LMFDB - Number field 21.3.214407920026380373514939807.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - x + 1)

gp: K = bnfinit(y^21 - y^20 - y^19 + y^18 - y^17 + 6*y^16 - 7*y^15 + 2*y^14 - 4*y^13 + 5*y^12 + 9*y^11 - 17*y^10 + 11*y^9 - 4*y^8 - 3*y^7 + 9*y^6 - 6*y^5 - y^4 + 3*y^3 - y^2 - y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - x + 1)

$ x^{21} - x^{20} - x^{19} + x^{18} - x^{17} + 6 x^{16} - 7 x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + \cdots + 1 $ x^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - x + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$21$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[3, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-214407920026380373514939807$ -214407920026380373514939807 $\medspace = -\,184607^{5}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.94$	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:	$184607^{1/2}\approx 429.65916724771506$
Ramified primes:	$184607$ 184607	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-184607}) $
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5358583}a^{20}-\frac{2662255}{5358583}a^{19}-\frac{443760}{5358583}a^{18}+\frac{399614}{5358583}a^{17}-\frac{2335469}{5358583}a^{16}-\frac{395015}{5358583}a^{15}-\frac{2367113}{5358583}a^{14}+\frac{1687214}{5358583}a^{13}+\frac{34779}{75473}a^{12}-\frac{2062332}{5358583}a^{11}-\frac{752710}{5358583}a^{10}-\frac{1207523}{5358583}a^{9}+\frac{1106327}{5358583}a^{8}+\frac{230556}{5358583}a^{7}+\frac{256508}{5358583}a^{6}-\frac{2348669}{5358583}a^{5}-\frac{229541}{5358583}a^{4}-\frac{1718490}{5358583}a^{3}+\frac{524140}{5358583}a^{2}+\frac{2635971}{5358583}a-\frac{2607503}{5358583}$ 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, a^18, a^19, 1/5358583*a^20 - 2662255/5358583*a^19 - 443760/5358583*a^18 + 399614/5358583*a^17 - 2335469/5358583*a^16 - 395015/5358583*a^15 - 2367113/5358583*a^14 + 1687214/5358583*a^13 + 34779/75473*a^12 - 2062332/5358583*a^11 - 752710/5358583*a^10 - 1207523/5358583*a^9 + 1106327/5358583*a^8 + 230556/5358583*a^7 + 256508/5358583*a^6 - 2348669/5358583*a^5 - 229541/5358583*a^4 - 1718490/5358583*a^3 + 524140/5358583*a^2 + 2635971/5358583*a - 2607503/5358583

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:	$a^{20}-a^{19}-a^{18}+a^{17}-a^{16}+6a^{15}-7a^{14}+2a^{13}-4a^{12}+5a^{11}+9a^{10}-17a^{9}+11a^{8}-4a^{7}-3a^{6}+9a^{5}-6a^{4}-a^{3}+3a^{2}-a-1$, $\frac{1042982}{5358583}a^{20}-\frac{298385}{5358583}a^{19}-\frac{2161444}{5358583}a^{18}-\frac{376792}{5358583}a^{17}-\frac{1054248}{5358583}a^{16}+\frac{6477808}{5358583}a^{15}-\frac{1663959}{5358583}a^{14}-\frac{3390720}{5358583}a^{13}-\frac{37755}{75473}a^{12}-\frac{2427743}{5358583}a^{11}+\frac{17655527}{5358583}a^{10}-\frac{13066845}{5358583}a^{9}+\frac{4752558}{5358583}a^{8}-\frac{654133}{5358583}a^{7}-\frac{10105168}{5358583}a^{6}+\frac{8500245}{5358583}a^{5}-\frac{12435737}{5358583}a^{4}+\frac{6138992}{5358583}a^{3}+\frac{2023569}{5358583}a^{2}+\frac{1070125}{5358583}a-\frac{1366952}{5358583}$, $\frac{130375}{5358583}a^{20}+\frac{1034}{5358583}a^{19}+\frac{1410651}{5358583}a^{18}-\frac{1827259}{5358583}a^{17}-\frac{1367649}{5358583}a^{16}+\frac{1260588}{5358583}a^{15}-\frac{845239}{5358583}a^{14}+\frac{11410266}{5358583}a^{13}-\frac{181188}{75473}a^{12}+\frac{6443274}{5358583}a^{11}-\frac{13552937}{5358583}a^{10}+\frac{15074581}{5358583}a^{9}+\frac{11121180}{5358583}a^{8}-\frac{18987879}{5358583}a^{7}+\frac{15389746}{5358583}a^{6}-\frac{18288255}{5358583}a^{5}+\frac{11995346}{5358583}a^{4}+\frac{4938646}{5358583}a^{3}-\frac{3256499}{5358583}a^{2}+\frac{2715586}{5358583}a+\frac{660478}{5358583}$, $\frac{1542060}{5358583}a^{20}-\frac{1827259}{5358583}a^{19}-\frac{2779334}{5358583}a^{18}+\frac{2437006}{5358583}a^{17}+\frac{646581}{5358583}a^{16}+\frac{10808791}{5358583}a^{15}-\frac{11760427}{5358583}a^{14}-\frac{5093231}{5358583}a^{13}-\frac{35479}{75473}a^{12}+\frac{9665001}{5358583}a^{11}+\frac{25473945}{5358583}a^{10}-\frac{34986459}{5358583}a^{9}+\frac{5185327}{5358583}a^{8}-\frac{79524}{5358583}a^{7}-\frac{3794831}{5358583}a^{6}+\frac{24145730}{5358583}a^{5}-\frac{15511561}{5358583}a^{4}-\frac{2486912}{5358583}a^{3}+\frac{4178761}{5358583}a^{2}+\frac{2644031}{5358583}a-\frac{791887}{5358583}$, $\frac{2868683}{5358583}a^{20}-\frac{638322}{5358583}a^{19}-\frac{5714851}{5358583}a^{18}-\frac{1131411}{5358583}a^{17}-\frac{1781253}{5358583}a^{16}+\frac{17448931}{5358583}a^{15}-\frac{3990085}{5358583}a^{14}-\frac{9748341}{5358583}a^{13}-\frac{172306}{75473}a^{12}-\frac{1036108}{5358583}a^{11}+\frac{45734831}{5358583}a^{10}-\frac{25100604}{5358583}a^{9}+\frac{296846}{5358583}a^{8}-\frac{7104776}{5358583}a^{7}-\frac{21912928}{5358583}a^{6}+\frac{22143540}{5358583}a^{5}-\frac{1609714}{5358583}a^{4}-\frac{3143164}{5358583}a^{3}-\frac{89265}{5358583}a^{2}+\frac{795743}{5358583}a+\frac{66981}{5358583}$, $a^{20}-a^{19}-a^{18}+a^{17}-a^{16}+6a^{15}-7a^{14}+2a^{13}-4a^{12}+5a^{11}+9a^{10}-17a^{9}+11a^{8}-4a^{7}-3a^{6}+9a^{5}-6a^{4}-a^{3}+3a^{2}-a$, $\frac{744597}{5358583}a^{20}-\frac{1118462}{5358583}a^{19}-\frac{1419774}{5358583}a^{18}-\frac{11266}{5358583}a^{17}+\frac{219916}{5358583}a^{16}+\frac{5636915}{5358583}a^{15}-\frac{5476684}{5358583}a^{14}+\frac{1491323}{5358583}a^{13}-\frac{107643}{75473}a^{12}+\frac{8268689}{5358583}a^{11}+\frac{4663849}{5358583}a^{10}-\frac{6720244}{5358583}a^{9}+\frac{3517795}{5358583}a^{8}-\frac{6976222}{5358583}a^{7}-\frac{886593}{5358583}a^{6}-\frac{6177845}{5358583}a^{5}+\frac{7181974}{5358583}a^{4}-\frac{1105377}{5358583}a^{3}+\frac{2113107}{5358583}a^{2}+\frac{5034613}{5358583}a-\frac{1042982}{5358583}$, $\frac{4422433}{5358583}a^{20}-\frac{1576467}{5358583}a^{19}-\frac{3581658}{5358583}a^{18}+\frac{108879}{5358583}a^{17}-\frac{6145480}{5358583}a^{16}+\frac{24272335}{5358583}a^{15}-\frac{15578870}{5358583}a^{14}+\frac{10538980}{5358583}a^{13}-\frac{365272}{75473}a^{12}+\frac{8232730}{5358583}a^{11}+\frac{33312081}{5358583}a^{10}-\frac{34730396}{5358583}a^{9}+\frac{40335522}{5358583}a^{8}-\frac{18069075}{5358583}a^{7}-\frac{17218553}{5358583}a^{6}+\frac{20935037}{5358583}a^{5}-\frac{10446899}{5358583}a^{4}+\frac{5266657}{5358583}a^{3}+\frac{11784310}{5358583}a^{2}-\frac{4628652}{5358583}a-\frac{2814872}{5358583}$, $\frac{3326477}{5358583}a^{20}-\frac{3527689}{5358583}a^{19}-\frac{1781595}{5358583}a^{18}+\frac{3095068}{5358583}a^{17}-\frac{4920730}{5358583}a^{16}+\frac{18052522}{5358583}a^{15}-\frac{25388798}{5358583}a^{14}+\frac{16619704}{5358583}a^{13}-\frac{263387}{75473}a^{12}+\frac{20562803}{5358583}a^{11}+\frac{11141408}{5358583}a^{10}-\frac{46538335}{5358583}a^{9}+\frac{46555903}{5358583}a^{8}-\frac{13319246}{5358583}a^{7}-\frac{11360272}{5358583}a^{6}+\frac{15246304}{5358583}a^{5}-\frac{13006804}{5358583}a^{4}-\frac{7908245}{5358583}a^{3}+\frac{12145487}{5358583}a^{2}-\frac{4312134}{5358583}a-\frac{5136572}{5358583}$, $\frac{1772478}{5358583}a^{20}-\frac{4152341}{5358583}a^{19}-\frac{590208}{5358583}a^{18}+\frac{4163969}{5358583}a^{17}-\frac{3110269}{5358583}a^{16}+\frac{13133359}{5358583}a^{15}-\frac{24550172}{5358583}a^{14}+\frac{15619903}{5358583}a^{13}-\frac{126470}{75473}a^{12}+\frac{15749248}{5358583}a^{11}+\frac{7362794}{5358583}a^{10}-\frac{52391713}{5358583}a^{9}+\frac{52556784}{5358583}a^{8}-\frac{22253310}{5358583}a^{7}-\frac{4904977}{5358583}a^{6}+\frac{22546424}{5358583}a^{5}-\frac{27392655}{5358583}a^{4}+\frac{15050802}{5358583}a^{3}+\frac{9084210}{5358583}a^{2}-\frac{6855975}{5358583}a-\frac{1375015}{5358583}$, $\frac{4942191}{5358583}a^{20}-\frac{4456999}{5358583}a^{19}-\frac{7262252}{5358583}a^{18}+\frac{4005211}{5358583}a^{17}+\frac{323591}{5358583}a^{16}+\frac{31532193}{5358583}a^{15}-\frac{32275056}{5358583}a^{14}-\frac{5387673}{5358583}a^{13}-\frac{315493}{75473}a^{12}+\frac{36337560}{5358583}a^{11}+\frac{57847063}{5358583}a^{10}-\frac{78663202}{5358583}a^{9}+\frac{11668326}{5358583}a^{8}-\frac{13376673}{5358583}a^{7}+\frac{4755803}{5358583}a^{6}+\frac{41660497}{5358583}a^{5}-\frac{12726065}{5358583}a^{4}-\frac{19685740}{5358583}a^{3}+\frac{2024127}{5358583}a^{2}+\frac{6677841}{5358583}a-\frac{1981118}{5358583}$ a^20 - a^19 - a^18 + a^17 - a^16 + 6a^15 - 7a^14 + 2a^13 - 4a^12 + 5a^11 + 9a^10 - 17a^9 + 11a^8 - 4a^7 - 3a^6 + 9a^5 - 6a^4 - a^3 + 3a^2 - a - 1, 1042982/5358583a^20 - 298385/5358583a^19 - 2161444/5358583a^18 - 376792/5358583a^17 - 1054248/5358583a^16 + 6477808/5358583a^15 - 1663959/5358583a^14 - 3390720/5358583a^13 - 37755/75473a^12 - 2427743/5358583a^11 + 17655527/5358583a^10 - 13066845/5358583a^9 + 4752558/5358583a^8 - 654133/5358583a^7 - 10105168/5358583a^6 + 8500245/5358583a^5 - 12435737/5358583a^4 + 6138992/5358583a^3 + 2023569/5358583a^2 + 1070125/5358583a - 1366952/5358583, 130375/5358583a^20 + 1034/5358583a^19 + 1410651/5358583a^18 - 1827259/5358583a^17 - 1367649/5358583a^16 + 1260588/5358583a^15 - 845239/5358583a^14 + 11410266/5358583a^13 - 181188/75473a^12 + 6443274/5358583a^11 - 13552937/5358583a^10 + 15074581/5358583a^9 + 11121180/5358583a^8 - 18987879/5358583a^7 + 15389746/5358583a^6 - 18288255/5358583a^5 + 11995346/5358583a^4 + 4938646/5358583a^3 - 3256499/5358583a^2 + 2715586/5358583a + 660478/5358583, 1542060/5358583a^20 - 1827259/5358583a^19 - 2779334/5358583a^18 + 2437006/5358583a^17 + 646581/5358583a^16 + 10808791/5358583a^15 - 11760427/5358583a^14 - 5093231/5358583a^13 - 35479/75473a^12 + 9665001/5358583a^11 + 25473945/5358583a^10 - 34986459/5358583a^9 + 5185327/5358583a^8 - 79524/5358583a^7 - 3794831/5358583a^6 + 24145730/5358583a^5 - 15511561/5358583a^4 - 2486912/5358583a^3 + 4178761/5358583a^2 + 2644031/5358583a - 791887/5358583, 2868683/5358583a^20 - 638322/5358583a^19 - 5714851/5358583a^18 - 1131411/5358583a^17 - 1781253/5358583a^16 + 17448931/5358583a^15 - 3990085/5358583a^14 - 9748341/5358583a^13 - 172306/75473a^12 - 1036108/5358583a^11 + 45734831/5358583a^10 - 25100604/5358583a^9 + 296846/5358583a^8 - 7104776/5358583a^7 - 21912928/5358583a^6 + 22143540/5358583a^5 - 1609714/5358583a^4 - 3143164/5358583a^3 - 89265/5358583a^2 + 795743/5358583a + 66981/5358583, a^20 - a^19 - a^18 + a^17 - a^16 + 6a^15 - 7a^14 + 2a^13 - 4a^12 + 5a^11 + 9a^10 - 17a^9 + 11a^8 - 4a^7 - 3a^6 + 9a^5 - 6a^4 - a^3 + 3a^2 - a, 744597/5358583a^20 - 1118462/5358583a^19 - 1419774/5358583a^18 - 11266/5358583a^17 + 219916/5358583a^16 + 5636915/5358583a^15 - 5476684/5358583a^14 + 1491323/5358583a^13 - 107643/75473a^12 + 8268689/5358583a^11 + 4663849/5358583a^10 - 6720244/5358583a^9 + 3517795/5358583a^8 - 6976222/5358583a^7 - 886593/5358583a^6 - 6177845/5358583a^5 + 7181974/5358583a^4 - 1105377/5358583a^3 + 2113107/5358583a^2 + 5034613/5358583a - 1042982/5358583, 4422433/5358583a^20 - 1576467/5358583a^19 - 3581658/5358583a^18 + 108879/5358583a^17 - 6145480/5358583a^16 + 24272335/5358583a^15 - 15578870/5358583a^14 + 10538980/5358583a^13 - 365272/75473a^12 + 8232730/5358583a^11 + 33312081/5358583a^10 - 34730396/5358583a^9 + 40335522/5358583a^8 - 18069075/5358583a^7 - 17218553/5358583a^6 + 20935037/5358583a^5 - 10446899/5358583a^4 + 5266657/5358583a^3 + 11784310/5358583a^2 - 4628652/5358583a - 2814872/5358583, 3326477/5358583a^20 - 3527689/5358583a^19 - 1781595/5358583a^18 + 3095068/5358583a^17 - 4920730/5358583a^16 + 18052522/5358583a^15 - 25388798/5358583a^14 + 16619704/5358583a^13 - 263387/75473a^12 + 20562803/5358583a^11 + 11141408/5358583a^10 - 46538335/5358583a^9 + 46555903/5358583a^8 - 13319246/5358583a^7 - 11360272/5358583a^6 + 15246304/5358583a^5 - 13006804/5358583a^4 - 7908245/5358583a^3 + 12145487/5358583a^2 - 4312134/5358583a - 5136572/5358583, 1772478/5358583a^20 - 4152341/5358583a^19 - 590208/5358583a^18 + 4163969/5358583a^17 - 3110269/5358583a^16 + 13133359/5358583a^15 - 24550172/5358583a^14 + 15619903/5358583a^13 - 126470/75473a^12 + 15749248/5358583a^11 + 7362794/5358583a^10 - 52391713/5358583a^9 + 52556784/5358583a^8 - 22253310/5358583a^7 - 4904977/5358583a^6 + 22546424/5358583a^5 - 27392655/5358583a^4 + 15050802/5358583a^3 + 9084210/5358583a^2 - 6855975/5358583a - 1375015/5358583, 4942191/5358583a^20 - 4456999/5358583a^19 - 7262252/5358583a^18 + 4005211/5358583a^17 + 323591/5358583a^16 + 31532193/5358583a^15 - 32275056/5358583a^14 - 5387673/5358583a^13 - 315493/75473a^12 + 36337560/5358583a^11 + 57847063/5358583a^10 - 78663202/5358583a^9 + 11668326/5358583a^8 - 13376673/5358583a^7 + 4755803/5358583a^6 + 41660497/5358583a^5 - 12726065/5358583a^4 - 19685740/5358583a^3 + 2024127/5358583a^2 + 6677841/5358583a - 1981118/5358583 (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:	$ 59489.8190642 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 59489.8190642 \cdot 1}{2\cdot\sqrt{214407920026380373514939807}}\cr\approx \mathstrut & 0.248028189270 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - 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^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - 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^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - 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^21 - x^20 - x^19 + x^18 - x^17 + 6*x^16 - 7*x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 9*x^11 - 17*x^10 + 11*x^9 - 4*x^8 - 3*x^7 + 9*x^6 - 6*x^5 - x^4 + 3*x^3 - x^2 - 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

$S_7$ (as 21T38):

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 5040

The 15 conjugacy class representatives for $S_7$

Character table for $S_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 7 sibling:	7.1.184607.1
Degree 14 sibling:	deg 14
Degree 30 sibling:	data not computed
Degree 35 sibling:	data not computed
Degree 42 siblings:	data not computed
Minimal sibling:	7.1.184607.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.7.0.1}{7} }^{3}$	${\href{/padicField/3.7.0.1}{7} }^{3}$	${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$	${\href{/padicField/7.7.0.1}{7} }^{3}$	${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$	${\href{/padicField/17.3.0.1}{3} }^{7}$	${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$	${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.3.0.1}{3} }^{7}$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }$	${\href{/padicField/59.7.0.1}{7} }^{3}$

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
$184607$ 184607	$\Q_{184607}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{184607}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{184607}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{184607}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{184607}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		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 $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$2$	$2$	$2$

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