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Properties

Label 21.21.7246641953...9889.4
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 7^{14}\cdot 43^{20}$
Root discriminant $569.17$
Ramified primes $3, 7, 43$
Class number $36$ (GRH)
Class group $[6, 6]$ (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![1731927086090117, 15884115372049848, 39769563390498270, 31976002108505938, 5000920037348550, -4000967262399996, -1405026978142459, 111760681819143, 94345678493136, 3238401809699, -2894978733948, -239871482424, 46019992039, 5297604816, -384550677, -57217692, 1567608, 322371, -2408, -903, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 903*x^19 - 2408*x^18 + 322371*x^17 + 1567608*x^16 - 57217692*x^15 - 384550677*x^14 + 5297604816*x^13 + 46019992039*x^12 - 239871482424*x^11 - 2894978733948*x^10 + 3238401809699*x^9 + 94345678493136*x^8 + 111760681819143*x^7 - 1405026978142459*x^6 - 4000967262399996*x^5 + 5000920037348550*x^4 + 31976002108505938*x^3 + 39769563390498270*x^2 + 15884115372049848*x + 1731927086090117)
 
gp: K = bnfinit(x^21 - 903*x^19 - 2408*x^18 + 322371*x^17 + 1567608*x^16 - 57217692*x^15 - 384550677*x^14 + 5297604816*x^13 + 46019992039*x^12 - 239871482424*x^11 - 2894978733948*x^10 + 3238401809699*x^9 + 94345678493136*x^8 + 111760681819143*x^7 - 1405026978142459*x^6 - 4000967262399996*x^5 + 5000920037348550*x^4 + 31976002108505938*x^3 + 39769563390498270*x^2 + 15884115372049848*x + 1731927086090117, 1)
 

Normalized defining polynomial

\( x^{21} - 903 x^{19} - 2408 x^{18} + 322371 x^{17} + 1567608 x^{16} - 57217692 x^{15} - 384550677 x^{14} + 5297604816 x^{13} + 46019992039 x^{12} - 239871482424 x^{11} - 2894978733948 x^{10} + 3238401809699 x^{9} + 94345678493136 x^{8} + 111760681819143 x^{7} - 1405026978142459 x^{6} - 4000967262399996 x^{5} + 5000920037348550 x^{4} + 31976002108505938 x^{3} + 39769563390498270 x^{2} + 15884115372049848 x + 1731927086090117 \)

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:  \(7246641953163409527953772394951784818867805657911594619889=3^{28}\cdot 7^{14}\cdot 43^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $569.17$
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:  $3, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2709=3^{2}\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2709}(64,·)$, $\chi_{2709}(1,·)$, $\chi_{2709}(1387,·)$, $\chi_{2709}(1285,·)$, $\chi_{2709}(970,·)$, $\chi_{2709}(655,·)$, $\chi_{2709}(2584,·)$, $\chi_{2709}(1948,·)$, $\chi_{2709}(1885,·)$, $\chi_{2709}(2080,·)$, $\chi_{2709}(1915,·)$, $\chi_{2709}(1444,·)$, $\chi_{2709}(1003,·)$, $\chi_{2709}(877,·)$, $\chi_{2709}(2482,·)$, $\chi_{2709}(310,·)$, $\chi_{2709}(2104,·)$, $\chi_{2709}(58,·)$, $\chi_{2709}(379,·)$, $\chi_{2709}(1726,·)$, $\chi_{2709}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{49} a^{6}$, $\frac{1}{49} a^{7}$, $\frac{1}{343} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{343} a^{9}$, $\frac{1}{343} a^{10}$, $\frac{1}{2401} a^{11} - \frac{1}{49} a^{5}$, $\frac{1}{2401} a^{12}$, $\frac{1}{16807} a^{13} - \frac{2}{343} a^{7} + \frac{1}{7} a$, $\frac{1}{621859} a^{14} - \frac{6}{621859} a^{13} + \frac{9}{88837} a^{12} - \frac{6}{88837} a^{11} + \frac{10}{12691} a^{10} - \frac{17}{12691} a^{9} - \frac{9}{12691} a^{8} - \frac{16}{12691} a^{7} + \frac{2}{1813} a^{6} + \frac{90}{1813} a^{5} - \frac{18}{259} a^{4} - \frac{3}{259} a^{3} - \frac{41}{259} a^{2} - \frac{20}{259} a + \frac{15}{37}$, $\frac{1}{4353013} a^{15} - \frac{12}{621859} a^{13} + \frac{48}{621859} a^{12} - \frac{11}{88837} a^{11} + \frac{80}{88837} a^{10} - \frac{1}{2401} a^{9} - \frac{10}{12691} a^{8} - \frac{13}{1813} a^{7} + \frac{65}{12691} a^{6} + \frac{16}{259} a^{5} + \frac{2}{49} a^{4} + \frac{15}{1813} a^{3} - \frac{16}{37} a^{2} + \frac{19}{259} a + \frac{90}{259}$, $\frac{1}{4353013} a^{16} + \frac{13}{621859} a^{13} - \frac{2}{12691} a^{12} + \frac{8}{88837} a^{11} + \frac{26}{88837} a^{10} + \frac{8}{12691} a^{9} - \frac{2}{1813} a^{8} + \frac{58}{12691} a^{7} - \frac{12}{1813} a^{6} + \frac{118}{1813} a^{5} + \frac{57}{1813} a^{4} + \frac{17}{37} a^{2} - \frac{113}{259} a - \frac{5}{37}$, $\frac{1}{30471091} a^{17} - \frac{1}{4353013} a^{14} - \frac{2}{621859} a^{13} + \frac{67}{621859} a^{12} + \frac{110}{621859} a^{11} - \frac{58}{88837} a^{10} + \frac{39}{88837} a^{9} - \frac{75}{88837} a^{8} + \frac{94}{12691} a^{7} + \frac{16}{12691} a^{6} + \frac{869}{12691} a^{5} + \frac{104}{1813} a^{4} + \frac{87}{1813} a^{3} - \frac{57}{1813} a^{2} + \frac{72}{259} a - \frac{62}{259}$, $\frac{1}{5271498743} a^{18} + \frac{81}{5271498743} a^{17} + \frac{85}{753071249} a^{16} + \frac{13}{753071249} a^{15} + \frac{3}{753071249} a^{14} - \frac{1203}{107581607} a^{13} - \frac{66}{15368801} a^{12} + \frac{14188}{107581607} a^{11} - \frac{9673}{15368801} a^{10} - \frac{4182}{15368801} a^{9} - \frac{10170}{15368801} a^{8} - \frac{10916}{2195543} a^{7} + \frac{7268}{2195543} a^{6} - \frac{80783}{2195543} a^{5} + \frac{916}{44807} a^{4} + \frac{20892}{313649} a^{3} - \frac{122350}{313649} a^{2} + \frac{1353}{6401} a + \frac{7942}{44807}$, $\frac{1}{36900491201} a^{19} - \frac{12}{5271498743} a^{17} + \frac{81}{753071249} a^{16} + \frac{23}{753071249} a^{15} + \frac{171}{753071249} a^{14} + \frac{6675}{753071249} a^{13} - \frac{18330}{107581607} a^{12} + \frac{2488}{107581607} a^{11} + \frac{11339}{15368801} a^{10} + \frac{4974}{15368801} a^{9} - \frac{9540}{15368801} a^{8} - \frac{37027}{15368801} a^{7} + \frac{4303}{2195543} a^{6} + \frac{57997}{2195543} a^{5} + \frac{4266}{313649} a^{4} - \frac{4647}{313649} a^{3} - \frac{154403}{313649} a^{2} - \frac{46103}{313649} a + \frac{2533}{44807}$, $\frac{1}{4055347676271788957200058187946219756048358240613968178670683143265511006559686981926613} a^{20} - \frac{49839660744163825776535153050001301860111908802137819679991372917997711921059}{4055347676271788957200058187946219756048358240613968178670683143265511006559686981926613} a^{19} - \frac{17480591679578684750366724243672073705041006417661089655441680189121348422347}{579335382324541279600008312563745679435479748659138311238669020466501572365669568846659} a^{18} - \frac{4204087659671504712380194889450735254433423299888456350107641046657540119386267}{579335382324541279600008312563745679435479748659138311238669020466501572365669568846659} a^{17} - \frac{160642466538384848516130055997329863733835242906042181375186032472571976825322}{11823171067847781216326700256402973049703668339982414515074877968704113721748358547891} a^{16} - \frac{8139484254676299294002850138573927633818106444066148811956177654653519635217201}{82762197474934468514286901794820811347925678379876901605524145780928796052238509835237} a^{15} + \frac{24313962126898315321252829758542162187834732699371955035612333049898227315487564}{82762197474934468514286901794820811347925678379876901605524145780928796052238509835237} a^{14} + \frac{2389973012716861013327298576434513461216204090175677360301763101656953833736023275}{82762197474934468514286901794820811347925678379876901605524145780928796052238509835237} a^{13} - \frac{1526488374560661294950868200912755690862174564016303360252586698773429141119587731}{11823171067847781216326700256402973049703668339982414515074877968704113721748358547891} a^{12} + \frac{1968743365507971399502632602399117608294122746676294812280566515907529474984926819}{11823171067847781216326700256402973049703668339982414515074877968704113721748358547891} a^{11} - \frac{970535028820576545395761763514379567172068285475995473991075878528006267595061597}{1689024438263968745189528608057567578529095477140344930724982566957730531678336935413} a^{10} + \frac{1760537554901461940269451929996255619656115329798447955124845705785351545428381698}{1689024438263968745189528608057567578529095477140344930724982566957730531678336935413} a^{9} + \frac{1828985585740713164904271500682324607535002154489509677103105773094656101308162683}{1689024438263968745189528608057567578529095477140344930724982566957730531678336935413} a^{8} - \frac{4094554316586675180613313866505943661063573443785649383626027589003217818887744531}{1689024438263968745189528608057567578529095477140344930724982566957730531678336935413} a^{7} - \frac{39158487328772322845121978371691154897258694243725298466436722223220389009319745}{6521329877467060792237562193272461693162530799769671547200704891728689311499370407} a^{6} - \frac{8792609080810379818316113550371213137913344033026516200360283549592390391064086804}{241289205466281249312789801151081082647013639591477847246426080993961504525476705059} a^{5} + \frac{2200540248370334794682908523256534305750656126683894209846343520795530162799224317}{34469886495183035616112828735868726092430519941639692463775154427708786360782386437} a^{4} + \frac{22667367049466288288630535026362125181271078593967709275270842193519072363596820}{703467071330266032981894463997320940661847345747748825791329682198138497158824213} a^{3} - \frac{6256299520069829391627479526891390307270855913063203085337096227618420714056842744}{34469886495183035616112828735868726092430519941639692463775154427708786360782386437} a^{2} - \frac{2769136437168921700904444653070788953393438352901277434077144096119248790405005154}{34469886495183035616112828735868726092430519941639692463775154427708786360782386437} a - \frac{38604350260693251346384104887981742604552427688501210254645857564454656398}{100685503148608560806560042682699101885656755287774800622623309127407473627}$

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

Class group and class number

$C_{6}\times C_{6}$, which has order $36$ (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:  \( 1538134121593772200000 \) (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

3.3.7338681.1, 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ R $21$ R $21$ $21$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{21}$ $21$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
43Data not computed