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Properties

Label 21.21.9835336749...2129.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{38}\cdot 31^{14}$
Root discriminant $333.78$
Ramified primes $7, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-330316678789, 2766193463894, 7245219154649, 1192289177177, -7304874384850, -5115634476204, 495742766778, 1524423979489, 392624985032, -78589262157, -45340006873, -1478858493, 1889261640, 209014400, -35693127, -5958638, 303702, 74949, -938, -441, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 441*x^19 - 938*x^18 + 74949*x^17 + 303702*x^16 - 5958638*x^15 - 35693127*x^14 + 209014400*x^13 + 1889261640*x^12 - 1478858493*x^11 - 45340006873*x^10 - 78589262157*x^9 + 392624985032*x^8 + 1524423979489*x^7 + 495742766778*x^6 - 5115634476204*x^5 - 7304874384850*x^4 + 1192289177177*x^3 + 7245219154649*x^2 + 2766193463894*x - 330316678789)
 
gp: K = bnfinit(x^21 - 441*x^19 - 938*x^18 + 74949*x^17 + 303702*x^16 - 5958638*x^15 - 35693127*x^14 + 209014400*x^13 + 1889261640*x^12 - 1478858493*x^11 - 45340006873*x^10 - 78589262157*x^9 + 392624985032*x^8 + 1524423979489*x^7 + 495742766778*x^6 - 5115634476204*x^5 - 7304874384850*x^4 + 1192289177177*x^3 + 7245219154649*x^2 + 2766193463894*x - 330316678789, 1)
 

Normalized defining polynomial

\( x^{21} - 441 x^{19} - 938 x^{18} + 74949 x^{17} + 303702 x^{16} - 5958638 x^{15} - 35693127 x^{14} + 209014400 x^{13} + 1889261640 x^{12} - 1478858493 x^{11} - 45340006873 x^{10} - 78589262157 x^{9} + 392624985032 x^{8} + 1524423979489 x^{7} + 495742766778 x^{6} - 5115634476204 x^{5} - 7304874384850 x^{4} + 1192289177177 x^{3} + 7245219154649 x^{2} + 2766193463894 x - 330316678789 \)

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:  \(98353367498957471525704156941061377491148627676882129=7^{38}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $333.78$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1519=7^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1519}(1,·)$, $\chi_{1519}(842,·)$, $\chi_{1519}(459,·)$, $\chi_{1519}(652,·)$, $\chi_{1519}(1493,·)$, $\chi_{1519}(1110,·)$, $\chi_{1519}(1303,·)$, $\chi_{1519}(408,·)$, $\chi_{1519}(25,·)$, $\chi_{1519}(218,·)$, $\chi_{1519}(1059,·)$, $\chi_{1519}(676,·)$, $\chi_{1519}(869,·)$, $\chi_{1519}(1327,·)$, $\chi_{1519}(625,·)$, $\chi_{1519}(242,·)$, $\chi_{1519}(435,·)$, $\chi_{1519}(1276,·)$, $\chi_{1519}(893,·)$, $\chi_{1519}(1086,·)$, $\chi_{1519}(191,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31} a^{18} + \frac{9}{31} a^{17} - \frac{15}{31} a^{16} + \frac{13}{31} a^{15} + \frac{12}{31} a^{14} - \frac{2}{31} a^{13} + \frac{5}{31} a^{12} + \frac{14}{31} a^{11} - \frac{1}{31} a^{10} - \frac{8}{31} a^{9} + \frac{2}{31} a^{8} - \frac{6}{31} a^{7} - \frac{12}{31} a^{6} + \frac{9}{31} a^{5} + \frac{4}{31} a^{4} + \frac{1}{31} a^{2} + \frac{2}{31} a - \frac{1}{31}$, $\frac{1}{31} a^{19} - \frac{3}{31} a^{17} - \frac{7}{31} a^{16} - \frac{12}{31} a^{15} + \frac{14}{31} a^{14} - \frac{8}{31} a^{13} - \frac{3}{31} a^{11} + \frac{1}{31} a^{10} + \frac{12}{31} a^{9} + \frac{7}{31} a^{8} + \frac{11}{31} a^{7} - \frac{7}{31} a^{6} - \frac{15}{31} a^{5} - \frac{5}{31} a^{4} + \frac{1}{31} a^{3} - \frac{7}{31} a^{2} + \frac{12}{31} a + \frac{9}{31}$, $\frac{1}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{20} + \frac{3822550873409959564699440437319433245479327930093872945196682948428954374002180056622493386143932004633994168969224479961}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{19} + \frac{10975632430158628359289222813355098865767232373509017968631240747882802450180224849911904740981732870655494307880651317538}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{18} - \frac{585993509553101672583462693802082620572607067332896857751898229480100117319246362220787516763854416924688077944697346687170}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{17} - \frac{593077963051099555297147647125811523470575654824380219728882970078532137433245461972419639406232982521348230153150750172669}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{16} + \frac{545475508705395040973406672053974792497793631911286956161723722663727461374515386069363798114912311037380655027700074887480}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{15} - \frac{877603224171839543619829662013003399843854134186130118626763145867224076275448244673076028885809881944469461149695952661096}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{14} + \frac{444142366848794756661764037899013907449150118527778469719557232239840334428031145889054251569214161742742885996671041446813}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{13} + \frac{169893622453463868836331273950459033694594380648219287900093326374913449376683694513390883649092505679520447951061600156049}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{12} + \frac{422526874720574915786648559922837417795177311748934667922509815235019024197520009742577696824509796481036352371531139764508}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{11} + \frac{571678650778160048335224643990261057098161991583599039028905436331904416757019372659774825780178397886348285806368699261830}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{10} - \frac{191675702989245211657294875739311042052247350982845721582608686263579717319942499379766531749137244477545251245416064520494}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{9} - \frac{710395406314748056410726395468114478794231773818982732446839005695472521581503949601640134213705218409010291796309352329110}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{8} + \frac{494895037510186085472702596112568513648186039240684425062964903514196989139863066201704054808447824590049504962677929364557}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{7} + \frac{562278605707035026375474903688468099509709632162494221362521783640481871848579587331078824678258158926091796668832177860320}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{6} + \frac{597367952017987082840282954809170385725317483570672838322495054092341542989931095097403988256411038704822752182698099672973}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{5} - \frac{838440482647512225879546203756829680686164384799196186283820430247251930758539483152439339276106837876408136998767536888235}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{4} - \frac{857112530619075512320629221430715653726188163590141643545607787413055649668720208687363926121511008106688544914215395137158}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{3} - \frac{194953165893248562890828889572923202366935138220051938082045667265950244652655783443066522028159909878184308076925518372699}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a^{2} + \frac{554006380333008567291945567516879947953710851180503120750746282221052903940915586962760301523981472355632969208540990196308}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447} a + \frac{449319926529035586878106381655653871030879820320964770951659506483000848732932766729693824726702079595102425053793361131689}{1861853095884801861252332720645752802184585580816366605312360920807641936445907068734688564021999865230845203009569178925447}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 17576256199406360000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.47089.2, 7.7.13841287201.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $21$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ R $21$ $21$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ R $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
7Data not computed
$31$31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$