↑

Properties

Label 21.21.2313617444...5952.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{35}\cdot 13^{12}$
Root discriminant $200.94$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:D_7:C_3$ (as 21T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-392454504, -1974848148, -3207628788, -763121359, 2791657596, 1920581572, -914967690, -994440486, 147566874, 258905465, -12480468, -39442403, 529074, 3686319, -8844, -211995, 0, 7238, 0, -133, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 133*x^19 + 7238*x^17 - 211995*x^15 - 8844*x^14 + 3686319*x^13 + 529074*x^12 - 39442403*x^11 - 12480468*x^10 + 258905465*x^9 + 147566874*x^8 - 994440486*x^7 - 914967690*x^6 + 1920581572*x^5 + 2791657596*x^4 - 763121359*x^3 - 3207628788*x^2 - 1974848148*x - 392454504)
 
gp: K = bnfinit(x^21 - 133*x^19 + 7238*x^17 - 211995*x^15 - 8844*x^14 + 3686319*x^13 + 529074*x^12 - 39442403*x^11 - 12480468*x^10 + 258905465*x^9 + 147566874*x^8 - 994440486*x^7 - 914967690*x^6 + 1920581572*x^5 + 2791657596*x^4 - 763121359*x^3 - 3207628788*x^2 - 1974848148*x - 392454504, 1)
 

Normalized defining polynomial

\( x^{21} - 133 x^{19} + 7238 x^{17} - 211995 x^{15} - 8844 x^{14} + 3686319 x^{13} + 529074 x^{12} - 39442403 x^{11} - 12480468 x^{10} + 258905465 x^{9} + 147566874 x^{8} - 994440486 x^{7} - 914967690 x^{6} + 1920581572 x^{5} + 2791657596 x^{4} - 763121359 x^{3} - 3207628788 x^{2} - 1974848148 x - 392454504 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2313617444278735447665222983483170933473170685952=2^{18}\cdot 7^{35}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.94$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a$, $\frac{1}{78} a^{14} - \frac{29}{78} a^{12} - \frac{1}{26} a^{10} + \frac{3}{26} a^{8} - \frac{5}{13} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{78} a^{15} - \frac{1}{26} a^{13} + \frac{23}{78} a^{11} + \frac{3}{26} a^{9} - \frac{5}{13} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{78} a^{16} + \frac{7}{39} a^{12} - \frac{5}{13} a^{9} - \frac{19}{39} a^{8} - \frac{2}{13} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{78} a^{17} + \frac{1}{78} a^{13} - \frac{1}{6} a^{11} - \frac{5}{13} a^{10} + \frac{1}{78} a^{9} - \frac{2}{13} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{6} a$, $\frac{1}{213954} a^{18} + \frac{4}{2743} a^{17} + \frac{1141}{213954} a^{16} + \frac{3}{2743} a^{15} - \frac{1147}{213954} a^{14} + \frac{191}{16458} a^{13} + \frac{88721}{213954} a^{12} + \frac{14621}{213954} a^{11} - \frac{545}{1266} a^{10} + \frac{519}{5486} a^{9} - \frac{1563}{5486} a^{8} + \frac{2531}{5486} a^{7} + \frac{7}{422} a^{6} + \frac{4547}{16458} a^{5} - \frac{397}{2743} a^{4} + \frac{119}{422} a^{3} + \frac{85}{633} a^{2} + \frac{317}{1266} a - \frac{30}{211}$, $\frac{1}{213954} a^{19} - \frac{33}{35659} a^{17} + \frac{32}{8229} a^{16} - \frac{47}{106977} a^{15} + \frac{1}{211} a^{14} + \frac{1334}{35659} a^{13} - \frac{29018}{106977} a^{12} + \frac{1727}{8229} a^{11} - \frac{43}{2743} a^{10} + \frac{2161}{8229} a^{9} + \frac{1106}{8229} a^{8} - \frac{3088}{8229} a^{7} - \frac{3284}{8229} a^{6} + \frac{8057}{16458} a^{5} - \frac{250}{633} a^{4} - \frac{38}{211} a^{3} - \frac{92}{633} a^{2} - \frac{56}{211} a + \frac{76}{211}$, $\frac{1}{746460253432170272830526243520446068699628725607368044} a^{20} - \frac{2562933498013603079164084570815992301310996420}{20735007039782507578625728986679057463878575711315779} a^{19} - \frac{852935231808548315753817644769426799282881952159}{746460253432170272830526243520446068699628725607368044} a^{18} - \frac{231185605675113210910007861776964804114495256542689}{62205021119347522735877186960037172391635727133947337} a^{17} - \frac{2190956053644927832705074285345329599206356665427111}{373230126716085136415263121760223034349814362803684022} a^{16} + \frac{211712002811884799093207050312576534574985489641413}{41470014079565015157251457973358114927757151422631558} a^{15} - \frac{969601711558551808900190957844741406594961981293907}{248820084477390090943508747840148689566542908535789348} a^{14} + \frac{1113829896575823128461903141596858823276802565938057}{41470014079565015157251457973358114927757151422631558} a^{13} - \frac{56917867802881927173839508669077795579211627221429941}{248820084477390090943508747840148689566542908535789348} a^{12} - \frac{6537211532882438758086224939256356578443703548108272}{62205021119347522735877186960037172391635727133947337} a^{11} - \frac{3325816062234390159942574028036994990778074112937249}{57420019494782328679271249501572774515356055815951388} a^{10} - \frac{737058074148793616114165377632511936272660233362823}{3190001083043462704403958305642931917519780878663966} a^{9} + \frac{22356849133863116280309237098830148320506694467645299}{57420019494782328679271249501572774515356055815951388} a^{8} + \frac{1767892794663571296266917209944997144775578453604433}{4785001624565194056605937458464397876279671317995949} a^{7} + \frac{20482130213399139043374998036005501472355082761567}{1595000541521731352201979152821465958759890439331983} a^{6} - \frac{1830176110352593642752273811560599575896746253170009}{4785001624565194056605937458464397876279671317995949} a^{5} + \frac{2971941947029725762763685195852505550258012971406798}{14355004873695582169817812375393193628839013953987847} a^{4} + \frac{224662080304012463219102148028907070818203958813851}{736154096086952931785528839763753519427641741230146} a^{3} + \frac{607133848554753998636685776636124462338630110434611}{4416924576521717590713173038582521116565850447380876} a^{2} - \frac{49217585426318637360242803606863607014566574607978}{368077048043476465892764419881876759713820870615073} a + \frac{42978929107774671730607846324339195027142679946864}{122692349347825488630921473293958919904606956871691}$

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:  $20$
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:  \( 773095802604000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:D_7:C_3$ (as 21T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 294
The 14 conjugacy class representatives for $C_7:D_7:C_3$
Character table for $C_7:D_7:C_3$

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 21 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$