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Properties

Label 21.3.20575045138...8087.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 769^{18}$
Root discriminant $1438.06$
Ramified primes $7, 769$
Class number Not computed
Class group Not computed
Galois group $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6295917838747831, -1410935175422024, -3177193347653934, -383224347675253, 56176184509851, 42625508333359, 23232341812268, -6211611183364, 126081016179, -68615906434, 29182239326, -5190026793, 728885410, -38250611, 519746, 197181, -66140, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 66140*x^16 + 197181*x^15 + 519746*x^14 - 38250611*x^13 + 728885410*x^12 - 5190026793*x^11 + 29182239326*x^10 - 68615906434*x^9 + 126081016179*x^8 - 6211611183364*x^7 + 23232341812268*x^6 + 42625508333359*x^5 + 56176184509851*x^4 - 383224347675253*x^3 - 3177193347653934*x^2 - 1410935175422024*x + 6295917838747831)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 66140*x^16 + 197181*x^15 + 519746*x^14 - 38250611*x^13 + 728885410*x^12 - 5190026793*x^11 + 29182239326*x^10 - 68615906434*x^9 + 126081016179*x^8 - 6211611183364*x^7 + 23232341812268*x^6 + 42625508333359*x^5 + 56176184509851*x^4 - 383224347675253*x^3 - 3177193347653934*x^2 - 1410935175422024*x + 6295917838747831, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 66140 x^{16} + 197181 x^{15} + 519746 x^{14} - 38250611 x^{13} + 728885410 x^{12} - 5190026793 x^{11} + 29182239326 x^{10} - 68615906434 x^{9} + 126081016179 x^{8} - 6211611183364 x^{7} + 23232341812268 x^{6} + 42625508333359 x^{5} + 56176184509851 x^{4} - 383224347675253 x^{3} - 3177193347653934 x^{2} - 1410935175422024 x + 6295917838747831 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2057504513883865290828006783622225388610315551093969843772003838087=-\,7^{17}\cdot 769^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1438.06$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 769$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{8} + \frac{3}{16} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{5}{16}$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{3}{32} a^{14} - \frac{3}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{3}{16} a^{10} + \frac{5}{32} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{5}{32} a^{6} - \frac{13}{32} a^{5} + \frac{5}{32} a^{4} + \frac{5}{16} a^{2} + \frac{7}{32} a - \frac{3}{32}$, $\frac{1}{224} a^{17} + \frac{1}{28} a^{14} + \frac{11}{112} a^{13} + \frac{3}{28} a^{12} + \frac{5}{224} a^{11} - \frac{25}{224} a^{10} + \frac{3}{32} a^{9} - \frac{3}{112} a^{8} + \frac{7}{32} a^{7} + \frac{1}{14} a^{6} + \frac{11}{56} a^{5} + \frac{83}{224} a^{4} + \frac{39}{112} a^{3} + \frac{29}{224} a^{2} + \frac{3}{8} a + \frac{9}{32}$, $\frac{1}{1205792} a^{18} + \frac{275}{172256} a^{17} - \frac{1719}{172256} a^{16} - \frac{12235}{1205792} a^{15} + \frac{107899}{1205792} a^{14} + \frac{146933}{1205792} a^{13} + \frac{21055}{602896} a^{12} + \frac{134907}{1205792} a^{11} + \frac{4377}{86128} a^{10} - \frac{68981}{602896} a^{9} + \frac{39405}{172256} a^{8} - \frac{281867}{1205792} a^{7} - \frac{111683}{1205792} a^{6} + \frac{93721}{301448} a^{5} - \frac{80337}{301448} a^{4} + \frac{566763}{1205792} a^{3} + \frac{12245}{172256} a^{2} + \frac{501}{6152} a + \frac{3621}{12304}$, $\frac{1}{8440544} a^{19} + \frac{1823}{1205792} a^{17} + \frac{3789}{2110136} a^{16} - \frac{19689}{2110136} a^{15} - \frac{210169}{2110136} a^{14} - \frac{950035}{8440544} a^{13} + \frac{991651}{8440544} a^{12} - \frac{28711}{301448} a^{11} - \frac{495711}{8440544} a^{10} + \frac{135}{86128} a^{9} - \frac{117731}{1055068} a^{8} - \frac{1669029}{8440544} a^{7} + \frac{206149}{8440544} a^{6} + \frac{638033}{4220272} a^{5} + \frac{2085031}{4220272} a^{4} + \frac{13147}{86128} a^{3} - \frac{3715}{10766} a^{2} - \frac{3321}{10766} a + \frac{5543}{24608}$, $\frac{1}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{20} - \frac{55565493552226015453978485157618946154591372144266852939064375684591684112977146552673122560888182876092393332825160550581059639}{1829968077084794332563336375599768986324338672883779636330946662574647738372565523780818074191180796435724922912246694397900774009697824} a^{19} - \frac{52600627923338045500933419685368837960887964380156572339007449002429459450369779454952483039295371968314844234719401881663400635}{522848022024226952160953250171362567521239620823937038951699046449899353820733006794519449768908798981635692260641912685114506859913664} a^{18} + \frac{6826892750594886899823696224084879966911275419578649230354788642324119185208228723156981478265721591933727765707798358555761916885663}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{17} + \frac{12371740712216750554313992512729021350466503963139118155819244210109722184993852525728103102467973768238090090964403620168493255064255}{1829968077084794332563336375599768986324338672883779636330946662574647738372565523780818074191180796435724922912246694397900774009697824} a^{16} - \frac{3588065428542508049431382420522246342028616244224678231323191501181425999930402639480208002981078461268938732913545192820569764945093}{261424011012113476080476625085681283760619810411968519475849523224949676910366503397259724884454399490817846130320956342557253429956832} a^{15} + \frac{114069113646983035981768814353674402997396475850533796407729450031194944313238387499687750526465630079549368727297799537738430234067367}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{14} + \frac{364219115619724105217967157685085659291485982188662561432637684363375000511332818609821102422735158290679782649184910132805473578899885}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{13} + \frac{202336100986662130725274371467440609971244616329273983809057888392214600270572471708878926188359106355217086946588122175281718237566297}{1829968077084794332563336375599768986324338672883779636330946662574647738372565523780818074191180796435724922912246694397900774009697824} a^{12} + \frac{16859461800266088061975984636130559378588083850854433358806626374313786821159799210737997616970358545722444067372017499037532464978403}{140766775160368794812564336584597614332641436375675356640842050967280595259428117213909082630090830495055763300942053415223136462284448} a^{11} + \frac{232803054623768965860115640873803108544131619886568708950962169633863906870010576276906735659798559800760649670128382412223773148058709}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{10} - \frac{622499554948760006870795503897881411127732386698643707220111872527755340715015213560997540121625617085655685610447535703612420565355833}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{9} + \frac{6255425033114649189875291158386975878546910569449672733450078386026473260768057450725772664605333264420522535062275272884060656353871}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{8} + \frac{14288555519332631013530689794587107409998553335813617069327264310422925992483531592195094810980448781952719897666343937497935515226875}{130712005506056738040238312542840641880309905205984259737924761612474838455183251698629862442227199745408923065160478171278626714978416} a^{7} + \frac{756856578336761584787868573006082501608722920770191095105926110160847538286976855535454940405589407202391399548847822539569666719173085}{1829968077084794332563336375599768986324338672883779636330946662574647738372565523780818074191180796435724922912246694397900774009697824} a^{6} + \frac{816548838692476461592380521723989942743802259339137875385724876034120413564755613661678891170318523454030489654110911296062137695224693}{1829968077084794332563336375599768986324338672883779636330946662574647738372565523780818074191180796435724922912246694397900774009697824} a^{5} + \frac{913192583786057275249044814313778578659172445018787789463276204319483265068916619459281120924524630413249913264526718755759619638345507}{3659936154169588665126672751199537972648677345767559272661893325149295476745131047561636148382361592871449845824493388795801548019395648} a^{4} + \frac{54734062051811005962104882621462463452278565709094615255468201037473705748727255803390304940290377233191228113854028087144084240629405}{130712005506056738040238312542840641880309905205984259737924761612474838455183251698629862442227199745408923065160478171278626714978416} a^{3} + \frac{32672830213620543330872652486393301956313408791392713192961987760604952091104048649292848726593487957104893662511468618007367305150705}{74692574574889564594421892881623223931605660117705291278814149492842764831533286684931349966986971283090813180091701812159215265701952} a^{2} + \frac{36948313287143018261505449697403308000161632850573897435676886964896626002131642118770678777237923865823335375281833622412099881506489}{74692574574889564594421892881623223931605660117705291278814149492842764831533286684931349966986971283090813180091701812159215265701952} a + \frac{15619298818248062274286187134116195770514227726422086443773768207497485814424193813939871210355309380946021990684453721096960149239}{50570463490108032900759575410713083230606404954438247311316282662723605166914886042607549063633697551178614204530603799701567546176}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), Deg 7

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
769Data not computed