Properties

Label 22.0.4514211640352695524009371.1
Degree 22
Signature $[0, 11]$
Discriminant $-\,11^{4}\cdot 181^{2}\cdot 101771\cdot 304099^{2}$
Ramified primes $11, 181, 101771, 304099$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T53

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 5, 12, 7, -33, -30, -3, 57, 37, 4, -55, -29, -7, 29, 16, 8, -9, -4, -5, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - 5*x^19 - 4*x^18 - 9*x^17 + 8*x^16 + 16*x^15 + 29*x^14 - 7*x^13 - 29*x^12 - 55*x^11 + 4*x^10 + 37*x^9 + 57*x^8 - 3*x^7 - 30*x^6 - 33*x^5 + 7*x^4 + 12*x^3 + 5*x^2 - 3*x + 1)
gp: K = bnfinit(x^22 + 2*x^20 - 5*x^19 - 4*x^18 - 9*x^17 + 8*x^16 + 16*x^15 + 29*x^14 - 7*x^13 - 29*x^12 - 55*x^11 + 4*x^10 + 37*x^9 + 57*x^8 - 3*x^7 - 30*x^6 - 33*x^5 + 7*x^4 + 12*x^3 + 5*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut +\mathstrut 2 x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut -\mathstrut 4 x^{18} \) \(\mathstrut -\mathstrut 9 x^{17} \) \(\mathstrut +\mathstrut 8 x^{16} \) \(\mathstrut +\mathstrut 16 x^{15} \) \(\mathstrut +\mathstrut 29 x^{14} \) \(\mathstrut -\mathstrut 7 x^{13} \) \(\mathstrut -\mathstrut 29 x^{12} \) \(\mathstrut -\mathstrut 55 x^{11} \) \(\mathstrut +\mathstrut 4 x^{10} \) \(\mathstrut +\mathstrut 37 x^{9} \) \(\mathstrut +\mathstrut 57 x^{8} \) \(\mathstrut -\mathstrut 3 x^{7} \) \(\mathstrut -\mathstrut 30 x^{6} \) \(\mathstrut -\mathstrut 33 x^{5} \) \(\mathstrut +\mathstrut 7 x^{4} \) \(\mathstrut +\mathstrut 12 x^{3} \) \(\mathstrut +\mathstrut 5 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $22$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 11]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-4514211640352695524009371=-\,11^{4}\cdot 181^{2}\cdot 101771\cdot 304099^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $11, 181, 101771, 304099$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{261110408789} a^{21} - \frac{120216289623}{261110408789} a^{20} + \frac{75317901722}{261110408789} a^{19} - \frac{15219657848}{261110408789} a^{18} + \frac{35198274088}{261110408789} a^{17} - \frac{67545294485}{261110408789} a^{16} + \frac{126318189061}{261110408789} a^{15} - \frac{92059699310}{261110408789} a^{14} - \frac{82219606505}{261110408789} a^{13} - \frac{19548395803}{261110408789} a^{12} - \frac{86392999637}{261110408789} a^{11} - \frac{38725479726}{261110408789} a^{10} + \frac{22921238314}{261110408789} a^{9} + \frac{77178351231}{261110408789} a^{8} - \frac{64764051861}{261110408789} a^{7} + \frac{57492278861}{261110408789} a^{6} + \frac{108489105613}{261110408789} a^{5} + \frac{101916220376}{261110408789} a^{4} - \frac{88485812731}{261110408789} a^{3} - \frac{110654111433}{261110408789} a^{2} - \frac{67141396227}{261110408789} a - \frac{42477049607}{261110408789}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, order 1 (assuming GRH)

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $10$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{23666157064}{261110408789} a^{21} + \frac{27315979445}{261110408789} a^{20} + \frac{109570962155}{261110408789} a^{19} - \frac{48217660640}{261110408789} a^{18} - \frac{17476201061}{261110408789} a^{17} - \frac{551560410800}{261110408789} a^{16} - \frac{149201390875}{261110408789} a^{15} - \frac{294768639260}{261110408789} a^{14} + \frac{1023329904632}{261110408789} a^{13} + \frac{680468452690}{261110408789} a^{12} + \frac{976413610442}{261110408789} a^{11} - \frac{1086543752928}{261110408789} a^{10} - \frac{477736671013}{261110408789} a^{9} - \frac{1366308767414}{261110408789} a^{8} + \frac{1114116990087}{261110408789} a^{7} - \frac{308446381108}{261110408789} a^{6} + \frac{789236795823}{261110408789} a^{5} - \frac{591809340034}{261110408789} a^{4} + \frac{629045555375}{261110408789} a^{3} + \frac{36602757869}{261110408789} a^{2} + \frac{181833040850}{261110408789} a - \frac{337100631656}{261110408789} \),  \( \frac{94906960682}{261110408789} a^{21} + \frac{30431727252}{261110408789} a^{20} + \frac{210402721042}{261110408789} a^{19} - \frac{422451924550}{261110408789} a^{18} - \frac{459751146985}{261110408789} a^{17} - \frac{1088425742730}{261110408789} a^{16} + \frac{531123663808}{261110408789} a^{15} + \frac{1500049758984}{261110408789} a^{14} + \frac{3364530852799}{261110408789} a^{13} + \frac{145057839476}{261110408789} a^{12} - \frac{2547946533411}{261110408789} a^{11} - \frac{6193550987644}{261110408789} a^{10} - \frac{1005231452314}{261110408789} a^{9} + \frac{3095827018420}{261110408789} a^{8} + \frac{6915057226484}{261110408789} a^{7} + \frac{980913422247}{261110408789} a^{6} - \frac{2598328295341}{261110408789} a^{5} - \frac{4326567931986}{261110408789} a^{4} + \frac{79767528335}{261110408789} a^{3} + \frac{1322297440599}{261110408789} a^{2} + \frac{947579777183}{261110408789} a - \frac{443905173075}{261110408789} \),  \( \frac{57929083768}{261110408789} a^{21} + \frac{80335933754}{261110408789} a^{20} + \frac{97155327998}{261110408789} a^{19} - \frac{123411106654}{261110408789} a^{18} - \frac{653798859128}{261110408789} a^{17} - \frac{765002616071}{261110408789} a^{16} - \frac{187151772297}{261110408789} a^{15} + \frac{1563519030889}{261110408789} a^{14} + \frac{2809512470880}{261110408789} a^{13} + \frac{1606725702735}{261110408789} a^{12} - \frac{2268589407257}{261110408789} a^{11} - \frac{5052274076984}{261110408789} a^{10} - \frac{3506260461008}{261110408789} a^{9} + \frac{2530230122909}{261110408789} a^{8} + \frac{5349792918563}{261110408789} a^{7} + \frac{3191161420128}{261110408789} a^{6} - \frac{1605442691490}{261110408789} a^{5} - \frac{3411229964209}{261110408789} a^{4} - \frac{1148577333701}{261110408789} a^{3} + \frac{838200896748}{261110408789} a^{2} + \frac{645322619527}{261110408789} a - \frac{72722926493}{261110408789} \),  \( \frac{101146482035}{261110408789} a^{21} + \frac{15708708644}{261110408789} a^{20} + \frac{171496539033}{261110408789} a^{19} - \frac{461628381403}{261110408789} a^{18} - \frac{569120917110}{261110408789} a^{17} - \frac{791071272574}{261110408789} a^{16} + \frac{678720053850}{261110408789} a^{15} + \frac{2082978167399}{261110408789} a^{14} + \frac{2930362021059}{261110408789} a^{13} - \frac{466281591740}{261110408789} a^{12} - \frac{3823642606088}{261110408789} a^{11} - \frac{5819601603325}{261110408789} a^{10} - \frac{203922586392}{261110408789} a^{9} + \frac{5151313457634}{261110408789} a^{8} + \frac{6063002495638}{261110408789} a^{7} + \frac{468201754708}{261110408789} a^{6} - \frac{4372425451419}{261110408789} a^{5} - \frac{3726696859630}{261110408789} a^{4} + \frac{293032687752}{261110408789} a^{3} + \frac{1751695177439}{261110408789} a^{2} + \frac{612028064380}{261110408789} a - \frac{111430747071}{261110408789} \),  \( \frac{27315979445}{261110408789} a^{21} + \frac{62238648027}{261110408789} a^{20} + \frac{70113124680}{261110408789} a^{19} + \frac{77188427195}{261110408789} a^{18} - \frac{338564997224}{261110408789} a^{17} - \frac{338530647387}{261110408789} a^{16} - \frac{673427152284}{261110408789} a^{15} + \frac{337011349776}{261110408789} a^{14} + \frac{846131552138}{261110408789} a^{13} + \frac{1662732165298}{261110408789} a^{12} + \frac{215094885592}{261110408789} a^{11} - \frac{572401299269}{261110408789} a^{10} - \frac{2241956578782}{261110408789} a^{9} - \frac{234853962561}{261110408789} a^{8} - \frac{237447909916}{261110408789} a^{7} + \frac{1499221507743}{261110408789} a^{6} + \frac{189173843078}{261110408789} a^{5} + \frac{463382455927}{261110408789} a^{4} - \frac{247391126899}{261110408789} a^{3} + \frac{63502255530}{261110408789} a^{2} - \frac{266102160464}{261110408789} a - \frac{23666157064}{261110408789} \),  \( \frac{7957448420}{261110408789} a^{21} + \frac{58112404482}{261110408789} a^{20} + \frac{65466933383}{261110408789} a^{19} + \frac{116317328330}{261110408789} a^{18} - \frac{136723266802}{261110408789} a^{17} - \frac{421108608370}{261110408789} a^{16} - \frac{613835845714}{261110408789} a^{15} - \frac{291882681304}{261110408789} a^{14} + \frac{781586122127}{261110408789} a^{13} + \frac{1570863079763}{261110408789} a^{12} + \frac{1232958701842}{261110408789} a^{11} - \frac{478035238396}{261110408789} a^{10} - \frac{1886630293352}{261110408789} a^{9} - \frac{1663961787057}{261110408789} a^{8} + \frac{342475789274}{261110408789} a^{7} + \frac{1029584609261}{261110408789} a^{6} + \frac{1178099748298}{261110408789} a^{5} - \frac{176816653541}{261110408789} a^{4} + \frac{91108162356}{261110408789} a^{3} - \frac{330803305125}{261110408789} a^{2} - \frac{10175658184}{261110408789} a - \frac{235954149621}{261110408789} \),  \( \frac{163421151847}{261110408789} a^{21} + \frac{111844675843}{261110408789} a^{20} + \frac{450826383860}{261110408789} a^{19} - \frac{508331490833}{261110408789} a^{18} - \frac{884334135603}{261110408789} a^{17} - \frac{2292053800489}{261110408789} a^{16} - \frac{412229574695}{261110408789} a^{15} + \frac{1854938905082}{261110408789} a^{14} + \frac{6135321365292}{261110408789} a^{13} + \frac{3597331794639}{261110408789} a^{12} - \frac{1078242053299}{261110408789} a^{11} - \frac{9433184262940}{261110408789} a^{10} - \frac{6243476801544}{261110408789} a^{9} - \frac{118432714461}{261110408789} a^{8} + \frac{8734003575598}{261110408789} a^{7} + \frac{5672208119788}{261110408789} a^{6} + \frac{24414741795}{261110408789} a^{5} - \frac{4762274436603}{261110408789} a^{4} - \frac{2183023287176}{261110408789} a^{3} + \frac{444149649124}{261110408789} a^{2} + \frac{908847359397}{261110408789} a - \frac{124933138681}{261110408789} \),  \( \frac{26092687701}{261110408789} a^{21} + \frac{33908951342}{261110408789} a^{20} + \frac{145869414827}{261110408789} a^{19} + \frac{13831713231}{261110408789} a^{18} - \frac{31467680563}{261110408789} a^{17} - \frac{601267557923}{261110408789} a^{16} - \frac{644377407754}{261110408789} a^{15} - \frac{494347565749}{261110408789} a^{14} + \frac{982195919431}{261110408789} a^{13} + \frac{1833621008836}{261110408789} a^{12} + \frac{2090368315385}{261110408789} a^{11} - \frac{542955562293}{261110408789} a^{10} - \frac{2480421016645}{261110408789} a^{9} - \frac{3288442656219}{261110408789} a^{8} - \frac{141449750029}{261110408789} a^{7} + \frac{2316158159416}{261110408789} a^{6} + \frac{2711625608328}{261110408789} a^{5} + \frac{240954660948}{261110408789} a^{4} - \frac{978618392865}{261110408789} a^{3} - \frac{884451626291}{261110408789} a^{2} - \frac{23345876491}{261110408789} a + \frac{195559691459}{261110408789} \),  \( \frac{37599162332}{261110408789} a^{21} + \frac{90746352931}{261110408789} a^{20} + \frac{37038896627}{261110408789} a^{19} + \frac{29424983875}{261110408789} a^{18} - \frac{711909000749}{261110408789} a^{17} - \frac{466638898434}{261110408789} a^{16} - \frac{615868683316}{261110408789} a^{15} + \frac{1572928539383}{261110408789} a^{14} + \frac{2156741838298}{261110408789} a^{13} + \frac{2346568649311}{261110408789} a^{12} - \frac{1970511348179}{261110408789} a^{11} - \frac{3836590335162}{261110408789} a^{10} - \frac{4812645865272}{261110408789} a^{9} + \frac{1965921514629}{261110408789} a^{8} + \frac{3605997864981}{261110408789} a^{7} + \frac{4758092557467}{261110408789} a^{6} - \frac{1056044516605}{261110408789} a^{5} - \frac{2231066480308}{261110408789} a^{4} - \frac{2018013872789}{261110408789} a^{3} + \frac{511233657546}{261110408789} a^{2} + \frac{537602668688}{261110408789} a + \frac{202561136690}{261110408789} \),  \( \frac{176335306185}{261110408789} a^{21} - \frac{32641373236}{261110408789} a^{20} + \frac{441138027073}{261110408789} a^{19} - \frac{941393364311}{261110408789} a^{18} - \frac{371482953907}{261110408789} a^{17} - \frac{1832405993323}{261110408789} a^{16} + \frac{1273189132764}{261110408789} a^{15} + \frac{1915692378004}{261110408789} a^{14} + \frac{4917568806128}{261110408789} a^{13} - \frac{665587060608}{261110408789} a^{12} - \frac{2754436878894}{261110408789} a^{11} - \frac{8707576483375}{261110408789} a^{10} + \frac{278418070549}{261110408789} a^{9} + \frac{2610145159315}{261110408789} a^{8} + \frac{8300284637636}{261110408789} a^{7} + \frac{281709858646}{261110408789} a^{6} - \frac{2306915797478}{261110408789} a^{5} - \frac{4082830554572}{261110408789} a^{4} + \frac{248592604903}{261110408789} a^{3} + \frac{903318531275}{261110408789} a^{2} + \frac{572247701251}{261110408789} a - \frac{133044547652}{261110408789} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1214.51296657 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

22T53:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 81749606400
Conjugacy class representatives for 22T53
Character table for 22T53

Intermediate fields

Deg 11

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

Sibling fields

Degree 22 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $18{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$11$11.7.0.1$x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
11.7.0.1$x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
181Data not computed
101771Data not computed
304099Data not computed