LMFDB - Number field 22.10.6172475179135960522642404800603172634624.2

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*x^2 - 1)

gp: K = bnfinit(y^22 - 11*y^20 - 165*y^18 - 165*y^16 + 1650*y^14 + 3234*y^12 - 1386*y^10 - 5610*y^8 - 1815*y^6 + 1265*y^4 + 495*y^2 - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*x^2 - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*x^2 - 1)

$ x^{22} - 11 x^{20} - 165 x^{18} - 165 x^{16} + 1650 x^{14} + 3234 x^{12} - 1386 x^{10} - 5610 x^{8} + \cdots - 1 $ x^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*x^2 - 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$22$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[10, 6]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$6172475179135960522642404800603172634624$ 6172475179135960522642404800603172634624 $\medspace = 2^{28}\cdot 7^{10}\cdot 11^{22}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$64.37$	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:	not computed
Ramified primes:	$2$, $7$, $11$ 2, 7, 11	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}a^{4}+\frac{3}{7}$, $\frac{1}{7}a^{7}-\frac{1}{7}a^{5}+\frac{3}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{4}+\frac{3}{7}a^{2}+\frac{3}{7}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{5}+\frac{3}{7}a^{3}+\frac{3}{7}a$, $\frac{1}{49}a^{10}+\frac{1}{49}a^{6}+\frac{1}{49}a^{4}+\frac{3}{49}a^{2}+\frac{6}{49}$, $\frac{1}{98}a^{11}-\frac{1}{98}a^{10}-\frac{1}{14}a^{9}-\frac{1}{14}a^{8}-\frac{3}{49}a^{7}+\frac{3}{49}a^{6}-\frac{17}{49}a^{5}+\frac{24}{49}a^{4}+\frac{31}{98}a^{3}+\frac{25}{98}a^{2}+\frac{13}{98}a+\frac{43}{98}$, $\frac{1}{98}a^{12}+\frac{1}{98}a^{8}-\frac{3}{49}a^{6}-\frac{39}{98}a^{4}+\frac{3}{49}a^{2}-\frac{3}{14}$, $\frac{1}{98}a^{13}+\frac{1}{98}a^{9}-\frac{3}{49}a^{7}-\frac{39}{98}a^{5}+\frac{3}{49}a^{3}-\frac{3}{14}a$, $\frac{1}{686}a^{14}+\frac{1}{686}a^{12}+\frac{3}{686}a^{10}-\frac{47}{686}a^{8}-\frac{43}{686}a^{6}-\frac{185}{686}a^{4}+\frac{257}{686}a^{2}+\frac{159}{686}$, $\frac{1}{686}a^{15}+\frac{1}{686}a^{13}+\frac{3}{686}a^{11}-\frac{47}{686}a^{9}-\frac{43}{686}a^{7}-\frac{185}{686}a^{5}+\frac{257}{686}a^{3}+\frac{159}{686}a$, $\frac{1}{1372}a^{16}+\frac{1}{686}a^{12}+\frac{3}{686}a^{10}+\frac{1}{343}a^{8}-\frac{1}{14}a^{7}+\frac{3}{343}a^{6}-\frac{3}{7}a^{5}+\frac{100}{343}a^{4}-\frac{1}{2}a^{3}+\frac{5}{98}a^{2}+\frac{2}{7}a-\frac{215}{1372}$, $\frac{1}{1372}a^{17}+\frac{1}{686}a^{13}+\frac{3}{686}a^{11}+\frac{1}{343}a^{9}-\frac{1}{14}a^{8}+\frac{3}{343}a^{7}+\frac{100}{343}a^{5}+\frac{1}{14}a^{4}+\frac{5}{98}a^{3}+\frac{2}{7}a^{2}-\frac{215}{1372}a+\frac{2}{7}$, $\frac{1}{9604}a^{18}+\frac{1}{4802}a^{16}+\frac{3}{4802}a^{14}-\frac{3}{686}a^{12}-\frac{3}{343}a^{10}-\frac{1}{14}a^{9}+\frac{13}{343}a^{8}+\frac{24}{343}a^{6}+\frac{1}{14}a^{5}+\frac{23}{4802}a^{4}+\frac{2}{7}a^{3}-\frac{699}{9604}a^{2}+\frac{2}{7}a-\frac{1213}{4802}$, $\frac{1}{9604}a^{19}+\frac{1}{4802}a^{17}+\frac{3}{4802}a^{15}-\frac{3}{686}a^{13}+\frac{1}{686}a^{11}-\frac{23}{686}a^{9}-\frac{1}{14}a^{8}+\frac{3}{343}a^{7}-\frac{1}{14}a^{6}-\frac{1643}{4802}a^{5}+\frac{1}{7}a^{4}+\frac{2339}{9604}a^{3}-\frac{3}{14}a^{2}-\frac{288}{2401}a+\frac{1}{14}$, $\frac{1}{9604}a^{20}+\frac{1}{4802}a^{16}+\frac{1}{4802}a^{14}-\frac{3}{686}a^{12}+\frac{1}{686}a^{10}-\frac{1}{14}a^{9}+\frac{23}{343}a^{8}-\frac{1}{14}a^{7}-\frac{187}{4802}a^{6}+\frac{1}{7}a^{5}-\frac{67}{1372}a^{4}-\frac{3}{14}a^{3}-\frac{1501}{4802}a^{2}+\frac{1}{14}a-\frac{334}{2401}$, $\frac{1}{9604}a^{21}+\frac{1}{4802}a^{17}+\frac{1}{4802}a^{15}-\frac{3}{686}a^{13}+\frac{1}{686}a^{11}-\frac{1}{98}a^{10}+\frac{23}{343}a^{9}-\frac{1}{14}a^{8}-\frac{187}{4802}a^{7}+\frac{3}{49}a^{6}-\frac{67}{1372}a^{5}-\frac{1}{98}a^{4}-\frac{1501}{4802}a^{3}+\frac{25}{98}a^{2}-\frac{334}{2401}a-\frac{3}{49}$ 1, a, a^2, a^3, a^4, a^5, 1/7*a^6 - 1/7*a^4 + 3/7, 1/7*a^7 - 1/7*a^5 + 3/7*a, 1/7*a^8 - 1/7*a^4 + 3/7*a^2 + 3/7, 1/7*a^9 - 1/7*a^5 + 3/7*a^3 + 3/7*a, 1/49*a^10 + 1/49*a^6 + 1/49*a^4 + 3/49*a^2 + 6/49, 1/98*a^11 - 1/98*a^10 - 1/14*a^9 - 1/14*a^8 - 3/49*a^7 + 3/49*a^6 - 17/49*a^5 + 24/49*a^4 + 31/98*a^3 + 25/98*a^2 + 13/98*a + 43/98, 1/98*a^12 + 1/98*a^8 - 3/49*a^6 - 39/98*a^4 + 3/49*a^2 - 3/14, 1/98*a^13 + 1/98*a^9 - 3/49*a^7 - 39/98*a^5 + 3/49*a^3 - 3/14*a, 1/686*a^14 + 1/686*a^12 + 3/686*a^10 - 47/686*a^8 - 43/686*a^6 - 185/686*a^4 + 257/686*a^2 + 159/686, 1/686*a^15 + 1/686*a^13 + 3/686*a^11 - 47/686*a^9 - 43/686*a^7 - 185/686*a^5 + 257/686*a^3 + 159/686*a, 1/1372*a^16 + 1/686*a^12 + 3/686*a^10 + 1/343*a^8 - 1/14*a^7 + 3/343*a^6 - 3/7*a^5 + 100/343*a^4 - 1/2*a^3 + 5/98*a^2 + 2/7*a - 215/1372, 1/1372*a^17 + 1/686*a^13 + 3/686*a^11 + 1/343*a^9 - 1/14*a^8 + 3/343*a^7 + 100/343*a^5 + 1/14*a^4 + 5/98*a^3 + 2/7*a^2 - 215/1372*a + 2/7, 1/9604*a^18 + 1/4802*a^16 + 3/4802*a^14 - 3/686*a^12 - 3/343*a^10 - 1/14*a^9 + 13/343*a^8 + 24/343*a^6 + 1/14*a^5 + 23/4802*a^4 + 2/7*a^3 - 699/9604*a^2 + 2/7*a - 1213/4802, 1/9604*a^19 + 1/4802*a^17 + 3/4802*a^15 - 3/686*a^13 + 1/686*a^11 - 23/686*a^9 - 1/14*a^8 + 3/343*a^7 - 1/14*a^6 - 1643/4802*a^5 + 1/7*a^4 + 2339/9604*a^3 - 3/14*a^2 - 288/2401*a + 1/14, 1/9604*a^20 + 1/4802*a^16 + 1/4802*a^14 - 3/686*a^12 + 1/686*a^10 - 1/14*a^9 + 23/343*a^8 - 1/14*a^7 - 187/4802*a^6 + 1/7*a^5 - 67/1372*a^4 - 3/14*a^3 - 1501/4802*a^2 + 1/14*a - 334/2401, 1/9604*a^21 + 1/4802*a^17 + 1/4802*a^15 - 3/686*a^13 + 1/686*a^11 - 1/98*a^10 + 23/343*a^9 - 1/14*a^8 - 187/4802*a^7 + 3/49*a^6 - 67/1372*a^5 - 1/98*a^4 - 1501/4802*a^3 + 25/98*a^2 - 334/2401*a - 3/49

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:	$15$	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:	$a$, $\frac{5}{686}a^{20}-\frac{213}{2401}a^{18}-\frac{2652}{2401}a^{16}+\frac{388}{2401}a^{14}+\frac{4365}{343}a^{12}+\frac{2983}{343}a^{10}-\frac{10264}{343}a^{8}-\frac{5052}{343}a^{6}+\frac{99523}{4802}a^{4}+\frac{4057}{2401}a^{2}-\frac{5700}{2401}$, $\frac{1201}{4802}a^{21}-\frac{13219}{4802}a^{19}-\frac{99035}{2401}a^{17}-\frac{98460}{2401}a^{15}+\frac{141556}{343}a^{13}+\frac{276390}{343}a^{11}-\frac{119850}{343}a^{9}-\frac{3350458}{2401}a^{7}-\frac{2152435}{4802}a^{5}+\frac{1484393}{4802}a^{3}+\frac{291124}{2401}a$, $\frac{255}{4802}a^{20}-\frac{2897}{4802}a^{18}-\frac{41033}{4802}a^{16}-\frac{27225}{4802}a^{14}+\frac{30783}{343}a^{12}+\frac{95451}{686}a^{10}-\frac{6112}{49}a^{8}-\frac{1208741}{4802}a^{6}-\frac{15879}{4802}a^{4}+\frac{160953}{2401}a^{2}+\frac{3765}{4802}$, $\frac{25}{4802}a^{20}-\frac{160}{2401}a^{18}-\frac{1786}{2401}a^{16}+\frac{2575}{4802}a^{14}+\frac{2873}{343}a^{12}+\frac{1335}{686}a^{10}-\frac{6638}{343}a^{8}-\frac{20599}{4802}a^{6}+\frac{82237}{4802}a^{4}+\frac{9645}{4802}a^{2}-\frac{11748}{2401}$, $\frac{405}{4802}a^{21}-\frac{2228}{2401}a^{19}-\frac{33402}{2401}a^{17}-\frac{66785}{4802}a^{15}+\frac{47640}{343}a^{13}+\frac{545}{2}a^{11}-\frac{39258}{343}a^{9}-\frac{2258617}{4802}a^{7}-\frac{751123}{4802}a^{5}+\frac{489533}{4802}a^{3}+\frac{98116}{2401}a$, $\frac{5}{686}a^{20}-\frac{186}{2401}a^{18}-\frac{2976}{2401}a^{16}-\frac{3755}{2401}a^{14}+\frac{4341}{343}a^{12}+\frac{1376}{49}a^{10}-\frac{4398}{343}a^{8}-\frac{18894}{343}a^{6}-\frac{50005}{4802}a^{4}+\frac{49458}{2401}a^{2}+\frac{13778}{2401}$, $\frac{145}{4802}a^{21}-\frac{1591}{4802}a^{19}-\frac{11981}{2401}a^{17}-\frac{24691}{4802}a^{15}+\frac{16987}{343}a^{13}+\frac{68333}{686}a^{11}-\frac{12905}{343}a^{9}-\frac{836781}{4802}a^{7}-\frac{295345}{4802}a^{5}+\frac{99265}{2401}a^{3}+\frac{40603}{2401}a$, $\frac{383}{2401}a^{21}-\frac{8569}{4802}a^{19}-\frac{62312}{2401}a^{17}-\frac{7512}{343}a^{15}+\frac{89909}{343}a^{13}+\frac{160477}{343}a^{11}-\frac{1768}{7}a^{9}-\frac{1938399}{2401}a^{7}-\frac{571097}{2401}a^{5}+\frac{855185}{4802}a^{3}+\frac{23858}{343}a$, $\frac{123}{2401}a^{20}-\frac{1430}{2401}a^{18}-\frac{38793}{4802}a^{16}-\frac{16395}{4802}a^{14}+\frac{29663}{343}a^{12}+\frac{76747}{686}a^{10}-\frac{47542}{343}a^{8}-\frac{970987}{4802}a^{6}+\frac{59555}{2401}a^{4}+\frac{243205}{4802}a^{2}+\frac{8417}{4802}$, $\frac{211}{4802}a^{21}+\frac{261}{9604}a^{20}-\frac{1137}{2401}a^{19}-\frac{435}{1372}a^{18}-\frac{35409}{4802}a^{17}-\frac{20501}{4802}a^{16}-\frac{20821}{2401}a^{15}-\frac{4024}{2401}a^{14}+\frac{50305}{686}a^{13}+\frac{15571}{343}a^{12}+\frac{15513}{98}a^{11}+\frac{19577}{343}a^{10}-\frac{19240}{343}a^{9}-\frac{48289}{686}a^{8}-\frac{1361069}{4802}a^{7}-\frac{457551}{4802}a^{6}-\frac{400711}{4802}a^{5}+\frac{2369}{196}a^{4}+\frac{181602}{2401}a^{3}+\frac{142991}{9604}a^{2}+\frac{25048}{2401}a-\frac{3121}{4802}$, $\frac{2027}{9604}a^{21}+\frac{65}{4802}a^{20}-\frac{22263}{9604}a^{19}-\frac{51}{343}a^{18}-\frac{334843}{9604}a^{17}-\frac{21489}{9604}a^{16}-\frac{84979}{2401}a^{15}-\frac{10783}{4802}a^{14}+\frac{119333}{343}a^{13}+\frac{15473}{686}a^{12}+\frac{236225}{343}a^{11}+\frac{2157}{49}a^{10}-\frac{194591}{686}a^{9}-\frac{7075}{343}a^{8}-\frac{2867378}{2401}a^{7}-\frac{186329}{2401}a^{6}-\frac{3844279}{9604}a^{5}-\frac{15455}{686}a^{4}+\frac{2584219}{9604}a^{3}+\frac{91471}{4802}a^{2}+\frac{1062525}{9604}a+\frac{71495}{9604}$, $\frac{9}{1372}a^{21}+\frac{15}{1372}a^{20}-\frac{743}{9604}a^{19}-\frac{313}{2401}a^{18}-\frac{9851}{9604}a^{17}-\frac{4035}{2401}a^{16}-\frac{509}{2401}a^{15}-\frac{1383}{4802}a^{14}+\frac{8047}{686}a^{13}+\frac{12259}{686}a^{12}+\frac{4138}{343}a^{11}+\frac{12987}{686}a^{10}-\frac{18371}{686}a^{9}-\frac{9691}{343}a^{8}-\frac{8398}{343}a^{7}-\frac{10992}{343}a^{6}+\frac{213867}{9604}a^{5}+\frac{42207}{9604}a^{4}+\frac{93771}{9604}a^{3}+\frac{18279}{2401}a^{2}-\frac{64235}{9604}a-\frac{303}{4802}$, $\frac{403}{9604}a^{21}-\frac{447}{9604}a^{20}-\frac{1105}{2401}a^{19}+\frac{5125}{9604}a^{18}-\frac{16663}{2401}a^{17}+\frac{17844}{2401}a^{16}-\frac{17121}{2401}a^{15}+\frac{20241}{4802}a^{14}+\frac{23754}{343}a^{13}-\frac{54101}{686}a^{12}+\frac{6768}{49}a^{11}-\frac{78171}{686}a^{10}-\frac{38789}{686}a^{9}+\frac{40826}{343}a^{8}-\frac{1162811}{4802}a^{7}+\frac{502902}{2401}a^{6}-\frac{764027}{9604}a^{5}-\frac{143139}{9604}a^{4}+\frac{294207}{4802}a^{3}-\frac{600437}{9604}a^{2}+\frac{124713}{4802}a+\frac{1023}{2401}$, $\frac{10761}{2401}a^{21}+\frac{967}{4802}a^{20}-\frac{473467}{9604}a^{19}-\frac{21255}{9604}a^{18}-\frac{1014631}{1372}a^{17}-\frac{79832}{2401}a^{16}-\frac{1776349}{2401}a^{15}-\frac{3287}{98}a^{14}+\frac{5072489}{686}a^{13}+\frac{4649}{14}a^{12}+\frac{4972418}{343}a^{11}+\frac{448879}{686}a^{10}-\frac{2126896}{343}a^{9}-\frac{188215}{686}a^{8}-\frac{120704155}{4802}a^{7}-\frac{5437967}{4802}a^{6}-\frac{39141433}{4802}a^{5}-\frac{1787183}{4802}a^{4}+\frac{7751035}{1372}a^{3}+\frac{2423861}{9604}a^{2}+\frac{21307249}{9604}a+\frac{34115}{343}$ a, 5/686a^20 - 213/2401a^18 - 2652/2401a^16 + 388/2401a^14 + 4365/343a^12 + 2983/343a^10 - 10264/343a^8 - 5052/343a^6 + 99523/4802a^4 + 4057/2401a^2 - 5700/2401, 1201/4802a^21 - 13219/4802a^19 - 99035/2401a^17 - 98460/2401a^15 + 141556/343a^13 + 276390/343a^11 - 119850/343a^9 - 3350458/2401a^7 - 2152435/4802a^5 + 1484393/4802a^3 + 291124/2401a, 255/4802a^20 - 2897/4802a^18 - 41033/4802a^16 - 27225/4802a^14 + 30783/343a^12 + 95451/686a^10 - 6112/49a^8 - 1208741/4802a^6 - 15879/4802a^4 + 160953/2401a^2 + 3765/4802, 25/4802a^20 - 160/2401a^18 - 1786/2401a^16 + 2575/4802a^14 + 2873/343a^12 + 1335/686a^10 - 6638/343a^8 - 20599/4802a^6 + 82237/4802a^4 + 9645/4802a^2 - 11748/2401, 405/4802a^21 - 2228/2401a^19 - 33402/2401a^17 - 66785/4802a^15 + 47640/343a^13 + 545/2a^11 - 39258/343a^9 - 2258617/4802a^7 - 751123/4802a^5 + 489533/4802a^3 + 98116/2401a, 5/686a^20 - 186/2401a^18 - 2976/2401a^16 - 3755/2401a^14 + 4341/343a^12 + 1376/49a^10 - 4398/343a^8 - 18894/343a^6 - 50005/4802a^4 + 49458/2401a^2 + 13778/2401, 145/4802a^21 - 1591/4802a^19 - 11981/2401a^17 - 24691/4802a^15 + 16987/343a^13 + 68333/686a^11 - 12905/343a^9 - 836781/4802a^7 - 295345/4802a^5 + 99265/2401a^3 + 40603/2401a, 383/2401a^21 - 8569/4802a^19 - 62312/2401a^17 - 7512/343a^15 + 89909/343a^13 + 160477/343a^11 - 1768/7a^9 - 1938399/2401a^7 - 571097/2401a^5 + 855185/4802a^3 + 23858/343a, 123/2401a^20 - 1430/2401a^18 - 38793/4802a^16 - 16395/4802a^14 + 29663/343a^12 + 76747/686a^10 - 47542/343a^8 - 970987/4802a^6 + 59555/2401a^4 + 243205/4802a^2 + 8417/4802, 211/4802a^21 + 261/9604a^20 - 1137/2401a^19 - 435/1372a^18 - 35409/4802a^17 - 20501/4802a^16 - 20821/2401a^15 - 4024/2401a^14 + 50305/686a^13 + 15571/343a^12 + 15513/98a^11 + 19577/343a^10 - 19240/343a^9 - 48289/686a^8 - 1361069/4802a^7 - 457551/4802a^6 - 400711/4802a^5 + 2369/196a^4 + 181602/2401a^3 + 142991/9604a^2 + 25048/2401a - 3121/4802, 2027/9604a^21 + 65/4802a^20 - 22263/9604a^19 - 51/343a^18 - 334843/9604a^17 - 21489/9604a^16 - 84979/2401a^15 - 10783/4802a^14 + 119333/343a^13 + 15473/686a^12 + 236225/343a^11 + 2157/49a^10 - 194591/686a^9 - 7075/343a^8 - 2867378/2401a^7 - 186329/2401a^6 - 3844279/9604a^5 - 15455/686a^4 + 2584219/9604a^3 + 91471/4802a^2 + 1062525/9604a + 71495/9604, 9/1372a^21 + 15/1372a^20 - 743/9604a^19 - 313/2401a^18 - 9851/9604a^17 - 4035/2401a^16 - 509/2401a^15 - 1383/4802a^14 + 8047/686a^13 + 12259/686a^12 + 4138/343a^11 + 12987/686a^10 - 18371/686a^9 - 9691/343a^8 - 8398/343a^7 - 10992/343a^6 + 213867/9604a^5 + 42207/9604a^4 + 93771/9604a^3 + 18279/2401a^2 - 64235/9604a - 303/4802, 403/9604a^21 - 447/9604a^20 - 1105/2401a^19 + 5125/9604a^18 - 16663/2401a^17 + 17844/2401a^16 - 17121/2401a^15 + 20241/4802a^14 + 23754/343a^13 - 54101/686a^12 + 6768/49a^11 - 78171/686a^10 - 38789/686a^9 + 40826/343a^8 - 1162811/4802a^7 + 502902/2401a^6 - 764027/9604a^5 - 143139/9604a^4 + 294207/4802a^3 - 600437/9604a^2 + 124713/4802a + 1023/2401, 10761/2401a^21 + 967/4802a^20 - 473467/9604a^19 - 21255/9604a^18 - 1014631/1372a^17 - 79832/2401a^16 - 1776349/2401a^15 - 3287/98a^14 + 5072489/686a^13 + 4649/14a^12 + 4972418/343a^11 + 448879/686a^10 - 2126896/343a^9 - 188215/686a^8 - 120704155/4802a^7 - 5437967/4802a^6 - 39141433/4802a^5 - 1787183/4802a^4 + 7751035/1372a^3 + 2423861/9604a^2 + 21307249/9604*a + 34115/343 (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:	$ 3989928450040 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 3989928450040 \cdot 1}{2\cdot\sqrt{6172475179135960522642404800603172634624}}\cr\approx \mathstrut & 1.59986966083025 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*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^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*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^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*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^22 - 11*x^20 - 165*x^18 - 165*x^16 + 1650*x^14 + 3234*x^12 - 1386*x^10 - 5610*x^8 - 1815*x^6 + 1265*x^4 + 495*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

$C_2^{10}.F_{11}$ (as 22T34):

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 112640

The 44 conjugacy class representatives for $C_2^{10}.F_{11}$

Character table for $C_2^{10}.F_{11}$

Intermediate fields

11.11.4910318845910094848.1

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

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

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

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

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

Sibling fields

Degree 22 sibling:	data not computed
Degree 44 siblings:	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	R	${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$	$20{,}\,{\href{/padicField/5.2.0.1}{2} }$	R	R	${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	${\href{/padicField/23.11.0.1}{11} }^{2}$	$20{,}\,{\href{/padicField/29.2.0.1}{2} }$	$20{,}\,{\href{/padicField/31.2.0.1}{2} }$	${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$	${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	$20{,}\,{\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2		Deg $22$	$22$	$1$	$28$
$7$ 7	$\Q_{7}$	$x + 4$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{7}$	$x + 4$	$1$	$1$	$0$	Trivial	$[\ ]$
	7.20.10.1	$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$	$2$	$10$	$10$	20T3	$[\ ]_{2}^{10}$
$11$ 11	11.11.11.6	$x^{11} + 11 x + 11$	$11$	$1$	$11$	$F_{11}$	$[11/10]_{10}$
$11$ 11	11.11.11.6	$x^{11} + 11 x + 11$	$11$	$1$	$11$	$F_{11}$	$[11/10]_{10}$

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