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Properties

Label 21.21.1951555699...5568.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 17^{12}$
Root discriminant $276.95$
Ramified primes $2, 3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T24

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2033235224, 3458771652, -5009255496, -9401708407, 4223489928, 8920956618, -1512309834, -3702309351, 263892258, 817061378, -23402064, -105527772, 1010310, 8311674, -16758, -399742, 0, 11277, 0, -168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 168*x^19 + 11277*x^17 - 399742*x^15 - 16758*x^14 + 8311674*x^13 + 1010310*x^12 - 105527772*x^11 - 23402064*x^10 + 817061378*x^9 + 263892258*x^8 - 3702309351*x^7 - 1512309834*x^6 + 8920956618*x^5 + 4223489928*x^4 - 9401708407*x^3 - 5009255496*x^2 + 3458771652*x + 2033235224)
 
gp: K = bnfinit(x^21 - 168*x^19 + 11277*x^17 - 399742*x^15 - 16758*x^14 + 8311674*x^13 + 1010310*x^12 - 105527772*x^11 - 23402064*x^10 + 817061378*x^9 + 263892258*x^8 - 3702309351*x^7 - 1512309834*x^6 + 8920956618*x^5 + 4223489928*x^4 - 9401708407*x^3 - 5009255496*x^2 + 3458771652*x + 2033235224, 1)
 

Normalized defining polynomial

\( x^{21} - 168 x^{19} + 11277 x^{17} - 399742 x^{15} - 16758 x^{14} + 8311674 x^{13} + 1010310 x^{12} - 105527772 x^{11} - 23402064 x^{10} + 817061378 x^{9} + 263892258 x^{8} - 3702309351 x^{7} - 1512309834 x^{6} + 8920956618 x^{5} + 4223489928 x^{4} - 9401708407 x^{3} - 5009255496 x^{2} + 3458771652 x + 2033235224 \)

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:  \(1951555699117213348169834432463016320890103201005568=2^{18}\cdot 3^{28}\cdot 7^{21}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $276.95$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} + \frac{2}{17} a^{9} + \frac{6}{17} a^{7} - \frac{4}{17} a^{5} + \frac{4}{17} a^{4}$, $\frac{1}{51} a^{12} - \frac{5}{17} a^{10} - \frac{1}{3} a^{9} + \frac{2}{17} a^{8} - \frac{4}{51} a^{6} + \frac{7}{17} a^{5} + \frac{1}{3}$, $\frac{1}{51} a^{13} - \frac{1}{3} a^{10} - \frac{5}{17} a^{9} - \frac{16}{51} a^{7} + \frac{7}{17} a^{6} - \frac{3}{17} a^{5} + \frac{3}{17} a^{4} + \frac{1}{3} a$, $\frac{1}{867} a^{14} + \frac{2}{867} a^{12} - \frac{1}{51} a^{11} + \frac{19}{289} a^{10} - \frac{2}{51} a^{9} + \frac{98}{867} a^{8} - \frac{129}{289} a^{7} - \frac{4}{51} a^{6} - \frac{3}{17} a^{5} - \frac{7}{17} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{1734} a^{15} + \frac{1}{867} a^{13} + \frac{1}{289} a^{11} + \frac{1}{3} a^{10} + \frac{287}{867} a^{9} + \frac{97}{289} a^{8} - \frac{11}{51} a^{7} + \frac{19}{51} a^{6} + \frac{2}{17} a^{5} - \frac{2}{17} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{1734} a^{16} + \frac{1}{867} a^{12} + \frac{230}{867} a^{10} - \frac{287}{867} a^{9} - \frac{95}{289} a^{8} - \frac{259}{867} a^{7} + \frac{10}{51} a^{6} + \frac{8}{17} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{29478} a^{17} + \frac{1}{14739} a^{15} - \frac{116}{14739} a^{13} + \frac{338}{14739} a^{11} + \frac{1447}{14739} a^{10} - \frac{376}{867} a^{9} - \frac{32}{867} a^{8} - \frac{16}{51} a^{7} - \frac{5}{51} a^{6} + \frac{22}{51} a^{5} + \frac{7}{17} a^{4} - \frac{35}{102} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{58956} a^{18} - \frac{5}{19652} a^{16} + \frac{1}{9826} a^{14} + \frac{45}{4913} a^{12} + \frac{97}{9826} a^{11} - \frac{273}{578} a^{10} + \frac{4}{17} a^{9} + \frac{199}{578} a^{8} + \frac{62}{289} a^{7} - \frac{2}{17} a^{6} + \frac{1}{34} a^{5} + \frac{23}{68} a^{4} - \frac{1}{3} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{1002252} a^{19} - \frac{5}{334084} a^{17} - \frac{133}{501126} a^{15} - \frac{1735}{250563} a^{13} + \frac{1447}{501126} a^{12} + \frac{1}{578} a^{11} - \frac{154}{867} a^{10} + \frac{131}{578} a^{9} + \frac{64}{289} a^{8} - \frac{13}{51} a^{7} - \frac{671}{1734} a^{6} + \frac{215}{1156} a^{5} + \frac{23}{51} a^{4} + \frac{7}{204} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{16883915483267168185283543681213895387682010146764} a^{20} + \frac{6204248856069311072305220206975044173222351}{16883915483267168185283543681213895387682010146764} a^{19} - \frac{1175077200181274487001112033451619143015313}{8441957741633584092641771840606947693841005073382} a^{18} + \frac{2309646969740409850438567378214188618678577}{5627971827755722728427847893737965129227336715588} a^{17} + \frac{2828063398485031820542542614885782678619310133}{16883915483267168185283543681213895387682010146764} a^{16} + \frac{989002864606626327505670966947744941813168148}{4220978870816792046320885920303473846920502536691} a^{15} + \frac{1063745037241257288552322337824883120116277553}{2813985913877861364213923946868982564613668357794} a^{14} + \frac{62171585306439670957921322672914642799318861377}{8441957741633584092641771840606947693841005073382} a^{13} + \frac{19062380688045666400194963364113634970924964089}{4220978870816792046320885920303473846920502536691} a^{12} + \frac{2275252755655617879694547570049387336116429986}{82764291584642981300409527849087722488637304641} a^{11} - \frac{34228703340529060004369728017609781589550907165}{82764291584642981300409527849087722488637304641} a^{10} + \frac{1993598835228833065844286339862535472053074663}{9736975480546233094165826805775026175133800546} a^{9} + \frac{1844872009074940365166478661533380253040236461}{9736975480546233094165826805775026175133800546} a^{8} - \frac{5943168455042854911227888170424256472256462477}{29210926441638699282497480417325078525401401638} a^{7} + \frac{3713735839561519960002104219417580949448243795}{58421852883277398564994960834650157050802803276} a^{6} - \frac{12662631360302915056175586145267956751068646091}{58421852883277398564994960834650157050802803276} a^{5} + \frac{35594397291989852942581480806336560209666793}{859144895342314684779337659333090544864747107} a^{4} - \frac{11360114069033145154874567266061220844731989}{1145526527123086246372450212444120726486329476} a^{3} - \frac{79280253317463918068406465710845013229688691}{202151740080544631712785331607786010556411084} a^{2} - \frac{48491550782991724903801751658039095683632643}{101075870040272315856392665803893005278205542} a + \frac{1321216729033921477979697162339248650794841}{16845978340045385976065444300648834213034257}$

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:  \( 26055225840600000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T24:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 882
The 26 conjugacy class representatives for t21n24
Character table for t21n24 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $21$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $21$

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
$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}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$