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Properties

Label 21.3.69865929557...4103.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 1063^{18}$
Root discriminant $1898.01$
Ramified primes $7, 1063$
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![135973981979723027, -90692100544127768, -25119733760908906, 2913800887665479, -1511275454987137, 403041892117935, 146378771353312, -29032774648716, -157374528081, 36609721794, 14713958422, -8271270553, 1782947986, -82474679, -1885902, 1091081, -150952, 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 - 150952*x^16 + 1091081*x^15 - 1885902*x^14 - 82474679*x^13 + 1782947986*x^12 - 8271270553*x^11 + 14713958422*x^10 + 36609721794*x^9 - 157374528081*x^8 - 29032774648716*x^7 + 146378771353312*x^6 + 403041892117935*x^5 - 1511275454987137*x^4 + 2913800887665479*x^3 - 25119733760908906*x^2 - 90692100544127768*x + 135973981979723027)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 150952*x^16 + 1091081*x^15 - 1885902*x^14 - 82474679*x^13 + 1782947986*x^12 - 8271270553*x^11 + 14713958422*x^10 + 36609721794*x^9 - 157374528081*x^8 - 29032774648716*x^7 + 146378771353312*x^6 + 403041892117935*x^5 - 1511275454987137*x^4 + 2913800887665479*x^3 - 25119733760908906*x^2 - 90692100544127768*x + 135973981979723027, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 150952 x^{16} + 1091081 x^{15} - 1885902 x^{14} - 82474679 x^{13} + 1782947986 x^{12} - 8271270553 x^{11} + 14713958422 x^{10} + 36609721794 x^{9} - 157374528081 x^{8} - 29032774648716 x^{7} + 146378771353312 x^{6} + 403041892117935 x^{5} - 1511275454987137 x^{4} + 2913800887665479 x^{3} - 25119733760908906 x^{2} - 90692100544127768 x + 135973981979723027 \)

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:  \(-698659295573481696443702685323246063670561579560848034212460416254103=-\,7^{17}\cdot 1063^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1898.01$
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:  $7, 1063$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{8} a^{9} - \frac{3}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{7}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a - \frac{1}{16}$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{3}{32} a^{14} - \frac{3}{32} a^{13} + \frac{1}{32} a^{12} - \frac{3}{32} a^{11} - \frac{1}{16} a^{10} + \frac{3}{32} a^{9} - \frac{3}{16} a^{8} - \frac{3}{16} a^{7} - \frac{9}{32} a^{6} - \frac{3}{32} a^{5} - \frac{3}{32} a^{4} - \frac{1}{8} a^{3} - \frac{7}{32} a - \frac{9}{32}$, $\frac{1}{32} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{16} a^{12} + \frac{1}{32} a^{11} - \frac{7}{32} a^{10} - \frac{7}{32} a^{9} - \frac{3}{16} a^{8} + \frac{7}{32} a^{7} + \frac{5}{16} a^{6} - \frac{1}{16} a^{5} + \frac{3}{32} a^{4} - \frac{1}{2} a^{3} + \frac{13}{32} a^{2} - \frac{3}{8} a + \frac{9}{32}$, $\frac{1}{2143008} a^{18} + \frac{20803}{2143008} a^{17} + \frac{21241}{2143008} a^{16} - \frac{21983}{2143008} a^{15} + \frac{209239}{2143008} a^{14} - \frac{12203}{2143008} a^{13} + \frac{55603}{535752} a^{12} + \frac{28529}{306144} a^{11} - \frac{23189}{153072} a^{10} + \frac{13297}{535752} a^{9} + \frac{325645}{2143008} a^{8} + \frac{490183}{2143008} a^{7} + \frac{835505}{2143008} a^{6} - \frac{18775}{153072} a^{5} - \frac{18901}{38268} a^{4} + \frac{94075}{306144} a^{3} - \frac{152623}{306144} a^{2} - \frac{20323}{51024} a - \frac{34133}{76536}$, $\frac{1}{49289184} a^{19} + \frac{1}{7041312} a^{18} - \frac{78263}{12322296} a^{17} + \frac{73351}{7041312} a^{16} - \frac{30563}{49289184} a^{15} + \frac{195811}{7041312} a^{14} - \frac{577945}{24644592} a^{13} + \frac{468581}{49289184} a^{12} - \frac{478039}{7041312} a^{11} - \frac{758021}{49289184} a^{10} - \frac{97343}{1760328} a^{9} + \frac{10193455}{49289184} a^{8} + \frac{330293}{1760328} a^{7} - \frac{5315503}{24644592} a^{6} + \frac{1116767}{3520656} a^{5} + \frac{355193}{880164} a^{4} + \frac{1828577}{7041312} a^{3} + \frac{334051}{782368} a^{2} - \frac{419677}{880164} a - \frac{320063}{2347104}$, $\frac{1}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{20} + \frac{26584669680593962819600729658290231327637675079259138552489406682602253838571316648156154987552086181454322692318502122480548032842138539}{2741093022516417829631593765680041536678586251884707409736035796342674537026371895624697717271372695990902630450338696619829029265175568661162528} a^{19} - \frac{19538284054905293292857828965327854014050495138311541766302808679975645717071265195788809205950902011137081962564658309394677199943721722629}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{18} + \frac{1878255046436317175538916024260412887610951218712465526407045190425327788806944455569921572585427497628433156508320797908834344814367117977440553}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{17} - \frac{11498461176842196605684168347515548637340030574509978615089403374237738893723043418216230071822638922007913483233881221825750570385977772683085}{7880642439734701260190832076330119417950935474168533802991102914485189293950819199921005937155196500973845062544723752782008459137379759900842268} a^{16} - \frac{219707897384084723331839812241789614720499841254318673886808758580385460361024284484447112207401765384223945932980640256588588611521880285149073}{7880642439734701260190832076330119417950935474168533802991102914485189293950819199921005937155196500973845062544723752782008459137379759900842268} a^{15} + \frac{164889170089825702347182576361903608509867472451644671202122499716727349314307530031124170143016746719121602468594435427999016998228200259523805}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{14} - \frac{8001993980494895495105120474771817291496762897856638012456477580917744595193331899554321776868668529785514971640584765583776093307573725373833057}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{13} - \frac{14488949436910581902685447388963434707386338434434941904584307521209416076072939029364211567818772092779151897232496257661430590852787255783323}{281451515704810759292529717011789979212533409791733350106825104088756760498243542854321640612685589320494466519454419742214587826334991425030081} a^{12} + \frac{362780472477845622472762630607962355147705787688703542390660256744130045050170319709636860883979563922525023946315829851905128624615533466690381}{4849626116759816160117435123895458103354421830257559263379140255067808796277427199951388269018582462137750807719830001712005205623002929169749088} a^{11} + \frac{1936495055287320545989222293273426848659391922272200463212806288474993737215822976305939912007276630323895561844385345623886011878084088031106287}{9699252233519632320234870247790916206708843660515118526758280510135617592554854399902776538037164924275501615439660003424010411246005858339498176} a^{10} - \frac{2364126320345683595646210856567906070110188930442067751066798737228970611667669233545961592450512695845852667125863325633683681099169599562204389}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{9} - \frac{1621890102435931381826825566650457739761342725019826376812083265246092352103847769518240321322796563860059207810964144155028253316662837200892437}{126090279035755220163053313221281910687214967586696540847857646631763028703213107198736094994483144015581521000715580044512135346198076158413476288} a^{8} + \frac{14867741232158282415302999826252819913031039927412358102453455949529235678187916668731212119222602689073359274663305895963576293437877134664651297}{63045139517877610081526656610640955343607483793348270423928823315881514351606553599368047497241572007790760500357790022256067673099038079206738144} a^{7} - \frac{275154093636953541627174294511333644727161798140144261570427969996373959236608782742179712547522016896151494169842873311028198010974243785596955}{562903031409621518585059434023579958425066819583466700213650208177513520996487085708643281225371178640988933038908839484429175652669982850060162} a^{6} + \frac{535381053004773767800569085032722029984432986039339303227441534208374593531275130094975681023007600460031669658898736056663467960783160828985345}{4503224251276972148680475472188639667400534556667733601709201665420108167971896685669146249802969429127911464311270715875433405221359862800481296} a^{5} - \frac{8633788050958092301602384441114019443292881999828490662167238029284564003944193474057144217506235057172626266554802932349191560692465950138326527}{18012897005107888594721901888754558669602138226670934406836806661680432671887586742676584999211877716511645857245082863501733620885439451201925184} a^{4} + \frac{668769173054950901778553325275567087771551201129136945336534971133964041880319933332125481777167701259846308432057575541146827649635376934537503}{1501074750425657382893491824062879889133511518889244533903067221806702722657298895223048749934323143042637154770423571958477801740453287600160432} a^{3} - \frac{5620279572775683276053271407154766450660605892201855372782927312920820447983539838944592138846550008880999799972860071597552171241630109234686899}{18012897005107888594721901888754558669602138226670934406836806661680432671887586742676584999211877716511645857245082863501733620885439451201925184} a^{2} - \frac{74192838954446732036085062717714511391069486082149048576924814506033107394751293688307107396505230692469233598790865559601364978110382322790005}{6004299001702629531573967296251519556534046075556978135612268887226810890629195580892194999737292572170548619081694287833911206961813150400641728} a - \frac{2358007470394720967032675526230890605801262148855697465753081015957453648627464061988757915178769613956930514678016333675099115890888924951167429}{6004299001702629531573967296251519556534046075556978135612268887226810890629195580892194999737292572170548619081694287833911206961813150400641728}$

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}$
1063Data not computed