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Properties

Label 21.9.66240253244...8544.1
Degree $21$
Signature $[9, 6]$
Discriminant $2^{18}\cdot 7^{38}\cdot 29^{6}\cdot 83^{6}$
Root discriminant $566.74$
Ramified primes $2, 7, 29, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T36

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4099725079976, 13197661027936, 13880363266378, 3195095162105, -2942729744920, -1225931184422, 200192438138, 56822241145, -10104570658, 7280679700, 577718400, -573732572, -13197254, 4202632, -39460, 428680, 0, -5887, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 70*x^19 - 5887*x^17 + 428680*x^15 - 39460*x^14 + 4202632*x^13 - 13197254*x^12 - 573732572*x^11 + 577718400*x^10 + 7280679700*x^9 - 10104570658*x^8 + 56822241145*x^7 + 200192438138*x^6 - 1225931184422*x^5 - 2942729744920*x^4 + 3195095162105*x^3 + 13880363266378*x^2 + 13197661027936*x + 4099725079976)
 
gp: K = bnfinit(x^21 - 70*x^19 - 5887*x^17 + 428680*x^15 - 39460*x^14 + 4202632*x^13 - 13197254*x^12 - 573732572*x^11 + 577718400*x^10 + 7280679700*x^9 - 10104570658*x^8 + 56822241145*x^7 + 200192438138*x^6 - 1225931184422*x^5 - 2942729744920*x^4 + 3195095162105*x^3 + 13880363266378*x^2 + 13197661027936*x + 4099725079976, 1)
 

Normalized defining polynomial

\( x^{21} - 70 x^{19} - 5887 x^{17} + 428680 x^{15} - 39460 x^{14} + 4202632 x^{13} - 13197254 x^{12} - 573732572 x^{11} + 577718400 x^{10} + 7280679700 x^{9} - 10104570658 x^{8} + 56822241145 x^{7} + 200192438138 x^{6} - 1225931184422 x^{5} - 2942729744920 x^{4} + 3195095162105 x^{3} + 13880363266378 x^{2} + 13197661027936 x + 4099725079976 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6624025324499884482282777372790865006831419904797075308544=2^{18}\cdot 7^{38}\cdot 29^{6}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $566.74$
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:  $2, 7, 29, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{25719172} a^{18} + \frac{107687}{443434} a^{17} - \frac{4075527}{25719172} a^{16} - \frac{8666}{221717} a^{15} - \frac{103414}{221717} a^{14} - \frac{20151}{221717} a^{13} - \frac{2712674}{6429793} a^{12} + \frac{874693}{6429793} a^{11} - \frac{54619}{6429793} a^{10} + \frac{4740315}{12859586} a^{9} + \frac{770111}{6429793} a^{8} - \frac{1434599}{12859586} a^{7} + \frac{1851829}{6429793} a^{6} - \frac{2353067}{6429793} a^{5} + \frac{1462113}{25719172} a^{4} + \frac{99366}{221717} a^{3} + \frac{13441}{886868} a^{2} - \frac{155273}{443434} a + \frac{18254}{221717}$, $\frac{1}{25719172} a^{19} - \frac{573777}{25719172} a^{17} - \frac{4405}{443434} a^{16} - \frac{42886}{221717} a^{15} - \frac{61760}{221717} a^{14} + \frac{2232580}{6429793} a^{13} + \frac{3145973}{6429793} a^{12} - \frac{2041380}{6429793} a^{11} - \frac{2588739}{12859586} a^{10} + \frac{2372709}{6429793} a^{9} - \frac{4617117}{12859586} a^{8} - \frac{672041}{6429793} a^{7} - \frac{1537558}{6429793} a^{6} + \frac{6221013}{25719172} a^{5} + \frac{189915}{443434} a^{4} + \frac{256231}{886868} a^{3} + \frac{49562}{221717} a^{2} - \frac{80183}{221717} a + \frac{86290}{221717}$, $\frac{1}{4324507302150824283916244847606803140041037936799879500804782508420263284349722922839989377620031197041543481636} a^{20} + \frac{319711634754471150599512142024298050284108825684148453606190042704734643219009134187054793187622636049}{26051248808137495686242438841004838193018300824095659643402304267591947496082668209879454082048380705069539046} a^{19} + \frac{70767440181843457124686430771054778328402458329170818422818762013976598013949665050406668926878763633469}{4324507302150824283916244847606803140041037936799879500804782508420263284349722922839989377620031197041543481636} a^{18} - \frac{5942402378279388870052973585998577677766581405405330420056408278410676246971881310161624405632035387315840881}{26051248808137495686242438841004838193018300824095659643402304267591947496082668209879454082048380705069539046} a^{17} - \frac{250660275620260167529098725140228372420174695601970756115607368880190123453039426439637113113395519661031279869}{1081126825537706070979061211901700785010259484199969875201195627105065821087430730709997344405007799260385870409} a^{16} - \frac{91878917444702986620785657506217869573588117609136369088889602487462428189749673203537615318429529201820873}{898318924418534334008359960034649592862700028417091711841458767847998189520092007237222554553392438105846174} a^{15} - \frac{63555209638799645773968470258403112990961239507956797445562669306752877431856489657318851595204883526392625307}{1081126825537706070979061211901700785010259484199969875201195627105065821087430730709997344405007799260385870409} a^{14} + \frac{464825820905640216262041805710252675519123654022695926971165409816198927486117173309242170922653674720574090471}{1081126825537706070979061211901700785010259484199969875201195627105065821087430730709997344405007799260385870409} a^{13} + \frac{164946959108463317437124169672210997554603683333436312315514052548992465079150926537020223113813399991511443714}{1081126825537706070979061211901700785010259484199969875201195627105065821087430730709997344405007799260385870409} a^{12} - \frac{65000356520962848582845653311896868702088852865040776623019877535002333378506746500041056282828665526284839511}{2162253651075412141958122423803401570020518968399939750402391254210131642174861461419994688810015598520771740818} a^{11} - \frac{11026724626001240859018207245364318441874113119523404382789102650137634708004130616368708366703914439545312424}{37280235363369174861346938341437958103802051179309306041420538865691924865083818300344736013965786181392616221} a^{10} + \frac{675951930805532340930746286165546638828355671969697609120563971859285836045678498644228661805939136237814462539}{2162253651075412141958122423803401570020518968399939750402391254210131642174861461419994688810015598520771740818} a^{9} + \frac{171750544858853042461682673873078319032801245353372339087917895161664937872403786400853738163260099965883517846}{1081126825537706070979061211901700785010259484199969875201195627105065821087430730709997344405007799260385870409} a^{8} - \frac{366347160104916030999491517084281380019545979767170682197436278655714429394559268809862306967329566392602922701}{1081126825537706070979061211901700785010259484199969875201195627105065821087430730709997344405007799260385870409} a^{7} - \frac{9068715660538865782175306353882658945219847777811897371351455862290787036435590852168733882613813111397480893}{52102497616274991372484877682009676386036601648191319286804608535183894992165336419758908164096761410139078092} a^{6} - \frac{3264910601676473690177110340557982640566900548063216443755150683241936307011285087329069169210975184327111513}{13025624404068747843121219420502419096509150412047829821701152133795973748041334104939727041024190352534769523} a^{5} - \frac{18939933087016399014390376798491650445828567817529988945270054902849922459130652192621259076235701496606294969}{52102497616274991372484877682009676386036601648191319286804608535183894992165336419758908164096761410139078092} a^{4} - \frac{164243416541152978064107315180636175670625750053773235179600380187968711339012528005567649043262765920170163}{449159462209267167004179980017324796431350014208545855920729383923999094760046003618611277276696219052923087} a^{3} - \frac{99111959883849968663428905944070096172457168103004142219627247937174830317443799082737349694166736916913717}{449159462209267167004179980017324796431350014208545855920729383923999094760046003618611277276696219052923087} a^{2} + \frac{334599029627609514182992704343533164622343250402941573266021102226442212037135022426514852006400846938903185}{898318924418534334008359960034649592862700028417091711841458767847998189520092007237222554553392438105846174} a + \frac{170754134224265094995207463163636770600180422286633916508458251526799570682635099722902945147048688067158121}{449159462209267167004179980017324796431350014208545855920729383923999094760046003618611277276696219052923087}$

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

Class group and class number

Trivial group, which has 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:  $14$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1662065239090000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T36:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4116
The 53 conjugacy class representatives for t21n36 are not computed
Character table for t21n36 is not computed

Intermediate fields

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

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

Sibling fields

Degree 28 siblings: data not computed
Degree 42 siblings: 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 R $21$ $21$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $21$ $21$ $21$ R $21$ $21$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $21$ $21$ $21$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$
83Data not computed