↑

Properties

Label 21.9.16179629887...6096.1
Degree $21$
Signature $[9, 6]$
Discriminant $2^{18}\cdot 7^{38}\cdot 41^{6}$
Root discriminant $177.03$
Ramified primes $2, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T36

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25712576, -169256528, 233927960, -37360225, 79733192, -25531520, -29660876, -5532374, -3575796, 2259145, 318976, -73549, 28532, -53753, -1472, 5075, 0, 182, 0, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 35*x^19 + 182*x^17 + 5075*x^15 - 1472*x^14 - 53753*x^13 + 28532*x^12 - 73549*x^11 + 318976*x^10 + 2259145*x^9 - 3575796*x^8 - 5532374*x^7 - 29660876*x^6 - 25531520*x^5 + 79733192*x^4 - 37360225*x^3 + 233927960*x^2 - 169256528*x + 25712576)
 
gp: K = bnfinit(x^21 - 35*x^19 + 182*x^17 + 5075*x^15 - 1472*x^14 - 53753*x^13 + 28532*x^12 - 73549*x^11 + 318976*x^10 + 2259145*x^9 - 3575796*x^8 - 5532374*x^7 - 29660876*x^6 - 25531520*x^5 + 79733192*x^4 - 37360225*x^3 + 233927960*x^2 - 169256528*x + 25712576, 1)
 

Normalized defining polynomial

\( x^{21} - 35 x^{19} + 182 x^{17} + 5075 x^{15} - 1472 x^{14} - 53753 x^{13} + 28532 x^{12} - 73549 x^{11} + 318976 x^{10} + 2259145 x^{9} - 3575796 x^{8} - 5532374 x^{7} - 29660876 x^{6} - 25531520 x^{5} + 79733192 x^{4} - 37360225 x^{3} + 233927960 x^{2} - 169256528 x + 25712576 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(161796298875473600212450076579403530770109956096=2^{18}\cdot 7^{38}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.03$
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:  $2, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{131938} a^{18} + \frac{79}{1609} a^{17} + \frac{24565}{131938} a^{16} + \frac{717}{3218} a^{15} + \frac{11571}{65969} a^{14} - \frac{194}{1609} a^{13} - \frac{29299}{65969} a^{12} - \frac{28780}{65969} a^{11} + \frac{7133}{65969} a^{10} + \frac{17095}{65969} a^{9} - \frac{16418}{65969} a^{8} + \frac{17915}{65969} a^{7} - \frac{29354}{65969} a^{6} + \frac{31026}{65969} a^{5} + \frac{18411}{131938} a^{4} - \frac{400}{1609} a^{3} - \frac{1091}{3218} a^{2} - \frac{91}{3218} a + \frac{715}{1609}$, $\frac{1}{263876} a^{19} + \frac{16365}{263876} a^{17} - \frac{627}{3218} a^{16} - \frac{12045}{131938} a^{15} - \frac{294}{1609} a^{14} + \frac{15817}{263876} a^{13} + \frac{21649}{65969} a^{12} - \frac{34811}{263876} a^{11} + \frac{26895}{65969} a^{10} - \frac{57723}{263876} a^{9} - \frac{17139}{65969} a^{8} - \frac{20537}{263876} a^{7} + \frac{31790}{65969} a^{6} - \frac{1332}{65969} a^{5} + \frac{638}{1609} a^{4} + \frac{969}{3218} a^{3} - \frac{465}{3218} a^{2} + \frac{425}{6436} a - \frac{534}{1609}$, $\frac{1}{1736494583989836157700070388418262987816724438778471804988306397594206256} a^{20} - \frac{145927489640922038668770327776868035916636380691803229738841337155}{217061822998729519712508798552282873477090554847308975623538299699275782} a^{19} + \frac{4441302066461142301725493587912989820574144506443421843871166149789}{1736494583989836157700070388418262987816724438778471804988306397594206256} a^{18} - \frac{9595681350082345208887822812645241128430021800125071430766592901908130}{108530911499364759856254399276141436738545277423654487811769149849637891} a^{17} - \frac{171042906997814582258291630916756343765100218434532340811559524258687145}{868247291994918078850035194209131493908362219389235902494153198797103128} a^{16} + \frac{53423740189308318928653263088891099077223185812393537164593286736618225}{217061822998729519712508798552282873477090554847308975623538299699275782} a^{15} - \frac{41697086510946852459406300810263476630578986587207029981418692701969829}{1736494583989836157700070388418262987816724438778471804988306397594206256} a^{14} - \frac{16924547302938619528659614218602418874787941604609180920993922501444275}{217061822998729519712508798552282873477090554847308975623538299699275782} a^{13} - \frac{279287234682269523812669279063665280896259642235462206992669471871169841}{1736494583989836157700070388418262987816724438778471804988306397594206256} a^{12} - \frac{166433526986514527788659017884333196782916576444890490076321529436515049}{434123645997459039425017597104565746954181109694617951247076599398551564} a^{11} + \frac{422030752366650959401952348728065794571381179276291374038244688972720875}{1736494583989836157700070388418262987816724438778471804988306397594206256} a^{10} + \frac{38569402786541918719422408692718184781206439276024726221876402298371559}{217061822998729519712508798552282873477090554847308975623538299699275782} a^{9} + \frac{410644210019275712541662925222335506804888819312095453353177126083644993}{1736494583989836157700070388418262987816724438778471804988306397594206256} a^{8} - \frac{42036598080529542002079354045653826395438409387767893975643301773146691}{434123645997459039425017597104565746954181109694617951247076599398551564} a^{7} + \frac{367238211916471365103575669462478407518887451828180925034690006206498401}{868247291994918078850035194209131493908362219389235902494153198797103128} a^{6} - \frac{88031469927074855315191686663030201987162820450151837973658842731538579}{434123645997459039425017597104565746954181109694617951247076599398551564} a^{5} + \frac{89615278761922691443753490417424338723800043834956173066920497745934727}{217061822998729519712508798552282873477090554847308975623538299699275782} a^{4} - \frac{950635929051839055764379399828098590092949852750346424145256189035479}{2647095402423530728201326811613205774110860424967182629555345118283851} a^{3} + \frac{12482436457570121999952198640693264364862584718539768062916688403829623}{42353526438776491651221228985811292385773766799474922072885521892541616} a^{2} + \frac{1389024455462496485727419591589728844506665218990602299343635547123405}{5294190804847061456402653623226411548221720849934365259110690236567702} a + \frac{987685864367368310667555231161712905030384135351917570147380073714937}{2647095402423530728201326811613205774110860424967182629555345118283851}$

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:  $14$
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:  \( 10593517930800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T36:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4116
The 53 conjugacy class representatives for t21n36 are not computed
Character table for t21n36 is not computed

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 28 siblings: 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 $21$ $21$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $21$ $21$ R ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $21$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
7Data not computed
41Data not computed