↑

Properties

Label 21.1.20714269215...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 41^{7}$
Root discriminant $831.42$
Ramified primes $2, 3, 5, 7, 41$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1393873588, -7005312580, 11145838172, -4210424932, -2441792668, 3217907756, -1385748588, 84516280, 183654100, -41136060, -27422584, 13782440, -407106, -1086554, 209059, 38241, -16051, 161, 539, -49, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588)
 
gp: K = bnfinit(x^21 - 7*x^20 - 49*x^19 + 539*x^18 + 161*x^17 - 16051*x^16 + 38241*x^15 + 209059*x^14 - 1086554*x^13 - 407106*x^12 + 13782440*x^11 - 27422584*x^10 - 41136060*x^9 + 183654100*x^8 + 84516280*x^7 - 1385748588*x^6 + 3217907756*x^5 - 2441792668*x^4 - 4210424932*x^3 + 11145838172*x^2 - 7005312580*x + 1393873588, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 49 x^{19} + 539 x^{18} + 161 x^{17} - 16051 x^{16} + 38241 x^{15} + 209059 x^{14} - 1086554 x^{13} - 407106 x^{12} + 13782440 x^{11} - 27422584 x^{10} - 41136060 x^{9} + 183654100 x^{8} + 84516280 x^{7} - 1385748588 x^{6} + 3217907756 x^{5} - 2441792668 x^{4} - 4210424932 x^{3} + 11145838172 x^{2} - 7005312580 x + 1393873588 \)

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:  \(20714269215444879542778783404872527378677760000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $831.42$
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:  $2, 3, 5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{150} a^{14} - \frac{3}{50} a^{13} - \frac{7}{150} a^{12} - \frac{1}{50} a^{11} + \frac{3}{50} a^{10} + \frac{1}{50} a^{9} + \frac{13}{150} a^{8} - \frac{17}{50} a^{7} - \frac{32}{75} a^{6} + \frac{12}{25} a^{5} + \frac{2}{25} a^{4} - \frac{3}{25} a^{3} - \frac{37}{75} a^{2} - \frac{6}{25} a + \frac{34}{75}$, $\frac{1}{150} a^{15} + \frac{1}{75} a^{13} - \frac{1}{25} a^{12} + \frac{2}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{15} a^{9} + \frac{1}{25} a^{8} - \frac{43}{150} a^{7} - \frac{9}{25} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{75} a^{3} - \frac{12}{25} a^{2} - \frac{23}{75} a + \frac{2}{25}$, $\frac{1}{150} a^{16} + \frac{2}{25} a^{13} - \frac{2}{75} a^{12} - \frac{4}{75} a^{10} - \frac{3}{50} a^{8} - \frac{12}{25} a^{7} + \frac{4}{75} a^{6} - \frac{4}{25} a^{5} + \frac{4}{15} a^{4} - \frac{6}{25} a^{3} + \frac{12}{25} a^{2} - \frac{1}{25} a + \frac{37}{75}$, $\frac{1}{750} a^{17} - \frac{1}{750} a^{16} - \frac{1}{750} a^{15} + \frac{2}{25} a^{13} + \frac{32}{375} a^{12} + \frac{8}{375} a^{11} - \frac{2}{375} a^{10} - \frac{11}{150} a^{9} + \frac{3}{50} a^{8} + \frac{11}{50} a^{7} + \frac{31}{75} a^{6} - \frac{29}{75} a^{5} + \frac{2}{75} a^{4} - \frac{34}{75} a^{3} - \frac{8}{125} a^{2} + \frac{3}{125} a - \frac{31}{375}$, $\frac{1}{750} a^{18} - \frac{1}{375} a^{16} - \frac{1}{750} a^{15} + \frac{32}{375} a^{13} + \frac{1}{15} a^{12} + \frac{7}{125} a^{11} + \frac{1}{750} a^{10} - \frac{4}{75} a^{9} + \frac{1}{25} a^{8} - \frac{13}{150} a^{7} + \frac{11}{75} a^{6} + \frac{12}{25} a^{5} + \frac{31}{75} a^{4} - \frac{29}{375} a^{3} + \frac{7}{25} a^{2} + \frac{158}{375} a - \frac{46}{375}$, $\frac{1}{949500} a^{19} - \frac{49}{189900} a^{18} - \frac{121}{316500} a^{17} + \frac{191}{189900} a^{16} - \frac{2729}{949500} a^{15} + \frac{283}{316500} a^{14} + \frac{6577}{189900} a^{13} - \frac{64867}{949500} a^{12} - \frac{299}{15825} a^{11} + \frac{34997}{474750} a^{10} - \frac{103}{47475} a^{9} - \frac{3127}{31650} a^{8} + \frac{12919}{47475} a^{7} + \frac{21299}{94950} a^{6} - \frac{25}{1266} a^{5} + \frac{66523}{237375} a^{4} - \frac{8602}{47475} a^{3} + \frac{7144}{26375} a^{2} - \frac{21442}{47475} a - \frac{31712}{237375}$, $\frac{1}{5563531093227642609019602403592295109206992882783066225784896320996441500} a^{20} + \frac{795634915016435474250414740771844959347712166094852276575673795603}{5563531093227642609019602403592295109206992882783066225784896320996441500} a^{19} - \frac{210755050998628801587034697286562560765228320452765633279252312933083}{618170121469738067668844711510255012134110320309229580642766257888493500} a^{18} + \frac{420115060448205230419643840167455378615596068306858714162006579414249}{5563531093227642609019602403592295109206992882783066225784896320996441500} a^{17} - \frac{17172700923428702670433304246385359951755502376892215005880875904406769}{5563531093227642609019602403592295109206992882783066225784896320996441500} a^{16} + \frac{5102373489281323058512389066470466635531947774430217786455726258004731}{1854510364409214203006534134530765036402330960927688741928298773665480500} a^{15} - \frac{11437103418474512666906113735058814057555344918179093909384480096547833}{5563531093227642609019602403592295109206992882783066225784896320996441500} a^{14} + \frac{131954048153481327559161882590566175585447226684469078638573347338971527}{5563531093227642609019602403592295109206992882783066225784896320996441500} a^{13} + \frac{71690927805739990616342977387921174343923351149891782567889375419681941}{927255182204607101503267067265382518201165480463844370964149386832740250} a^{12} - \frac{76433629049978947819516537515875157877150018903182355900487227223046569}{1390882773306910652254900600898073777301748220695766556446224080249110375} a^{11} - \frac{136170107469367870805615670650843677775623272199953306003434379292231987}{2781765546613821304509801201796147554603496441391533112892448160498220750} a^{10} - \frac{4567131034052639312628303509321753224481181802412323651359259811130097}{61817012146973806766884471151025501213411032030922958064276625788849350} a^{9} + \frac{7727196603737426302909562653538637716053910896747892005691867159021869}{111270621864552852180392048071845902184139857655661324515697926419928830} a^{8} - \frac{56389035385287305620551279112879328160945106000742309296484815755096452}{278176554661382130450980120179614755460349644139153311289244816049822075} a^{7} - \frac{88533024698643828922769547311857606375293919257397048699954494827265607}{185451036440921420300653413453076503640233096092768874192829877366548050} a^{6} - \frac{60490786579297247135256686868725653166373713613335147068563060021711767}{1390882773306910652254900600898073777301748220695766556446224080249110375} a^{5} - \frac{609143568833631652539007549737875554451867180905602383582197172279043986}{1390882773306910652254900600898073777301748220695766556446224080249110375} a^{4} - \frac{57449748491757628557686351828193895857266949452747059851436040890266077}{463627591102303550751633533632691259100582740231922185482074693416370125} a^{3} - \frac{690140893331697594836092492931715657817252819467167923311979622674212203}{1390882773306910652254900600898073777301748220695766556446224080249110375} a^{2} + \frac{592849549141068754907239000675700938750563999025031797626220553553337093}{1390882773306910652254900600898073777301748220695766556446224080249110375} a + \frac{207205660343175230164525338185356209531451800801531369736593815706946529}{463627591102303550751633533632691259100582740231922185482074693416370125}$

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

3.1.4920.1, 7.1.600362847000000.35

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ R $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$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$$[\ ]_{7}^{6}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.1$x^{14} - 5$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$