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Properties

Label 21.1.18508389408...0448.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{18}\cdot 7^{17}\cdot 17^{19}$
Root discriminant $477.97$
Ramified primes $2, 3, 7, 17$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3093525184, 13796853952, 136118128608, 54391211280, -8314660648, 5927602716, 589911294, 627665995, 670558140, -73278177, 8335586, 12925563, -1568000, 228095, -62238, 16121, -4116, 1029, -98, 49, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 49*x^19 - 98*x^18 + 1029*x^17 - 4116*x^16 + 16121*x^15 - 62238*x^14 + 228095*x^13 - 1568000*x^12 + 12925563*x^11 + 8335586*x^10 - 73278177*x^9 + 670558140*x^8 + 627665995*x^7 + 589911294*x^6 + 5927602716*x^5 - 8314660648*x^4 + 54391211280*x^3 + 136118128608*x^2 + 13796853952*x + 3093525184)
 
gp: K = bnfinit(x^21 + 49*x^19 - 98*x^18 + 1029*x^17 - 4116*x^16 + 16121*x^15 - 62238*x^14 + 228095*x^13 - 1568000*x^12 + 12925563*x^11 + 8335586*x^10 - 73278177*x^9 + 670558140*x^8 + 627665995*x^7 + 589911294*x^6 + 5927602716*x^5 - 8314660648*x^4 + 54391211280*x^3 + 136118128608*x^2 + 13796853952*x + 3093525184, 1)
 

Normalized defining polynomial

\( x^{21} + 49 x^{19} - 98 x^{18} + 1029 x^{17} - 4116 x^{16} + 16121 x^{15} - 62238 x^{14} + 228095 x^{13} - 1568000 x^{12} + 12925563 x^{11} + 8335586 x^{10} - 73278177 x^{9} + 670558140 x^{8} + 627665995 x^{7} + 589911294 x^{6} + 5927602716 x^{5} - 8314660648 x^{4} + 54391211280 x^{3} + 136118128608 x^{2} + 13796853952 x + 3093525184 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(185083894086285956343696257354334893311234724948971880448=2^{33}\cdot 3^{18}\cdot 7^{17}\cdot 17^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $477.97$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{6} + \frac{5}{28} a^{5} - \frac{3}{28} a^{4} + \frac{13}{28} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{28} a^{7} + \frac{1}{4} a^{3} - \frac{3}{7}$, $\frac{1}{56} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{2}{7} a$, $\frac{1}{56} a^{9} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{56} a^{10} - \frac{1}{56} a^{6} + \frac{1}{28} a^{5} - \frac{1}{14} a^{4} + \frac{5}{28} a^{3} + \frac{3}{14} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{784} a^{11} - \frac{3}{392} a^{10} + \frac{3}{784} a^{9} + \frac{1}{196} a^{8} - \frac{5}{784} a^{7} - \frac{1}{56} a^{6} - \frac{17}{112} a^{5} + \frac{9}{98} a^{4} + \frac{8}{49} a^{3} + \frac{10}{49} a^{2} + \frac{11}{49} a + \frac{2}{49}$, $\frac{1}{1568} a^{12} + \frac{9}{1568} a^{10} - \frac{3}{784} a^{9} - \frac{9}{1568} a^{8} + \frac{3}{392} a^{7} + \frac{1}{224} a^{6} + \frac{57}{784} a^{5} - \frac{3}{28} a^{4} - \frac{5}{98} a^{3} - \frac{41}{98} a^{2} + \frac{6}{49} a + \frac{13}{49}$, $\frac{1}{1568} a^{13} - \frac{1}{1568} a^{11} - \frac{1}{784} a^{10} - \frac{11}{1568} a^{9} + \frac{1}{1568} a^{7} - \frac{13}{784} a^{6} + \frac{15}{112} a^{5} - \frac{39}{392} a^{4} - \frac{25}{196} a^{3} + \frac{3}{98} a^{2} + \frac{1}{7} a - \frac{10}{49}$, $\frac{1}{26656} a^{14} + \frac{1}{6664} a^{13} - \frac{5}{26656} a^{12} - \frac{1}{1904} a^{11} - \frac{33}{3808} a^{10} + \frac{5}{3332} a^{9} - \frac{3}{1568} a^{8} + \frac{69}{13328} a^{7} - \frac{1}{784} a^{6} - \frac{139}{833} a^{5} + \frac{26}{119} a^{4} + \frac{52}{119} a^{3} - \frac{113}{833} a^{2} - \frac{393}{833} a + \frac{26}{833}$, $\frac{1}{53312} a^{15} - \frac{1}{13328} a^{13} + \frac{3}{26656} a^{12} - \frac{11}{26656} a^{11} - \frac{45}{26656} a^{10} - \frac{13}{1904} a^{9} - \frac{29}{3808} a^{8} + \frac{485}{53312} a^{7} - \frac{235}{26656} a^{6} + \frac{643}{3332} a^{5} + \frac{124}{833} a^{4} + \frac{925}{3332} a^{3} + \frac{20}{119} a^{2} + \frac{8}{49} a - \frac{409}{833}$, $\frac{1}{746368} a^{16} - \frac{3}{746368} a^{15} - \frac{5}{373184} a^{14} + \frac{1}{26656} a^{13} + \frac{9}{53312} a^{12} + \frac{31}{53312} a^{11} - \frac{3}{392} a^{10} - \frac{773}{93296} a^{9} - \frac{5199}{746368} a^{8} + \frac{12105}{746368} a^{7} + \frac{39}{53312} a^{6} + \frac{1283}{26656} a^{5} - \frac{695}{13328} a^{4} + \frac{2411}{6664} a^{3} - \frac{3443}{23324} a^{2} + \frac{232}{5831} a + \frac{3495}{11662}$, $\frac{1}{1492736} a^{17} - \frac{5}{1492736} a^{15} - \frac{1}{746368} a^{14} - \frac{23}{106624} a^{13} - \frac{1}{26656} a^{12} - \frac{31}{106624} a^{11} + \frac{2899}{373184} a^{10} - \frac{925}{213248} a^{9} + \frac{81}{93296} a^{8} + \frac{12711}{1492736} a^{7} + \frac{1669}{106624} a^{6} + \frac{1109}{53312} a^{5} + \frac{1661}{26656} a^{4} + \frac{1641}{5488} a^{3} - \frac{851}{6664} a^{2} + \frac{1079}{23324} a - \frac{3809}{23324}$, $\frac{1}{2985472} a^{18} - \frac{1}{2985472} a^{17} - \frac{1}{2985472} a^{16} - \frac{9}{2985472} a^{15} + \frac{1}{93296} a^{14} - \frac{65}{213248} a^{13} + \frac{5}{213248} a^{12} + \frac{807}{1492736} a^{11} + \frac{1809}{2985472} a^{10} - \frac{9853}{2985472} a^{9} + \frac{22035}{2985472} a^{8} - \frac{32485}{2985472} a^{7} - \frac{3511}{213248} a^{6} - \frac{535}{6272} a^{5} + \frac{13535}{373184} a^{4} + \frac{29853}{186592} a^{3} - \frac{40909}{93296} a^{2} - \frac{2570}{5831} a + \frac{4797}{46648}$, $\frac{1}{101506048} a^{19} + \frac{3}{50753024} a^{18} - \frac{13}{50753024} a^{17} + \frac{1}{3172064} a^{16} - \frac{925}{101506048} a^{15} - \frac{285}{50753024} a^{14} + \frac{387}{1812608} a^{13} - \frac{2929}{12688256} a^{12} - \frac{35193}{101506048} a^{11} + \frac{36761}{50753024} a^{10} + \frac{305607}{50753024} a^{9} + \frac{3979}{1586032} a^{8} - \frac{494331}{101506048} a^{7} + \frac{101831}{7250432} a^{6} + \frac{4320961}{25376512} a^{5} + \frac{2316261}{12688256} a^{4} + \frac{2359151}{6344128} a^{3} + \frac{1450383}{3172064} a^{2} - \frac{45905}{93296} a + \frac{467469}{1586032}$, $\frac{1}{107590761829113066535313931251456648238263109129683026991553536} a^{20} - \frac{30942427432031450122750549572358398531122845715915049}{53795380914556533267656965625728324119131554564841513495776768} a^{19} - \frac{2026308152077470105499505507282551379273466071017049357}{26897690457278266633828482812864162059565777282420756747888384} a^{18} + \frac{16311049683199806048854400958209619818067721165772827829}{53795380914556533267656965625728324119131554564841513495776768} a^{17} - \frac{38194198044229686367325294373865577923134197024900038347}{107590761829113066535313931251456648238263109129683026991553536} a^{16} + \frac{51296392537563926583771227378950835209793590583215075123}{6724422614319566658457120703216040514891444320605189186972096} a^{15} + \frac{110258188380733271321987459750768635123421958634314376805}{13448845228639133316914241406432081029782888641210378373944192} a^{14} + \frac{6322590566798443172070226061816718637456338895578480529225}{26897690457278266633828482812864162059565777282420756747888384} a^{13} - \frac{10591404353078888695668997077431453189629158365157151256165}{107590761829113066535313931251456648238263109129683026991553536} a^{12} + \frac{12540988344804393734383552805946600678130545508605629119687}{53795380914556533267656965625728324119131554564841513495776768} a^{11} - \frac{180772258441247334814487211449095481956558923460748149328203}{26897690457278266633828482812864162059565777282420756747888384} a^{10} + \frac{418160323221764279002709846635705444405982770143130652495809}{53795380914556533267656965625728324119131554564841513495776768} a^{9} + \frac{429669818791316380648369593063219757063846185190953363087727}{107590761829113066535313931251456648238263109129683026991553536} a^{8} - \frac{38195496186422810472716116919808876748944770549491467511907}{26897690457278266633828482812864162059565777282420756747888384} a^{7} - \frac{5406647337936091371752041963620984441259000355991918443145}{791108542861125489230249494496004766457816978894728139643776} a^{6} + \frac{94934374359104641249842316130903315621237796426664224240245}{395554271430562744615124747248002383228908489447364069821888} a^{5} - \frac{244096392935006861064593026331343809742240968378384648843195}{1681105653579891664614280175804010128722861080151297296743024} a^{4} + \frac{828306856810212202602522027862271672037186711205759751635635}{1681105653579891664614280175804010128722861080151297296743024} a^{3} - \frac{43719693357440619901354127338694206801884136871465912373601}{210138206697486458076785021975501266090357635018912162092878} a^{2} - \frac{807922487267648225606106027751095809845505796004164432010187}{1681105653579891664614280175804010128722861080151297296743024} a + \frac{251895949375974857940771640285609179017603750104827815391359}{840552826789945832307140087902005064361430540075648648371512}$

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:  $10$
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:  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

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.6664.1, 7.1.18927411780570048.3

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

Sibling fields

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} - 4 x^{13} - 4 x^{12} + 8 x^{11} + 8 x^{10} - 6 x^{8} + 8 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 6$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
7Data not computed
$17$17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
17.14.13.1$x^{14} - 17$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$