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Properties

Label 21.3.24011181716...4512.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{18}\cdot 7^{17}\cdot 13^{14}$
Root discriminant $48.39$
Ramified primes $2, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2484, 21492, -16944, -82628, 97712, 106640, -69004, -180216, 130100, 51928, -42058, -24814, 17006, 5248, -4907, 75, 661, -185, -19, 27, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 27*x^19 - 19*x^18 - 185*x^17 + 661*x^16 + 75*x^15 - 4907*x^14 + 5248*x^13 + 17006*x^12 - 24814*x^11 - 42058*x^10 + 51928*x^9 + 130100*x^8 - 180216*x^7 - 69004*x^6 + 106640*x^5 + 97712*x^4 - 82628*x^3 - 16944*x^2 + 21492*x + 2484)
 
gp: K = bnfinit(x^21 - 7*x^20 + 27*x^19 - 19*x^18 - 185*x^17 + 661*x^16 + 75*x^15 - 4907*x^14 + 5248*x^13 + 17006*x^12 - 24814*x^11 - 42058*x^10 + 51928*x^9 + 130100*x^8 - 180216*x^7 - 69004*x^6 + 106640*x^5 + 97712*x^4 - 82628*x^3 - 16944*x^2 + 21492*x + 2484, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 27 x^{19} - 19 x^{18} - 185 x^{17} + 661 x^{16} + 75 x^{15} - 4907 x^{14} + 5248 x^{13} + 17006 x^{12} - 24814 x^{11} - 42058 x^{10} + 51928 x^{9} + 130100 x^{8} - 180216 x^{7} - 69004 x^{6} + 106640 x^{5} + 97712 x^{4} - 82628 x^{3} - 16944 x^{2} + 21492 x + 2484 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-240111817160475801918451219385024512=-\,2^{18}\cdot 7^{17}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.39$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14}$, $\frac{1}{12} a^{19} - \frac{1}{12} a^{18} - \frac{1}{12} a^{16} + \frac{1}{12} a^{15} + \frac{1}{12} a^{14} + \frac{1}{12} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{29095435539320506827471415385070792544488057380922024} a^{20} + \frac{5380435712785759783525725507648403165653014140091}{7273858884830126706867853846267698136122014345230506} a^{19} + \frac{19448307112203556658194931785962478676994060286559}{9698478513106835609157138461690264181496019126974008} a^{18} + \frac{728550894244701176335802737271268796002255122582567}{7273858884830126706867853846267698136122014345230506} a^{17} + \frac{2230544872073638941585407076732347696777649319689127}{29095435539320506827471415385070792544488057380922024} a^{16} - \frac{168337755665081324440496239518988620619162918763878}{3636929442415063353433926923133849068061007172615253} a^{15} - \frac{1659466749293369272045576707576007039038015598021537}{9698478513106835609157138461690264181496019126974008} a^{14} + \frac{17014856029212287021238408907573560868073310664094}{3636929442415063353433926923133849068061007172615253} a^{13} - \frac{659375330277154898190004226731994815420762827522055}{3636929442415063353433926923133849068061007172615253} a^{12} + \frac{816514846491064762274370972654504459687776363375497}{3636929442415063353433926923133849068061007172615253} a^{11} + \frac{1226363672188458845033871566895797712026007846737755}{14547717769660253413735707692535396272244028690461012} a^{10} - \frac{592547703578954240156481910153029108537857653950584}{3636929442415063353433926923133849068061007172615253} a^{9} + \frac{257918894823776539010024141121282518928986773988117}{3636929442415063353433926923133849068061007172615253} a^{8} - \frac{3335142137695159699323792165288853761056688925948093}{7273858884830126706867853846267698136122014345230506} a^{7} - \frac{589537620677092345382048687251947549496353972894278}{1212309814138354451144642307711283022687002390871751} a^{6} - \frac{418447724146271069660490896642051649978731876276531}{3636929442415063353433926923133849068061007172615253} a^{5} + \frac{1661206735511957641850522109518277802425118957022561}{7273858884830126706867853846267698136122014345230506} a^{4} - \frac{3165795860971712825036514859742813694746266396415755}{7273858884830126706867853846267698136122014345230506} a^{3} - \frac{606276159843357499791826064782152156371143894916776}{3636929442415063353433926923133849068061007172615253} a^{2} + \frac{443874533073912813658297549634117232310032718625605}{1212309814138354451144642307711283022687002390871751} a - \frac{271399777136149939112162445397918688890383753566883}{808206542758902967429761538474188681791334927247834}$

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:  $11$
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:  \( 5499663708.5292225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.8281.1, 7.1.626971072.1

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

Sibling fields

Degree 21 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$