↑

Properties

Label 21.3.24344854108...2807.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 601^{18}$
Root discriminant $1164.18$
Ramified primes $7, 601$
Class number Not computed
Class group Not computed
Galois group $F_7$ (as 21T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-84881631489088, -215779097824640, -9595175213344, -19183703267008, -35643871943428, -1175582555784, 6911521990801, -1285942710366, -48950465208, -2937883611, 7475278120, -1168389061, 111348616, -16216613, 891810, -57451, -34864, 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 - 34864*x^16 - 57451*x^15 + 891810*x^14 - 16216613*x^13 + 111348616*x^12 - 1168389061*x^11 + 7475278120*x^10 - 2937883611*x^9 - 48950465208*x^8 - 1285942710366*x^7 + 6911521990801*x^6 - 1175582555784*x^5 - 35643871943428*x^4 - 19183703267008*x^3 - 9595175213344*x^2 - 215779097824640*x - 84881631489088)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 - 34864*x^16 - 57451*x^15 + 891810*x^14 - 16216613*x^13 + 111348616*x^12 - 1168389061*x^11 + 7475278120*x^10 - 2937883611*x^9 - 48950465208*x^8 - 1285942710366*x^7 + 6911521990801*x^6 - 1175582555784*x^5 - 35643871943428*x^4 - 19183703267008*x^3 - 9595175213344*x^2 - 215779097824640*x - 84881631489088, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 34864 x^{16} - 57451 x^{15} + 891810 x^{14} - 16216613 x^{13} + 111348616 x^{12} - 1168389061 x^{11} + 7475278120 x^{10} - 2937883611 x^{9} - 48950465208 x^{8} - 1285942710366 x^{7} + 6911521990801 x^{6} - 1175582555784 x^{5} - 35643871943428 x^{4} - 19183703267008 x^{3} - 9595175213344 x^{2} - 215779097824640 x - 84881631489088 \)

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:  \(-24344854108509137895346177549498986990410149857662202745929382807=-\,7^{17}\cdot 601^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1164.18$
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:  $7, 601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{296643984} a^{18} - \frac{6552773}{296643984} a^{17} + \frac{27239827}{296643984} a^{16} + \frac{8746027}{74160996} a^{15} - \frac{29884331}{296643984} a^{14} - \frac{8828971}{148321992} a^{13} + \frac{67481329}{296643984} a^{12} - \frac{13406989}{74160996} a^{11} + \frac{52529339}{296643984} a^{10} - \frac{7136041}{148321992} a^{9} - \frac{65826353}{296643984} a^{8} + \frac{11789483}{148321992} a^{7} - \frac{116088055}{296643984} a^{6} - \frac{4899589}{21188856} a^{5} - \frac{10157299}{21188856} a^{4} + \frac{365815}{3259824} a^{3} + \frac{7422751}{21188856} a^{2} - \frac{514151}{1765738} a + \frac{2024975}{5297214}$, $\frac{1}{818144107872} a^{19} + \frac{257}{818144107872} a^{18} + \frac{91130472653}{818144107872} a^{17} - \frac{520304069}{409072053936} a^{16} - \frac{5243013871}{62934162144} a^{15} - \frac{463437083}{25567003371} a^{14} - \frac{136307000755}{818144107872} a^{13} - \frac{1411822025}{10489027024} a^{12} - \frac{121650841517}{818144107872} a^{11} + \frac{3765660277}{34089337828} a^{10} + \frac{14434638047}{62934162144} a^{9} - \frac{3432794759}{14609716212} a^{8} + \frac{22251713759}{272714702624} a^{7} - \frac{14711064377}{34089337828} a^{6} - \frac{27606682949}{58438864848} a^{5} - \frac{28625372401}{116877729696} a^{4} - \frac{1869999051}{4869905404} a^{3} - \frac{14596236001}{29219432424} a^{2} - \frac{987207907}{2087102316} a + \frac{2494767853}{7304858106}$, $\frac{1}{95075597657614518849522038652501145918340533968472286724133829638394289418345641020542512379356779147057095507508121491380416} a^{20} + \frac{11913268649353154256827467723481350840753421154616766419798436677117875033497906816397533469848470709878380630381}{31691865885871506283174012884167048639446844656157428908044609879464763139448547006847504126452259715685698502502707163793472} a^{19} - \frac{37149835398360721755351950477510609840801841559718068448721027180905981303039157342913397824530218968703584563049219}{31691865885871506283174012884167048639446844656157428908044609879464763139448547006847504126452259715685698502502707163793472} a^{18} - \frac{441168734184297828539857712520029693429310206997370334413175891672096966024824129202411996984646072525952276834597806121427}{3961483235733938285396751610520881079930855582019678613505576234933095392431068375855938015806532464460712312812838395474184} a^{17} - \frac{708720531699126230524811025886272380376868870427273548810881264730210400306420261248574953719212242944192107370802683512857}{31691865885871506283174012884167048639446844656157428908044609879464763139448547006847504126452259715685698502502707163793472} a^{16} + \frac{12163118908604443517765333901556934714461060667264820118723759154965679510167429655096192007071361300258379088233066542181}{323386386590525574318102172287418863667824945470994172531067447749640440198454561294362287004614895058017331658190889426464} a^{15} + \frac{648229146054996312877941787379745151031948194094256054692390590019557129379405964009508110546760626256299580553707750067027}{31691865885871506283174012884167048639446844656157428908044609879464763139448547006847504126452259715685698502502707163793472} a^{14} - \frac{2017660186369371824349353037933519589483909678993066328516933898783029867467469782192162620903571441256899072717497886554673}{11884449707201814856190254831562643239792566746059035840516728704799286177293205127567814047419597393382136938438515186422552} a^{13} - \frac{263647941381171552269307747544341677284016401061908559300996430121224694293508258228497148753553957407699337823403717886515}{2437835837374731252551847144935926818418988050473648377541893067651135626111426692834423394342481516591207577115592858753344} a^{12} + \frac{6329719723750201960428101640082705961482864998925224642625851965906045736165940292910561747381131414344467261128050755957565}{47537798828807259424761019326250572959170266984236143362066914819197144709172820510271256189678389573528547753754060745690208} a^{11} - \frac{2862979123470069553690888586533064913833109476837500003142247583994041011902919668154712832940611593291581394620413379262515}{31691865885871506283174012884167048639446844656157428908044609879464763139448547006847504126452259715685698502502707163793472} a^{10} + \frac{191804358667358202358444911529229256564469996299460849497963293233645218120370719915833764591090718170399586692631344781039}{1218917918687365626275923572467963409209494025236824188770946533825567813055713346417211697171240758295603788557796429376672} a^{9} - \frac{8310816941851097748014248733525850221145532514302556220731418570358756790950387148776395586809048266976952569002875789346927}{95075597657614518849522038652501145918340533968472286724133829638394289418345641020542512379356779147057095507508121491380416} a^{8} - \frac{2547622597916573454407090531745151276679506719919238644117007337197215584105721800337261707358844457625669121152623878959937}{47537798828807259424761019326250572959170266984236143362066914819197144709172820510271256189678389573528547753754060745690208} a^{7} - \frac{2720638334605058589163951756569619368301778529037216573856187334307032350223400865753072037681231869116861376006425237469831}{15845932942935753141587006442083524319723422328078714454022304939732381569724273503423752063226129857842849251251353581896736} a^{6} + \frac{854314194330450461143852040218471674528561850717839369815909120317315689346355223043385913845862450175630171832834306571017}{4527409412267358040453430412023864091349549236593918415434944268494966162778363858121072018064608530812242643214672451970496} a^{5} + \frac{279510779399006312341540190324409896357774982025306811261774822807201320908087184238249536623669391294809254357042520121433}{6791114118401037060680145618035796137024323854890877623152416402742449244167545787181608027096912796218363964822008677955744} a^{4} + \frac{12004628420813845495726239213806096792319340209407099201142468343899084977010673729144863912945783792344567250381819754615}{848889264800129632585018202254474517128040481861359702894052050342806155520943223397701003387114099527295495602751084744468} a^{3} + \frac{144736773448634699669698210110599584417355663420234225878156255032377426877713468307030020094460693295012266687603247192813}{1697778529600259265170036404508949034256080963722719405788104100685612311041886446795402006774228199054590991205502169488936} a^{2} + \frac{13484721289819955155911862303614911972412781399274502726475661717215097518419711502070771954075630209638046839269153854361}{212222316200032408146254550563618629282010120465339925723513012585701538880235805849425250846778524881823873900687771186117} a - \frac{351980358544767876453500993693780649915054966358981336207347326033033955430387775760833236116910548937837150503004764994}{5441597851282882260160373091374836648256669755521536557013154168864142022570148867933980790943039099533945484633019774003}$

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}$
601Data not computed