a^{11}+\frac{11424997}{1864301611}a^{9}-\frac{1}{2}a^{8}-\frac{53047185}{3728603222}a^{7}-\frac{1}{2}a^{6}+\frac{696298591}{1864301611}a^{5}-\frac{1}{2}a^{4}-\frac{1845338417}{3728603222}a^{3}-\frac{1}{2}a^{2}+\frac{125475136}{1864301611}a-\frac{1}{2}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a - 1/2, 1/2*a^11 - 1/2*a^8 - 1/2*a^7 - 1/2*a^3 - 1/2*a^2 - 1/2, 1/2*a^12 - 1/2*a^9 - 1/2*a^8 - 1/2*a^4 - 1/2*a^3 - 1/2*a, 1/2*a^13 - 1/2*a^8 - 1/2*a^6 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^14 - 1/2*a^9 - 1/2*a^7 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^15 - 1/2*a^9 - 1/2*a^6 - 1/2*a^2 - 1/2*a - 1/2, 1/82*a^16 - 7/82*a^14 - 5/41*a^12 + 8/41*a^10 + 23/82*a^8 + 21/82*a^6 - 1/2*a^5 + 11/41*a^4 - 1/2*a^3 + 18/41*a^2 - 1/2*a + 31/82, 1/82*a^17 - 7/82*a^15 - 5/41*a^13 + 8/41*a^11 + 23/82*a^9 + 21/82*a^7 - 1/2*a^6 + 11/41*a^5 - 1/2*a^4 + 18/41*a^3 - 1/2*a^2 + 31/82*a, 1/3728603222*a^18 - 18208461/3728603222*a^16 + 907902983/3728603222*a^14 + 715922473/3728603222*a^12 + 338098780/1864301611*a^10 - 1/2*a^9 - 1841451617/3728603222*a^8 - 53047185/3728603222*a^6 - 471704429/3728603222*a^4 + 9481597/1864301611*a^2 - 1/2*a - 1613351339/3728603222, 1/3728603222*a^19 - 18208461/3728603222*a^17 + 907902983/3728603222*a^15 + 715922473/3728603222*a^13 + 338098780/1864301611*a^11 + 11424997/1864301611*a^9 - 1/2*a^8 - 53047185/3728603222*a^7 - 1/2*a^6 + 696298591/1864301611*a^5 - 1/2*a^4 - 1845338417/3728603222*a^3 - 1/2*a^2 + 125475136/1864301611*a - 1/2

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:	$11$	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:	$\frac{213298957}{1864301611}a^{18}-\frac{2350081062}{1864301611}a^{16}+\frac{13476379852}{1864301611}a^{14}-\frac{46712147252}{1864301611}a^{12}+\frac{105847596099}{1864301611}a^{10}-\frac{177844900139}{1864301611}a^{8}+\frac{244633657402}{1864301611}a^{6}-\frac{248952889545}{1864301611}a^{4}+\frac{39998830722}{1864301611}a^{2}-\frac{2801688836}{1864301611}$, $\frac{118730942}{1864301611}a^{18}-\frac{1283828392}{1864301611}a^{16}+\frac{7238018593}{1864301611}a^{14}-\frac{24509641377}{1864301611}a^{12}+\frac{53851069473}{1864301611}a^{10}-\frac{87857292825}{1864301611}a^{8}+\frac{118172345094}{1864301611}a^{6}-\frac{114938968226}{1864301611}a^{4}-\frac{1169369876}{1864301611}a^{2}-\frac{989764940}{1864301611}$, $\frac{1337496541}{3728603222}a^{19}+\frac{166097159}{1864301611}a^{18}-\frac{14710818431}{3728603222}a^{17}-\frac{1804912601}{1864301611}a^{16}+\frac{84243802219}{3728603222}a^{15}+\frac{20447841315}{3728603222}a^{14}-\frac{291453124379}{3728603222}a^{13}-\frac{34847440216}{1864301611}a^{12}+\frac{658879625515}{3728603222}a^{11}+\frac{154432446325}{3728603222}a^{10}-\frac{552298873286}{1864301611}a^{9}-\frac{253747199809}{3728603222}a^{8}+\frac{37009516061}{90941542}a^{7}+\frac{171177492198}{1864301611}a^{6}-\frac{1541844304347}{3728603222}a^{5}-\frac{335695406123}{3728603222}a^{4}+\frac{116406183472}{1864301611}a^{3}+\frac{10934505039}{3728603222}a^{2}-\frac{10340532272}{1864301611}a+\frac{615903701}{3728603222}$, $\frac{1337496541}{3728603222}a^{19}-\frac{37282564}{1864301611}a^{18}-\frac{14710818431}{3728603222}a^{17}+\frac{418098664}{1864301611}a^{16}+\frac{84243802219}{3728603222}a^{15}-\frac{4867103547}{3728603222}a^{14}-\frac{291453124379}{3728603222}a^{13}+\frac{8600059709}{1864301611}a^{12}+\frac{658879625515}{3728603222}a^{11}-\frac{39893221581}{3728603222}a^{10}-\frac{552298873286}{1864301611}a^{9}+\frac{68321798165}{3728603222}a^{8}+\frac{37009516061}{90941542}a^{7}-\frac{47626106963}{1864301611}a^{6}-\frac{1541844304347}{3728603222}a^{5}+\frac{100188371499}{3728603222}a^{4}+\frac{116406183472}{1864301611}a^{3}-\frac{27936211831}{3728603222}a^{2}-\frac{10340532272}{1864301611}a+\frac{485990109}{3728603222}$, $\frac{14679}{99794}a^{19}-\frac{80896}{49897}a^{17}+\frac{928715}{99794}a^{15}-\frac{1612199}{49897}a^{13}+\frac{3664538}{49897}a^{11}-\frac{6185115}{49897}a^{9}+\frac{8553503}{49897}a^{7}-\frac{8790986}{49897}a^{5}+\frac{1651370}{49897}a^{3}-\frac{565605}{99794}a-\frac{1}{2}$, $\frac{161794883}{1864301611}a^{19}-\frac{101607652}{1864301611}a^{18}-\frac{3549403773}{3728603222}a^{17}+\frac{2247095517}{3728603222}a^{16}+\frac{20283169713}{3728603222}a^{15}-\frac{12924029561}{3728603222}a^{14}-\frac{69990218697}{3728603222}a^{13}+\frac{22487204061}{1864301611}a^{12}+\frac{78896007754}{1864301611}a^{11}-\frac{51214526600}{1864301611}a^{10}-\frac{264190193583}{3728603222}a^{9}+\frac{86494746071}{1864301611}a^{8}+\frac{363066155945}{3728603222}a^{7}-\frac{239254617261}{3728603222}a^{6}-\frac{369014319121}{3728603222}a^{5}+\frac{246138340459}{3728603222}a^{4}+\frac{27715804397}{1864301611}a^{3}-\frac{23665157121}{1864301611}a^{2}-\frac{15991170217}{3728603222}a+\frac{1239049535}{1864301611}$, $\frac{203215304}{1864301611}a^{19}-\frac{2247095517}{1864301611}a^{17}+\frac{12924029561}{1864301611}a^{15}-\frac{44974408122}{1864301611}a^{13}+\frac{102429053200}{1864301611}a^{11}-\frac{172989492142}{1864301611}a^{9}+\frac{239254617261}{1864301611}a^{7}-\frac{246138340459}{1864301611}a^{5}+\frac{47330314242}{1864301611}a^{3}-\frac{4342400681}{1864301611}a$, $\frac{49446237}{1864301611}a^{19}-\frac{410489105}{1864301611}a^{18}-\frac{514844399}{1864301611}a^{17}+\frac{8990363757}{3728603222}a^{16}+\frac{5600203751}{3728603222}a^{15}-\frac{25635467681}{1864301611}a^{14}-\frac{17980346723}{3728603222}a^{13}+\frac{88183341817}{1864301611}a^{12}+\frac{36481545181}{3728603222}a^{11}-\frac{395647894055}{3728603222}a^{10}-\frac{54236490521}{3728603222}a^{9}+\frac{329159797534}{1864301611}a^{8}+\frac{33256114306}{1864301611}a^{7}-\frac{10966285142}{45470771}a^{6}-\frac{26458483379}{1864301611}a^{5}+\frac{451598562063}{1864301611}a^{4}-\frac{40432012839}{3728603222}a^{3}-\frac{49293123913}{1864301611}a^{2}-\frac{20716549}{90941542}a+\frac{6872250325}{3728603222}$, $\frac{1743927149}{3728603222}a^{19}+\frac{47201798}{1864301611}a^{18}-\frac{468414865}{90941542}a^{17}-\frac{545168461}{1864301611}a^{16}+\frac{110091861341}{3728603222}a^{15}+\frac{6504918389}{3728603222}a^{14}-\frac{381401940623}{3728603222}a^{13}-\frac{11864707036}{1864301611}a^{12}+\frac{863737731915}{3728603222}a^{11}+\frac{57262745873}{3728603222}a^{10}-\frac{725288365428}{1864301611}a^{9}-\frac{101942600469}{3728603222}a^{8}+\frac{1995899393023}{3728603222}a^{7}+\frac{73456165204}{1864301611}a^{6}-\frac{2034120985265}{3728603222}a^{5}-\frac{162210372967}{3728603222}a^{4}+\frac{163736497714}{1864301611}a^{3}+\frac{69063156405}{3728603222}a^{2}-\frac{14682932953}{1864301611}a-\frac{6219281373}{3728603222}$, $\frac{937387225}{3728603222}a^{19}-\frac{235982783}{3728603222}a^{18}-\frac{5048980259}{1864301611}a^{17}+\frac{1262546334}{1864301611}a^{16}+\frac{56705314113}{3728603222}a^{15}-\frac{14088358497}{3728603222}a^{14}-\frac{95437017599}{1864301611}a^{13}+\frac{46986205359}{3728603222}a^{12}+\frac{415419754673}{3728603222}a^{11}-\frac{50462013432}{1864301611}a^{10}-\frac{8163068768}{45470771}a^{9}+\frac{160416183879}{3728603222}a^{8}+\frac{443982187343}{1864301611}a^{7}-\frac{105127239249}{1864301611}a^{6}-\frac{839923361923}{3728603222}a^{5}+\frac{96872237184}{1864301611}a^{4}-\frac{80223189045}{3728603222}a^{3}+\frac{18820029188}{1864301611}a^{2}+\frac{9224698693}{1864301611}a-\frac{1882662069}{1864301611}$, $\frac{554247941}{1864301611}a^{19}-\frac{374272080}{1864301611}a^{18}-\frac{6026255895}{1864301611}a^{17}+\frac{8231086033}{3728603222}a^{16}+\frac{68309381421}{3728603222}a^{15}-\frac{23557403929}{1864301611}a^{14}-\frac{116520189852}{1864301611}a^{13}+\frac{162908344003}{3728603222}a^{12}+\frac{6308482238}{45470771}a^{11}-\frac{367975314871}{3728603222}a^{10}-\frac{426270914366}{1864301611}a^{9}+\frac{616536585245}{3728603222}a^{8}+\frac{1155677789049}{3728603222}a^{7}-\frac{846694277159}{3728603222}a^{6}-\frac{570962329356}{1864301611}a^{5}+\frac{858983087775}{3728603222}a^{4}+\frac{31024675549}{1864301611}a^{3}-\frac{63079302905}{1864301611}a^{2}-\frac{24808500167}{3728603222}a+\frac{10935543293}{3728603222}$ 213298957/1864301611a^18 - 2350081062/1864301611a^16 + 13476379852/1864301611a^14 - 46712147252/1864301611a^12 + 105847596099/1864301611a^10 - 177844900139/1864301611a^8 + 244633657402/1864301611a^6 - 248952889545/1864301611a^4 + 39998830722/1864301611a^2 - 2801688836/1864301611, 118730942/1864301611a^18 - 1283828392/1864301611a^16 + 7238018593/1864301611a^14 - 24509641377/1864301611a^12 + 53851069473/1864301611a^10 - 87857292825/1864301611a^8 + 118172345094/1864301611a^6 - 114938968226/1864301611a^4 - 1169369876/1864301611a^2 - 989764940/1864301611, 1337496541/3728603222a^19 + 166097159/1864301611a^18 - 14710818431/3728603222a^17 - 1804912601/1864301611a^16 + 84243802219/3728603222a^15 + 20447841315/3728603222a^14 - 291453124379/3728603222a^13 - 34847440216/1864301611a^12 + 658879625515/3728603222a^11 + 154432446325/3728603222a^10 - 552298873286/1864301611a^9 - 253747199809/3728603222a^8 + 37009516061/90941542a^7 + 171177492198/1864301611a^6 - 1541844304347/3728603222a^5 - 335695406123/3728603222a^4 + 116406183472/1864301611a^3 + 10934505039/3728603222a^2 - 10340532272/1864301611a + 615903701/3728603222, 1337496541/3728603222a^19 - 37282564/1864301611a^18 - 14710818431/3728603222a^17 + 418098664/1864301611a^16 + 84243802219/3728603222a^15 - 4867103547/3728603222a^14 - 291453124379/3728603222a^13 + 8600059709/1864301611a^12 + 658879625515/3728603222a^11 - 39893221581/3728603222a^10 - 552298873286/1864301611a^9 + 68321798165/3728603222a^8 + 37009516061/90941542a^7 - 47626106963/1864301611a^6 - 1541844304347/3728603222a^5 + 100188371499/3728603222a^4 + 116406183472/1864301611a^3 - 27936211831/3728603222a^2 - 10340532272/1864301611a + 485990109/3728603222, 14679/99794a^19 - 80896/49897a^17 + 928715/99794a^15 - 1612199/49897a^13 + 3664538/49897a^11 - 6185115/49897a^9 + 8553503/49897a^7 - 8790986/49897a^5 + 1651370/49897a^3 - 565605/99794a - 1/2, 161794883/1864301611a^19 - 101607652/1864301611a^18 - 3549403773/3728603222a^17 + 2247095517/3728603222a^16 + 20283169713/3728603222a^15 - 12924029561/3728603222a^14 - 69990218697/3728603222a^13 + 22487204061/1864301611a^12 + 78896007754/1864301611a^11 - 51214526600/1864301611a^10 - 264190193583/3728603222a^9 + 86494746071/1864301611a^8 + 363066155945/3728603222a^7 - 239254617261/3728603222a^6 - 369014319121/3728603222a^5 + 246138340459/3728603222a^4 + 27715804397/1864301611a^3 - 23665157121/1864301611a^2 - 15991170217/3728603222a + 1239049535/1864301611, 203215304/1864301611a^19 - 2247095517/1864301611a^17 + 12924029561/1864301611a^15 - 44974408122/1864301611a^13 + 102429053200/1864301611a^11 - 172989492142/1864301611a^9 + 239254617261/1864301611a^7 - 246138340459/1864301611a^5 + 47330314242/1864301611a^3 - 4342400681/1864301611a, 49446237/1864301611a^19 - 410489105/1864301611a^18 - 514844399/1864301611a^17 + 8990363757/3728603222a^16 + 5600203751/3728603222a^15 - 25635467681/1864301611a^14 - 17980346723/3728603222a^13 + 88183341817/1864301611a^12 + 36481545181/3728603222a^11 - 395647894055/3728603222a^10 - 54236490521/3728603222a^9 + 329159797534/1864301611a^8 + 33256114306/1864301611a^7 - 10966285142/45470771a^6 - 26458483379/1864301611a^5 + 451598562063/1864301611a^4 - 40432012839/3728603222a^3 - 49293123913/1864301611a^2 - 20716549/90941542a + 6872250325/3728603222, 1743927149/3728603222a^19 + 47201798/1864301611a^18 - 468414865/90941542a^17 - 545168461/1864301611a^16 + 110091861341/3728603222a^15 + 6504918389/3728603222a^14 - 381401940623/3728603222a^13 - 11864707036/1864301611a^12 + 863737731915/3728603222a^11 + 57262745873/3728603222a^10 - 725288365428/1864301611a^9 - 101942600469/3728603222a^8 + 1995899393023/3728603222a^7 + 73456165204/1864301611a^6 - 2034120985265/3728603222a^5 - 162210372967/3728603222a^4 + 163736497714/1864301611a^3 + 69063156405/3728603222a^2 - 14682932953/1864301611a - 6219281373/3728603222, 937387225/3728603222a^19 - 235982783/3728603222a^18 - 5048980259/1864301611a^17 + 1262546334/1864301611a^16 + 56705314113/3728603222a^15 - 14088358497/3728603222a^14 - 95437017599/1864301611a^13 + 46986205359/3728603222a^12 + 415419754673/3728603222a^11 - 50462013432/1864301611a^10 - 8163068768/45470771a^9 + 160416183879/3728603222a^8 + 443982187343/1864301611a^7 - 105127239249/1864301611a^6 - 839923361923/3728603222a^5 + 96872237184/1864301611a^4 - 80223189045/3728603222a^3 + 18820029188/1864301611a^2 + 9224698693/1864301611a - 1882662069/1864301611, 554247941/1864301611a^19 - 374272080/1864301611a^18 - 6026255895/1864301611a^17 + 8231086033/3728603222a^16 + 68309381421/3728603222a^15 - 23557403929/1864301611a^14 - 116520189852/1864301611a^13 + 162908344003/3728603222a^12 + 6308482238/45470771a^11 - 367975314871/3728603222a^10 - 426270914366/1864301611a^9 + 616536585245/3728603222a^8 + 1155677789049/3728603222a^7 - 846694277159/3728603222a^6 - 570962329356/1864301611a^5 + 858983087775/3728603222a^4 + 31024675549/1864301611a^3 - 63079302905/1864301611a^2 - 24808500167/3728603222*a + 10935543293/3728603222 (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:	$ 23464.0125022 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 23464.0125022 \cdot 1}{2\cdot\sqrt{5597618628270744384765625}}\cr\approx \mathstrut & 0.192721231487 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 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 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 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 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 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 - 11*x^18 + 63*x^16 - 218*x^14 + 493*x^12 - 827*x^10 + 1137*x^8 - 1157*x^6 + 180*x^4 - 21*x^2 + 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

$C_2\wr S_5$ (as 20T288):

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 3840

The 36 conjugacy class representatives for $C_2\wr S_5$

Character table for $C_2\wr S_5$

Intermediate fields

$\Q(\sqrt{5}) $, 5.1.2665.1, 10.2.887778125.1, 10.2.2365928703125.1, 10.2.18927429625.2

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 10 siblings:	data not computed
Degree 20 siblings:	data not computed
Degree 30 siblings:	data not computed
Degree 32 siblings:	data not computed
Degree 40 siblings:	data not computed
Minimal sibling:	10.2.2365928703125.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.10.0.1}{10} }^{2}$	${\href{/padicField/3.10.0.1}{10} }^{2}$	R	${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$	${\href{/padicField/11.5.0.1}{5} }^{4}$	R	${\href{/padicField/17.10.0.1}{10} }^{2}$	${\href{/padicField/19.5.0.1}{5} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$	${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$	${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$	R	${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$	${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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	5.6.3.1	$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.6.3.1	$x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	5.8.6.2	$x^{8} + 10 x^{4} - 25$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$13$ 13	13.6.0.1	$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	13.6.0.1	$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	13.8.6.1	$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$41$ 41	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.1.2	$x^{2} + 123$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.1	$x^{2} + 41$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.1	$x^{2} + 41$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.1	$x^{2} + 41$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.2	$x^{2} + 123$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.1	$x^{2} + 41$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} + 38 x + 6$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$

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