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Properties

Label 21.21.3167682693...8849.1
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![8771, -313502, -5181015, -12941586, 39416874, 29458695, -62993456, -33331528, 41742773, 21408857, -12696230, -7173629, 1687220, 1161058, -101273, -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 - 101273*x^14 + 1161058*x^13 + 1687220*x^12 - 7173629*x^11 - 12696230*x^10 + 21408857*x^9 + 41742773*x^8 - 33331528*x^7 - 62993456*x^6 + 29458695*x^5 + 39416874*x^4 - 12941586*x^3 - 5181015*x^2 - 313502*x + 8771)
 
gp: K = bnfinit(x^21 - x^20 - 106*x^19 + 19*x^18 + 4514*x^17 + 2212*x^16 - 98469*x^15 - 101273*x^14 + 1161058*x^13 + 1687220*x^12 - 7173629*x^11 - 12696230*x^10 + 21408857*x^9 + 41742773*x^8 - 33331528*x^7 - 62993456*x^6 + 29458695*x^5 + 39416874*x^4 - 12941586*x^3 - 5181015*x^2 - 313502*x + 8771, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 106 x^{19} + 19 x^{18} + 4514 x^{17} + 2212 x^{16} - 98469 x^{15} - 101273 x^{14} + 1161058 x^{13} + 1687220 x^{12} - 7173629 x^{11} - 12696230 x^{10} + 21408857 x^{9} + 41742773 x^{8} - 33331528 x^{7} - 62993456 x^{6} + 29458695 x^{5} + 39416874 x^{4} - 12941586 x^{3} - 5181015 x^{2} - 313502 x + 8771 \)

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$$\chi_{301}(64,·)$, $\chi_{301}(1,·)$, $\chi_{301}(67,·)$, $\chi_{301}(9,·)$, $\chi_{301}(74,·)$, $\chi_{301}(78,·)$, $\chi_{301}(79,·)$, $\chi_{301}(81,·)$, $\chi_{301}(274,·)$, $\chi_{301}(275,·)$, $\chi_{301}(142,·)$, $\chi_{301}(221,·)$, $\chi_{301}(176,·)$, $\chi_{301}(100,·)$, $\chi_{301}(298,·)$, $\chi_{301}(109,·)$, $\chi_{301}(240,·)$, $\chi_{301}(53,·)$, $\chi_{301}(183,·)$, $\chi_{301}(58,·)$, $\chi_{301}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{259} a^{17} - \frac{11}{259} a^{16} - \frac{4}{259} a^{15} + \frac{29}{259} a^{14} + \frac{36}{259} a^{13} + \frac{16}{37} a^{12} - \frac{7}{37} a^{11} + \frac{81}{259} a^{10} - \frac{18}{37} a^{9} + \frac{17}{37} a^{8} - \frac{92}{259} a^{7} + \frac{25}{259} a^{6} + \frac{15}{259} a^{5} + \frac{108}{259} a^{4} + \frac{80}{259} a^{3} - \frac{16}{37} a^{2} + \frac{13}{37} a - \frac{1}{37}$, $\frac{1}{259} a^{18} - \frac{2}{37} a^{16} - \frac{15}{259} a^{15} - \frac{15}{259} a^{14} + \frac{27}{259} a^{13} + \frac{110}{259} a^{12} + \frac{97}{259} a^{11} - \frac{86}{259} a^{10} - \frac{46}{259} a^{9} + \frac{107}{259} a^{8} + \frac{123}{259} a^{7} + \frac{31}{259} a^{6} + \frac{2}{37} a^{5} - \frac{64}{259} a^{4} - \frac{46}{259} a^{3} - \frac{15}{37} a^{2} - \frac{6}{37} a - \frac{11}{37}$, $\frac{1}{46361} a^{19} - \frac{89}{46361} a^{18} - \frac{29}{46361} a^{17} - \frac{3007}{46361} a^{16} + \frac{1417}{46361} a^{15} + \frac{9141}{46361} a^{14} - \frac{7902}{46361} a^{13} + \frac{2995}{6623} a^{12} - \frac{11462}{46361} a^{11} + \frac{15865}{46361} a^{10} + \frac{3254}{6623} a^{9} + \frac{18008}{46361} a^{8} + \frac{1122}{6623} a^{7} - \frac{2565}{46361} a^{6} + \frac{981}{46361} a^{5} + \frac{12725}{46361} a^{4} + \frac{11632}{46361} a^{3} + \frac{570}{6623} a^{2} + \frac{1919}{6623} a + \frac{7}{37}$, $\frac{1}{23042888227585618953648125010721868624302924421880832373} a^{20} - \frac{33251770014845524420944201956932268473994011574062}{23042888227585618953648125010721868624302924421880832373} a^{19} + \frac{39489711195406488600649322149067460173276520030209858}{23042888227585618953648125010721868624302924421880832373} a^{18} + \frac{2920191044244242273552278326321919120853209720029363}{3291841175369374136235446430103124089186132060268690339} a^{17} + \frac{1462324448307454713959220337853651764018484996102459784}{23042888227585618953648125010721868624302924421880832373} a^{16} - \frac{65311351185723206702707711065337124961817741742117671}{23042888227585618953648125010721868624302924421880832373} a^{15} - \frac{6038730309545596487092472731760369924943877406089413103}{23042888227585618953648125010721868624302924421880832373} a^{14} - \frac{9027301461535988489149247395722943887396333154720161619}{23042888227585618953648125010721868624302924421880832373} a^{13} - \frac{151487961701744874096738632901937465566824271624540}{12709811487912641452646511313139475247822903707601121} a^{12} + \frac{10587578623926910180659037774221038271722675069548303843}{23042888227585618953648125010721868624302924421880832373} a^{11} + \frac{7507084776165429334709538576724745105612422717758859565}{23042888227585618953648125010721868624302924421880832373} a^{10} - \frac{6692849036358411751219037899495188851583781217266846241}{23042888227585618953648125010721868624302924421880832373} a^{9} + \frac{3294500147680115771587542247477341553791097915890178290}{23042888227585618953648125010721868624302924421880832373} a^{8} + \frac{9166256653150271681695343694506790308300247024628456289}{23042888227585618953648125010721868624302924421880832373} a^{7} + \frac{3211320838488695252280375286966005924438742487191975042}{23042888227585618953648125010721868624302924421880832373} a^{6} + \frac{174174738919848127900153097809194054682725183788762272}{3291841175369374136235446430103124089186132060268690339} a^{5} + \frac{445626710974047500102386411035397938067202394257267643}{3291841175369374136235446430103124089186132060268690339} a^{4} - \frac{840430053564130200933484086304670385260285835895336370}{3291841175369374136235446430103124089186132060268690339} a^{3} + \frac{87459119812086951665085090152074312032650406174994693}{470263025052767733747920918586160584169447437181241477} a^{2} + \frac{181143209651691821432115033102209095234773497633359129}{470263025052767733747920918586160584169447437181241477} a + \frac{903870911839728720684555762487370644571072766855598}{2627167737724959406412966025621008850108644900453863}$

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:  \( 2941850439885216.0 \) (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

3.3.90601.2, 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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ R $21$ $21$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ R $21$ $21$ $21$

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$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
43Data not computed