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Properties

Label 21.1.14098230443...8961.1
Degree $21$
Signature $[1, 10]$
Discriminant $3^{28}\cdot 151^{10}$
Root discriminant $47.18$
Ramified primes $3, 151$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{21}$ (as 21T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![270459, -546750, 176418, 4131, 41958, 419256, -240039, 255240, -28809, -29471, 30675, 1746, -5479, 2043, 792, -517, 33, 90, -1, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^19 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459)
 
gp: K = bnfinit(x^21 - 9*x^19 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459, 1)
 

Normalized defining polynomial

\( x^{21} - 9 x^{19} - x^{18} + 90 x^{17} + 33 x^{16} - 517 x^{15} + 792 x^{14} + 2043 x^{13} - 5479 x^{12} + 1746 x^{11} + 30675 x^{10} - 29471 x^{9} - 28809 x^{8} + 255240 x^{7} - 240039 x^{6} + 419256 x^{5} + 41958 x^{4} + 4131 x^{3} + 176418 x^{2} - 546750 x + 270459 \)

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:  \(140982304436617562370493756640408961=3^{28}\cdot 151^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.18$
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:  $3, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{10} + \frac{4}{9} a^{9} - \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3}$, $\frac{1}{9} a^{13} + \frac{4}{9} a^{10} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{14} + \frac{4}{27} a^{11} + \frac{1}{3} a^{10} + \frac{1}{9} a^{9} + \frac{7}{27} a^{8} - \frac{1}{3} a^{7} - \frac{2}{9} a^{6} + \frac{1}{27} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{189} a^{15} - \frac{1}{63} a^{14} - \frac{2}{63} a^{13} + \frac{1}{27} a^{12} - \frac{1}{9} a^{11} + \frac{5}{63} a^{10} + \frac{64}{189} a^{9} + \frac{5}{63} a^{8} + \frac{17}{63} a^{7} + \frac{67}{189} a^{6} + \frac{8}{63} a^{5} + \frac{2}{21} a^{4} + \frac{25}{63} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{189} a^{16} - \frac{1}{189} a^{14} + \frac{10}{189} a^{13} + \frac{8}{189} a^{11} - \frac{59}{189} a^{10} + \frac{20}{63} a^{9} + \frac{68}{189} a^{8} + \frac{52}{189} a^{7} - \frac{16}{63} a^{6} - \frac{22}{189} a^{5} + \frac{29}{63} a^{4} + \frac{13}{63} a^{3} - \frac{8}{21} a^{2} - \frac{2}{7} a$, $\frac{1}{189} a^{17} - \frac{2}{63} a^{13} - \frac{2}{63} a^{12} + \frac{2}{21} a^{11} - \frac{17}{63} a^{10} + \frac{1}{7} a^{9} - \frac{5}{21} a^{8} - \frac{20}{63} a^{7} - \frac{20}{63} a^{6} + \frac{41}{189} a^{5} + \frac{19}{63} a^{4} - \frac{20}{63} a^{3} - \frac{8}{21} a^{2} - \frac{3}{7} a$, $\frac{1}{567} a^{18} - \frac{1}{567} a^{15} - \frac{1}{189} a^{14} - \frac{1}{27} a^{13} - \frac{31}{567} a^{12} - \frac{31}{189} a^{11} + \frac{20}{63} a^{10} - \frac{88}{567} a^{9} - \frac{4}{189} a^{8} - \frac{44}{189} a^{7} + \frac{247}{567} a^{6} - \frac{31}{189} a^{5} + \frac{8}{21} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{7} a$, $\frac{1}{60669} a^{19} + \frac{26}{60669} a^{18} + \frac{5}{6741} a^{17} - \frac{130}{60669} a^{16} - \frac{143}{60669} a^{15} + \frac{152}{20223} a^{14} - \frac{1579}{60669} a^{13} + \frac{412}{8667} a^{12} - \frac{326}{2247} a^{11} - \frac{22357}{60669} a^{10} + \frac{100}{8667} a^{9} + \frac{1423}{20223} a^{8} - \frac{24959}{60669} a^{7} + \frac{5018}{60669} a^{6} - \frac{3085}{6741} a^{5} + \frac{1331}{6741} a^{4} - \frac{2873}{6741} a^{3} - \frac{1097}{2247} a^{2} - \frac{302}{749} a + \frac{35}{107}$, $\frac{1}{98808932051461655065818490554737114724207912393} a^{20} + \frac{29573302435925082157408147217619346968256}{3659590075980061298734018168693967212007700459} a^{19} - \frac{7197326254059267389564990542378605932019413}{32936310683820551688606163518245704908069304131} a^{18} - \frac{196006671136729881741555462399201935885786383}{98808932051461655065818490554737114724207912393} a^{17} + \frac{501181911164259162039495679236077245123115}{477337835997399299834871935047039201566221799} a^{16} + \frac{2527412166429464638376654319302772220869757}{2533562360293888591431243347557361916005331087} a^{15} - \frac{38248182250925558292353497878623357111155531}{98808932051461655065818490554737114724207912393} a^{14} + \frac{93481537541903760733586519818679298435817303}{3659590075980061298734018168693967212007700459} a^{13} - \frac{1275950957420688850097961312737437026742076977}{32936310683820551688606163518245704908069304131} a^{12} - \frac{15273244789548629923141323065913472895594186143}{98808932051461655065818490554737114724207912393} a^{11} + \frac{1397968655196076010574240026445781943797780943}{10978770227940183896202054506081901636023101377} a^{10} + \frac{1025477986676041364766083351988967259230605760}{3659590075980061298734018168693967212007700459} a^{9} + \frac{5990055289481518873189323464228288604521316901}{14115561721637379295116927222105302103458273199} a^{8} - \frac{12646247417563880571575253624467565852972523}{36718295076723023064220918080541477043555523} a^{7} - \frac{13577946434278881803889026828332491547825921259}{32936310683820551688606163518245704908069304131} a^{6} + \frac{1007258599183525178095201143119020369169398805}{10978770227940183896202054506081901636023101377} a^{5} + \frac{480255610955096427030989058213180174465756560}{3659590075980061298734018168693967212007700459} a^{4} - \frac{187337069277691237917916897497549602331589306}{522798582282865899819145452670566744572528637} a^{3} + \frac{28102265579316104204929752525068655826657790}{406621119553340144303779796521551912445300051} a^{2} + \frac{82850896649112170612750133042456533311209047}{406621119553340144303779796521551912445300051} a + \frac{7445109499083331471456406969232234791179369}{58088731364762877757682828074507416063614293}$

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:  $10$
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:  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:  \( 2150750435.99 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{21}$ (as 21T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 12 conjugacy class representatives for $D_{21}$
Character table for $D_{21}$

Intermediate fields

3.1.12231.1, 7.1.3442951.1

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

Sibling fields

Galois closure: 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 $21$ R $21$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $21$ $21$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $21$ $21$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ $21$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $21$

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
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$