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Properties

Label 21.21.3167682693...8849.2
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 43^{20}$
Root discriminant $131.55$
Ramified primes $7, 43$
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![910567, -678013, -14175798, 9727326, 58173689, -9722475, -97849256, -26783273, 55808804, 24767415, -13739797, -7722653, 1663441, 1181526, -98564, -98469, 2212, 4514, 19, -106, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 106*x^19 + 19*x^18 + 4514*x^17 + 2212*x^16 - 98469*x^15 - 98564*x^14 + 1181526*x^13 + 1663441*x^12 - 7722653*x^11 - 13739797*x^10 + 24767415*x^9 + 55808804*x^8 - 26783273*x^7 - 97849256*x^6 - 9722475*x^5 + 58173689*x^4 + 9727326*x^3 - 14175798*x^2 - 678013*x + 910567)
 
gp: K = bnfinit(x^21 - x^20 - 106*x^19 + 19*x^18 + 4514*x^17 + 2212*x^16 - 98469*x^15 - 98564*x^14 + 1181526*x^13 + 1663441*x^12 - 7722653*x^11 - 13739797*x^10 + 24767415*x^9 + 55808804*x^8 - 26783273*x^7 - 97849256*x^6 - 9722475*x^5 + 58173689*x^4 + 9727326*x^3 - 14175798*x^2 - 678013*x + 910567, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 106 x^{19} + 19 x^{18} + 4514 x^{17} + 2212 x^{16} - 98469 x^{15} - 98564 x^{14} + 1181526 x^{13} + 1663441 x^{12} - 7722653 x^{11} - 13739797 x^{10} + 24767415 x^{9} + 55808804 x^{8} - 26783273 x^{7} - 97849256 x^{6} - 9722475 x^{5} + 58173689 x^{4} + 9727326 x^{3} - 14175798 x^{2} - 678013 x + 910567 \)

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:  \(316768269303064912141617448027213301889478849=7^{14}\cdot 43^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.55$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(301=7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{301}(64,·)$, $\chi_{301}(1,·)$, $\chi_{301}(135,·)$, $\chi_{301}(268,·)$, $\chi_{301}(78,·)$, $\chi_{301}(144,·)$, $\chi_{301}(274,·)$, $\chi_{301}(212,·)$, $\chi_{301}(23,·)$, $\chi_{301}(25,·)$, $\chi_{301}(282,·)$, $\chi_{301}(95,·)$, $\chi_{301}(289,·)$, $\chi_{301}(228,·)$, $\chi_{301}(165,·)$, $\chi_{301}(296,·)$, $\chi_{301}(176,·)$, $\chi_{301}(183,·)$, $\chi_{301}(186,·)$, $\chi_{301}(60,·)$, $\chi_{301}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{18} - \frac{3}{7} a^{14} + \frac{2}{7} a^{13} - \frac{3}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{5136817} a^{19} - \frac{73709}{5136817} a^{18} - \frac{2768}{65023} a^{17} - \frac{30600}{733831} a^{16} + \frac{23505}{5136817} a^{15} + \frac{1458392}{5136817} a^{14} - \frac{2433659}{5136817} a^{13} - \frac{639140}{5136817} a^{12} - \frac{358647}{733831} a^{11} + \frac{134977}{5136817} a^{10} + \frac{1521392}{5136817} a^{9} + \frac{1959645}{5136817} a^{8} + \frac{822715}{5136817} a^{7} - \frac{1465508}{5136817} a^{6} - \frac{1586651}{5136817} a^{5} - \frac{122973}{733831} a^{4} + \frac{172528}{733831} a^{3} + \frac{337815}{733831} a^{2} + \frac{16245}{104833} a + \frac{39869}{104833}$, $\frac{1}{76266232648653897227885001012751688377767407099136620461} a^{20} - \frac{6827376682012723783992746175403699683412545999141}{76266232648653897227885001012751688377767407099136620461} a^{19} + \frac{583581096449980878064812363310891365591237244378957200}{76266232648653897227885001012751688377767407099136620461} a^{18} + \frac{465377080039427050301858435215584458295690342279668873}{10895176092664842461126428716107384053966772442733802923} a^{17} - \frac{3841535779690380580597787010929587253119358655066432865}{76266232648653897227885001012751688377767407099136620461} a^{16} - \frac{1006435210721205638138189869096631753826590380060057940}{76266232648653897227885001012751688377767407099136620461} a^{15} - \frac{13172924301670293856631918031505404533026934679132484214}{76266232648653897227885001012751688377767407099136620461} a^{14} + \frac{27620864497261879304190176947537680683904910134063242108}{76266232648653897227885001012751688377767407099136620461} a^{13} + \frac{2666265565319865255461718224836196791323767238571458667}{10895176092664842461126428716107384053966772442733802923} a^{12} + \frac{8175465623161270979376174322167642615371278740483113095}{76266232648653897227885001012751688377767407099136620461} a^{11} - \frac{5295113983547166718435378009532185107472739688327317170}{76266232648653897227885001012751688377767407099136620461} a^{10} - \frac{25203536526367914440848460299842642845755273831370282090}{76266232648653897227885001012751688377767407099136620461} a^{9} + \frac{1336700644293277810093656690508302127925601996799894765}{76266232648653897227885001012751688377767407099136620461} a^{8} + \frac{5528651939578510835730171162996347001856113205268725430}{76266232648653897227885001012751688377767407099136620461} a^{7} + \frac{36389888435212125815722214088745374019163019556555820705}{76266232648653897227885001012751688377767407099136620461} a^{6} - \frac{2381743918706164645332889232034574666414789109536031733}{10895176092664842461126428716107384053966772442733802923} a^{5} - \frac{5022202843862169225711230742834466403168216252456600112}{10895176092664842461126428716107384053966772442733802923} a^{4} - \frac{2923628462092869678937312112298210703045592183437211305}{10895176092664842461126428716107384053966772442733802923} a^{3} - \frac{719242005450930772240564602515968935100200891671191242}{1556453727523548923018061245158197721995253206104828989} a^{2} - \frac{261940977997972275533917285488860407855188501296907354}{1556453727523548923018061245158197721995253206104828989} a + \frac{28099754250196801482388025261463389296257851209582626}{1556453727523548923018061245158197721995253206104828989}$

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:  \( 2201310310593683.5 \) (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

3.3.90601.1, 7.7.6321363049.1

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 $21$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ $21$ R $21$ $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ R $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $21$

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
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
43Data not computed