LMFDB - Number field 22.0.2988811416117414420033765003.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 2*x + 1)

gp: K = bnfinit(y^22 - y^21 - 4*y^20 + 3*y^19 + 11*y^18 - 6*y^17 - 18*y^16 + 2*y^15 + 26*y^14 - 8*y^13 - 18*y^12 + 23*y^11 + 12*y^10 - 27*y^9 - 30*y^8 + 23*y^7 + 19*y^6 + 3*y^5 + 5*y^4 - 26*y^3 + 8*y^2 + 2*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 2*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 2*x + 1)

$ x^{22} - x^{21} - 4 x^{20} + 3 x^{19} + 11 x^{18} - 6 x^{17} - 18 x^{16} + 2 x^{15} + 26 x^{14} + \cdots + 1 $ x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 2*x + 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:	$[0, 11]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-2988811416117414420033765003$ -2988811416117414420033765003 $\medspace = -\,3^{11}\cdot 167^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.74$	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:	$3^{1/2}167^{1/2}\approx 22.38302928559939$
Ramified primes:	$3$, $167$ 3, 167	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-3}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}+\frac{2}{5}a^{13}-\frac{2}{5}a^{12}-\frac{2}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{85}a^{18}+\frac{8}{85}a^{17}+\frac{6}{85}a^{16}+\frac{1}{5}a^{15}-\frac{1}{5}a^{14}+\frac{13}{85}a^{13}+\frac{12}{85}a^{12}-\frac{33}{85}a^{11}-\frac{31}{85}a^{9}+\frac{2}{85}a^{8}+\frac{19}{85}a^{7}-\frac{18}{85}a^{6}-\frac{31}{85}a^{5}+\frac{41}{85}a^{4}-\frac{41}{85}a^{3}+\frac{29}{85}a^{2}+\frac{6}{17}a+\frac{4}{17}$, $\frac{1}{85}a^{19}-\frac{7}{85}a^{17}+\frac{3}{85}a^{16}-\frac{2}{5}a^{15}-\frac{38}{85}a^{14}-\frac{7}{85}a^{13}+\frac{7}{85}a^{12}+\frac{9}{85}a^{11}+\frac{3}{85}a^{10}+\frac{12}{85}a^{9}+\frac{37}{85}a^{8}-\frac{2}{5}a^{7}-\frac{6}{85}a^{6}+\frac{22}{85}a^{4}+\frac{1}{5}a^{3}-\frac{32}{85}a^{2}+\frac{18}{85}a-\frac{41}{85}$, $\frac{1}{85}a^{20}+\frac{8}{85}a^{17}+\frac{8}{85}a^{16}-\frac{21}{85}a^{15}+\frac{2}{17}a^{14}-\frac{4}{85}a^{13}-\frac{26}{85}a^{12}-\frac{41}{85}a^{11}-\frac{39}{85}a^{10}+\frac{41}{85}a^{9}+\frac{31}{85}a^{8}+\frac{42}{85}a^{7}-\frac{24}{85}a^{6}-\frac{5}{17}a^{5}-\frac{19}{85}a^{4}-\frac{13}{85}a^{3}-\frac{18}{85}a-\frac{13}{85}$, $\frac{1}{10500305}a^{21}-\frac{1946}{617665}a^{20}+\frac{6791}{10500305}a^{19}+\frac{53459}{10500305}a^{18}+\frac{4252}{2100061}a^{17}+\frac{218369}{10500305}a^{16}-\frac{10972}{51221}a^{15}+\frac{3597096}{10500305}a^{14}+\frac{4053216}{10500305}a^{13}+\frac{15072}{617665}a^{12}-\frac{152688}{456535}a^{11}-\frac{128189}{456535}a^{10}+\frac{262911}{617665}a^{9}-\frac{1293033}{10500305}a^{8}+\frac{403505}{2100061}a^{7}+\frac{911804}{10500305}a^{6}-\frac{829627}{10500305}a^{5}+\frac{4011368}{10500305}a^{4}-\frac{1286802}{10500305}a^{3}+\frac{237819}{617665}a^{2}+\frac{206699}{617665}a+\frac{1171814}{10500305}$ 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, 1/5*a^16 - 2/5*a^15 + 1/5*a^14 + 2/5*a^13 - 2/5*a^12 - 2/5*a^11 + 2/5*a^10 + 2/5*a^9 - 1/5*a^7 + 2/5*a^6 - 1/5*a^5 + 2/5*a^4 - 1/5*a^3 + 1/5*a^2 - 1/5*a - 1/5, 1/5*a^17 + 2/5*a^15 - 1/5*a^14 + 2/5*a^13 - 1/5*a^12 - 2/5*a^11 + 1/5*a^10 - 1/5*a^9 - 1/5*a^8 - 2/5*a^6 - 2/5*a^4 - 1/5*a^3 + 1/5*a^2 + 2/5*a - 2/5, 1/85*a^18 + 8/85*a^17 + 6/85*a^16 + 1/5*a^15 - 1/5*a^14 + 13/85*a^13 + 12/85*a^12 - 33/85*a^11 - 31/85*a^9 + 2/85*a^8 + 19/85*a^7 - 18/85*a^6 - 31/85*a^5 + 41/85*a^4 - 41/85*a^3 + 29/85*a^2 + 6/17*a + 4/17, 1/85*a^19 - 7/85*a^17 + 3/85*a^16 - 2/5*a^15 - 38/85*a^14 - 7/85*a^13 + 7/85*a^12 + 9/85*a^11 + 3/85*a^10 + 12/85*a^9 + 37/85*a^8 - 2/5*a^7 - 6/85*a^6 + 22/85*a^4 + 1/5*a^3 - 32/85*a^2 + 18/85*a - 41/85, 1/85*a^20 + 8/85*a^17 + 8/85*a^16 - 21/85*a^15 + 2/17*a^14 - 4/85*a^13 - 26/85*a^12 - 41/85*a^11 - 39/85*a^10 + 41/85*a^9 + 31/85*a^8 + 42/85*a^7 - 24/85*a^6 - 5/17*a^5 - 19/85*a^4 - 13/85*a^3 - 18/85*a - 13/85, 1/10500305*a^21 - 1946/617665*a^20 + 6791/10500305*a^19 + 53459/10500305*a^18 + 4252/2100061*a^17 + 218369/10500305*a^16 - 10972/51221*a^15 + 3597096/10500305*a^14 + 4053216/10500305*a^13 + 15072/617665*a^12 - 152688/456535*a^11 - 128189/456535*a^10 + 262911/617665*a^9 - 1293033/10500305*a^8 + 403505/2100061*a^7 + 911804/10500305*a^6 - 829627/10500305*a^5 + 4011368/10500305*a^4 - 1286802/10500305*a^3 + 237819/617665*a^2 + 206699/617665*a + 1171814/10500305

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$

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:	$10$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{252771}{2100061} a^{21} + \frac{1971991}{10500305} a^{20} + \frac{3815992}{10500305} a^{19} - \frac{7438603}{10500305} a^{18} - \frac{10147709}{10500305} a^{17} + \frac{3741150}{2100061} a^{16} + \frac{22072}{15065} a^{15} - \frac{23383672}{10500305} a^{14} - \frac{29887762}{10500305} a^{13} + \frac{40929768}{10500305} a^{12} + \frac{870837}{456535} a^{11} - \frac{2069523}{456535} a^{10} + \frac{1134992}{2100061} a^{9} + \frac{45113928}{10500305} a^{8} - \frac{3164372}{10500305} a^{7} - \frac{69379161}{10500305} a^{6} + \frac{263964}{10500305} a^{5} + \frac{32881823}{10500305} a^{4} + \frac{351687}{617665} a^{3} + \frac{36666409}{10500305} a^{2} - \frac{31325554}{10500305} a + \frac{2406049}{10500305} $ -(252771)/(2100061)a^(21) + (1971991)/(10500305)a^(20) + (3815992)/(10500305)a^(19) - (7438603)/(10500305)a^(18) - (10147709)/(10500305)a^(17) + (3741150)/(2100061)a^(16) + (22072)/(15065)a^(15) - (23383672)/(10500305)a^(14) - (29887762)/(10500305)a^(13) + (40929768)/(10500305)a^(12) + (870837)/(456535)a^(11) - (2069523)/(456535)a^(10) + (1134992)/(2100061)a^(9) + (45113928)/(10500305)a^(8) - (3164372)/(10500305)a^(7) - (69379161)/(10500305)a^(6) + (263964)/(10500305)a^(5) + (32881823)/(10500305)a^(4) + (351687)/(617665)a^(3) + (36666409)/(10500305)a^(2) - (31325554)/(10500305)*a + (2406049)/(10500305) (order $6$)	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:	$\frac{1484586}{10500305}a^{21}+\frac{187824}{10500305}a^{20}-\frac{5387122}{10500305}a^{19}-\frac{2998866}{10500305}a^{18}+\frac{11476028}{10500305}a^{17}+\frac{9914507}{10500305}a^{16}-\frac{14712}{15065}a^{15}-\frac{4706088}{2100061}a^{14}-\frac{167741}{10500305}a^{13}+\frac{1911143}{2100061}a^{12}+\frac{409194}{456535}a^{11}+\frac{862249}{456535}a^{10}+\frac{18198626}{10500305}a^{9}+\frac{10257928}{10500305}a^{8}-\frac{7120513}{2100061}a^{7}-\frac{48732187}{10500305}a^{6}-\frac{35166671}{10500305}a^{5}+\frac{29000413}{10500305}a^{4}+\frac{3484368}{617665}a^{3}+\frac{4591382}{10500305}a^{2}+\frac{106640}{2100061}a-\frac{13837578}{10500305}$, $\frac{726973}{10500305}a^{21}+\frac{671918}{10500305}a^{20}-\frac{2593362}{10500305}a^{19}-\frac{639397}{2100061}a^{18}+\frac{1034734}{2100061}a^{17}+\frac{8832636}{10500305}a^{16}-\frac{91968}{256105}a^{15}-\frac{15651828}{10500305}a^{14}-\frac{3034703}{10500305}a^{13}+\frac{9196207}{10500305}a^{12}-\frac{2632}{456535}a^{11}+\frac{28289}{91307}a^{10}+\frac{17818221}{10500305}a^{9}+\frac{17737142}{10500305}a^{8}-\frac{14683459}{10500305}a^{7}-\frac{37865663}{10500305}a^{6}-\frac{18316898}{10500305}a^{5}+\frac{7694941}{10500305}a^{4}+\frac{27062836}{10500305}a^{3}+\frac{2432069}{10500305}a^{2}+\frac{419181}{10500305}a+\frac{3083063}{10500305}$, $\frac{3487487}{10500305}a^{21}-\frac{546995}{2100061}a^{20}-\frac{2737528}{2100061}a^{19}+\frac{7864116}{10500305}a^{18}+\frac{2155909}{617665}a^{17}-\frac{15101982}{10500305}a^{16}-\frac{1402341}{256105}a^{15}+\frac{266683}{2100061}a^{14}+\frac{79401052}{10500305}a^{13}-\frac{25140139}{10500305}a^{12}-\frac{2526569}{456535}a^{11}+\frac{3385733}{456535}a^{10}+\frac{9683025}{2100061}a^{9}-\frac{77819293}{10500305}a^{8}-\frac{98674863}{10500305}a^{7}+\frac{62150026}{10500305}a^{6}+\frac{46530132}{10500305}a^{5}+\frac{389349}{2100061}a^{4}+\frac{26160493}{10500305}a^{3}-\frac{66852439}{10500305}a^{2}+\frac{26496148}{10500305}a+\frac{6421846}{10500305}$, $\frac{41552}{256105}a^{21}+\frac{39908}{256105}a^{20}-\frac{42156}{51221}a^{19}-\frac{240896}{256105}a^{18}+\frac{30602}{15065}a^{17}+\frac{745321}{256105}a^{16}-\frac{727324}{256105}a^{15}-\frac{1542433}{256105}a^{14}+\frac{365032}{256105}a^{13}+\frac{98032}{15065}a^{12}-\frac{11519}{11135}a^{11}-\frac{19591}{11135}a^{10}+\frac{1703594}{256105}a^{9}+\frac{742243}{256105}a^{8}-\frac{2769621}{256105}a^{7}-\frac{2834264}{256105}a^{6}+\frac{876683}{256105}a^{5}+\frac{2143192}{256105}a^{4}+\frac{1541097}{256105}a^{3}-\frac{35083}{256105}a^{2}-\frac{1261221}{256105}a-\frac{5538}{51221}$, $\frac{3658278}{10500305}a^{21}-\frac{3678747}{10500305}a^{20}-\frac{13944362}{10500305}a^{19}+\frac{11685678}{10500305}a^{18}+\frac{37279309}{10500305}a^{17}-\frac{25496306}{10500305}a^{16}-\frac{1446397}{256105}a^{15}+\frac{17170701}{10500305}a^{14}+\frac{87378016}{10500305}a^{13}-\frac{48064612}{10500305}a^{12}-\frac{32447}{5371}a^{11}+\frac{868456}{91307}a^{10}+\frac{2248914}{617665}a^{9}-\frac{98757302}{10500305}a^{8}-\frac{84029782}{10500305}a^{7}+\frac{98462298}{10500305}a^{6}+\frac{39632459}{10500305}a^{5}-\frac{22331391}{10500305}a^{4}+\frac{20749568}{10500305}a^{3}-\frac{88229969}{10500305}a^{2}+\frac{44008931}{10500305}a+\frac{10958273}{10500305}$, $\frac{39908}{256105}a^{21}-\frac{16896}{51221}a^{20}-\frac{154772}{256105}a^{19}+\frac{304058}{256105}a^{18}+\frac{474399}{256105}a^{17}-\frac{724709}{256105}a^{16}-\frac{898213}{256105}a^{15}+\frac{827113}{256105}a^{14}+\frac{1633928}{256105}a^{13}-\frac{236709}{51221}a^{12}-\frac{49624}{11135}a^{11}+\frac{71981}{11135}a^{10}+\frac{160553}{256105}a^{9}-\frac{2265304}{256105}a^{8}-\frac{1020339}{256105}a^{7}+\frac{2921459}{256105}a^{6}+\frac{1141853}{256105}a^{5}-\frac{161971}{51221}a^{4}-\frac{495828}{256105}a^{3}-\frac{1558554}{256105}a^{2}+\frac{1150427}{256105}a-\frac{13862}{256105}$, $\frac{2438466}{10500305}a^{21}-\frac{1542014}{10500305}a^{20}-\frac{2144683}{2100061}a^{19}+\frac{3955766}{10500305}a^{18}+\frac{30317498}{10500305}a^{17}-\frac{4973409}{10500305}a^{16}-\frac{1253921}{256105}a^{15}-\frac{10756942}{10500305}a^{14}+\frac{69332872}{10500305}a^{13}+\frac{3746609}{10500305}a^{12}-\frac{2404614}{456535}a^{11}+\frac{1816794}{456535}a^{10}+\frac{50234188}{10500305}a^{9}-\frac{66583159}{10500305}a^{8}-\frac{105033896}{10500305}a^{7}+\frac{37341648}{10500305}a^{6}+\frac{77520432}{10500305}a^{5}+\frac{23258677}{10500305}a^{4}+\frac{4277823}{2100061}a^{3}-\frac{51054467}{10500305}a^{2}-\frac{2828146}{10500305}a+\frac{1480867}{2100061}$, $\frac{1336584}{2100061}a^{21}-\frac{581665}{2100061}a^{20}-\frac{5157150}{2100061}a^{19}+\frac{3919549}{10500305}a^{18}+\frac{65836532}{10500305}a^{17}+\frac{533006}{10500305}a^{16}-\frac{2298521}{256105}a^{15}-\frac{43559421}{10500305}a^{14}+\frac{110986461}{10500305}a^{13}-\frac{3725886}{10500305}a^{12}-\frac{3167832}{456535}a^{11}+\frac{4800808}{456535}a^{10}+\frac{21915646}{2100061}a^{9}-\frac{76063232}{10500305}a^{8}-\frac{209481306}{10500305}a^{7}-\frac{2932843}{10500305}a^{6}+\frac{44869514}{10500305}a^{5}+\frac{60354603}{10500305}a^{4}+\frac{6485542}{617665}a^{3}-\frac{92764027}{10500305}a^{2}+\frac{31762123}{10500305}a-\frac{14070517}{10500305}$, $\frac{5121419}{10500305}a^{21}-\frac{5597513}{10500305}a^{20}-\frac{4259120}{2100061}a^{19}+\frac{16427376}{10500305}a^{18}+\frac{59542311}{10500305}a^{17}-\frac{31798863}{10500305}a^{16}-\frac{2420979}{256105}a^{15}+\frac{8304686}{10500305}a^{14}+\frac{142243064}{10500305}a^{13}-\frac{34531681}{10500305}a^{12}-\frac{3953123}{456535}a^{11}+\frac{5361318}{456535}a^{10}+\frac{59142078}{10500305}a^{9}-\frac{160261273}{10500305}a^{8}-\frac{175465567}{10500305}a^{7}+\frac{23689130}{2100061}a^{6}+\frac{6461427}{617665}a^{5}+\frac{34611199}{10500305}a^{4}+\frac{48285528}{10500305}a^{3}-\frac{127452284}{10500305}a^{2}+\frac{24562383}{10500305}a-\frac{1865279}{10500305}$, $\frac{76166}{2100061}a^{21}-\frac{1587451}{10500305}a^{20}-\frac{1676287}{10500305}a^{19}+\frac{1222311}{2100061}a^{18}+\frac{6146097}{10500305}a^{17}-\frac{14414427}{10500305}a^{16}-\frac{345463}{256105}a^{15}+\frac{17097167}{10500305}a^{14}+\frac{29431579}{10500305}a^{13}-\frac{17113603}{10500305}a^{12}-\frac{951616}{456535}a^{11}+\frac{902191}{456535}a^{10}-\frac{3730372}{10500305}a^{9}-\frac{41378741}{10500305}a^{8}-\frac{12431061}{10500305}a^{7}+\frac{60973998}{10500305}a^{6}+\frac{35245498}{10500305}a^{5}-\frac{26633913}{10500305}a^{4}-\frac{26795514}{10500305}a^{3}-\frac{16950863}{10500305}a^{2}+\frac{18863862}{10500305}a+\frac{1680846}{10500305}$ 1484586/10500305a^21 + 187824/10500305a^20 - 5387122/10500305a^19 - 2998866/10500305a^18 + 11476028/10500305a^17 + 9914507/10500305a^16 - 14712/15065a^15 - 4706088/2100061a^14 - 167741/10500305a^13 + 1911143/2100061a^12 + 409194/456535a^11 + 862249/456535a^10 + 18198626/10500305a^9 + 10257928/10500305a^8 - 7120513/2100061a^7 - 48732187/10500305a^6 - 35166671/10500305a^5 + 29000413/10500305a^4 + 3484368/617665a^3 + 4591382/10500305a^2 + 106640/2100061a - 13837578/10500305, 726973/10500305a^21 + 671918/10500305a^20 - 2593362/10500305a^19 - 639397/2100061a^18 + 1034734/2100061a^17 + 8832636/10500305a^16 - 91968/256105a^15 - 15651828/10500305a^14 - 3034703/10500305a^13 + 9196207/10500305a^12 - 2632/456535a^11 + 28289/91307a^10 + 17818221/10500305a^9 + 17737142/10500305a^8 - 14683459/10500305a^7 - 37865663/10500305a^6 - 18316898/10500305a^5 + 7694941/10500305a^4 + 27062836/10500305a^3 + 2432069/10500305a^2 + 419181/10500305a + 3083063/10500305, 3487487/10500305a^21 - 546995/2100061a^20 - 2737528/2100061a^19 + 7864116/10500305a^18 + 2155909/617665a^17 - 15101982/10500305a^16 - 1402341/256105a^15 + 266683/2100061a^14 + 79401052/10500305a^13 - 25140139/10500305a^12 - 2526569/456535a^11 + 3385733/456535a^10 + 9683025/2100061a^9 - 77819293/10500305a^8 - 98674863/10500305a^7 + 62150026/10500305a^6 + 46530132/10500305a^5 + 389349/2100061a^4 + 26160493/10500305a^3 - 66852439/10500305a^2 + 26496148/10500305a + 6421846/10500305, 41552/256105a^21 + 39908/256105a^20 - 42156/51221a^19 - 240896/256105a^18 + 30602/15065a^17 + 745321/256105a^16 - 727324/256105a^15 - 1542433/256105a^14 + 365032/256105a^13 + 98032/15065a^12 - 11519/11135a^11 - 19591/11135a^10 + 1703594/256105a^9 + 742243/256105a^8 - 2769621/256105a^7 - 2834264/256105a^6 + 876683/256105a^5 + 2143192/256105a^4 + 1541097/256105a^3 - 35083/256105a^2 - 1261221/256105a - 5538/51221, 3658278/10500305a^21 - 3678747/10500305a^20 - 13944362/10500305a^19 + 11685678/10500305a^18 + 37279309/10500305a^17 - 25496306/10500305a^16 - 1446397/256105a^15 + 17170701/10500305a^14 + 87378016/10500305a^13 - 48064612/10500305a^12 - 32447/5371a^11 + 868456/91307a^10 + 2248914/617665a^9 - 98757302/10500305a^8 - 84029782/10500305a^7 + 98462298/10500305a^6 + 39632459/10500305a^5 - 22331391/10500305a^4 + 20749568/10500305a^3 - 88229969/10500305a^2 + 44008931/10500305a + 10958273/10500305, 39908/256105a^21 - 16896/51221a^20 - 154772/256105a^19 + 304058/256105a^18 + 474399/256105a^17 - 724709/256105a^16 - 898213/256105a^15 + 827113/256105a^14 + 1633928/256105a^13 - 236709/51221a^12 - 49624/11135a^11 + 71981/11135a^10 + 160553/256105a^9 - 2265304/256105a^8 - 1020339/256105a^7 + 2921459/256105a^6 + 1141853/256105a^5 - 161971/51221a^4 - 495828/256105a^3 - 1558554/256105a^2 + 1150427/256105a - 13862/256105, 2438466/10500305a^21 - 1542014/10500305a^20 - 2144683/2100061a^19 + 3955766/10500305a^18 + 30317498/10500305a^17 - 4973409/10500305a^16 - 1253921/256105a^15 - 10756942/10500305a^14 + 69332872/10500305a^13 + 3746609/10500305a^12 - 2404614/456535a^11 + 1816794/456535a^10 + 50234188/10500305a^9 - 66583159/10500305a^8 - 105033896/10500305a^7 + 37341648/10500305a^6 + 77520432/10500305a^5 + 23258677/10500305a^4 + 4277823/2100061a^3 - 51054467/10500305a^2 - 2828146/10500305a + 1480867/2100061, 1336584/2100061a^21 - 581665/2100061a^20 - 5157150/2100061a^19 + 3919549/10500305a^18 + 65836532/10500305a^17 + 533006/10500305a^16 - 2298521/256105a^15 - 43559421/10500305a^14 + 110986461/10500305a^13 - 3725886/10500305a^12 - 3167832/456535a^11 + 4800808/456535a^10 + 21915646/2100061a^9 - 76063232/10500305a^8 - 209481306/10500305a^7 - 2932843/10500305a^6 + 44869514/10500305a^5 + 60354603/10500305a^4 + 6485542/617665a^3 - 92764027/10500305a^2 + 31762123/10500305a - 14070517/10500305, 5121419/10500305a^21 - 5597513/10500305a^20 - 4259120/2100061a^19 + 16427376/10500305a^18 + 59542311/10500305a^17 - 31798863/10500305a^16 - 2420979/256105a^15 + 8304686/10500305a^14 + 142243064/10500305a^13 - 34531681/10500305a^12 - 3953123/456535a^11 + 5361318/456535a^10 + 59142078/10500305a^9 - 160261273/10500305a^8 - 175465567/10500305a^7 + 23689130/2100061a^6 + 6461427/617665a^5 + 34611199/10500305a^4 + 48285528/10500305a^3 - 127452284/10500305a^2 + 24562383/10500305a - 1865279/10500305, 76166/2100061a^21 - 1587451/10500305a^20 - 1676287/10500305a^19 + 1222311/2100061a^18 + 6146097/10500305a^17 - 14414427/10500305a^16 - 345463/256105a^15 + 17097167/10500305a^14 + 29431579/10500305a^13 - 17113603/10500305a^12 - 951616/456535a^11 + 902191/456535a^10 - 3730372/10500305a^9 - 41378741/10500305a^8 - 12431061/10500305a^7 + 60973998/10500305a^6 + 35245498/10500305a^5 - 26633913/10500305a^4 - 26795514/10500305a^3 - 16950863/10500305a^2 + 18863862/10500305a + 1680846/10500305 (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:	$ 138999.076948 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 138999.076948 \cdot 1}{6\cdot\sqrt{2988811416117414420033765003}}\cr\approx \mathstrut & 0.255323013187 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 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^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 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^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 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^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 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

$D_{22}$ (as 22T3):

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 44

The 14 conjugacy class representatives for $D_{22}$

Character table for $D_{22}$

Intermediate fields

$\Q(\sqrt{-3}) $, 11.1.129891985607.1

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

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

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

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

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

Sibling fields

Galois closure:	deg 44
Degree 22 sibling:	22.2.499131506491608208145638755501.1
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	$22$	R	${\href{/padicField/5.2.0.1}{2} }^{11}$	${\href{/padicField/7.11.0.1}{11} }^{2}$	$22$	${\href{/padicField/13.2.0.1}{2} }^{10}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.2.0.1}{2} }^{11}$	${\href{/padicField/19.11.0.1}{11} }^{2}$	${\href{/padicField/23.2.0.1}{2} }^{11}$	$22$	${\href{/padicField/31.11.0.1}{11} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{10}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{11}$	${\href{/padicField/43.2.0.1}{2} }^{10}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	$22$	${\href{/padicField/53.2.0.1}{2} }^{11}$	${\href{/padicField/59.2.0.1}{2} }^{11}$

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
$3$ 3	3.22.11.2	$x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$	$2$	$11$	$11$	22T1	$[\ ]_{2}^{11}$
$167$ 167	167.2.0.1	$x^{2} + 166 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	167.4.2.1	$x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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