Properties

Label 30.0.109...375.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.098\times 10^{48}$
Root discriminant $39.93$
Ramified primes $3, 5, 47$
Class number not computed
Class group not computed
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 7*x^29 + 48*x^28 - 257*x^27 + 1124*x^26 - 4108*x^25 + 12894*x^24 - 35106*x^23 + 82734*x^22 - 166828*x^21 + 288975*x^20 - 424231*x^19 + 515389*x^18 - 443551*x^17 + 184757*x^16 + 79835*x^15 - 29816*x^14 - 348911*x^13 + 993639*x^12 - 2148173*x^11 + 1765791*x^10 + 384577*x^9 - 1489368*x^8 + 2068027*x^7 - 1731374*x^6 - 444980*x^5 + 2213182*x^4 - 1190713*x^3 - 21588*x^2 + 127306*x + 52081)
 
gp: K = bnfinit(x^30 - 7*x^29 + 48*x^28 - 257*x^27 + 1124*x^26 - 4108*x^25 + 12894*x^24 - 35106*x^23 + 82734*x^22 - 166828*x^21 + 288975*x^20 - 424231*x^19 + 515389*x^18 - 443551*x^17 + 184757*x^16 + 79835*x^15 - 29816*x^14 - 348911*x^13 + 993639*x^12 - 2148173*x^11 + 1765791*x^10 + 384577*x^9 - 1489368*x^8 + 2068027*x^7 - 1731374*x^6 - 444980*x^5 + 2213182*x^4 - 1190713*x^3 - 21588*x^2 + 127306*x + 52081, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52081, 127306, -21588, -1190713, 2213182, -444980, -1731374, 2068027, -1489368, 384577, 1765791, -2148173, 993639, -348911, -29816, 79835, 184757, -443551, 515389, -424231, 288975, -166828, 82734, -35106, 12894, -4108, 1124, -257, 48, -7, 1]);
 

\( x^{30} - 7 x^{29} + 48 x^{28} - 257 x^{27} + 1124 x^{26} - 4108 x^{25} + 12894 x^{24} - 35106 x^{23} + 82734 x^{22} - 166828 x^{21} + 288975 x^{20} - 424231 x^{19} + 515389 x^{18} - 443551 x^{17} + 184757 x^{16} + 79835 x^{15} - 29816 x^{14} - 348911 x^{13} + 993639 x^{12} - 2148173 x^{11} + 1765791 x^{10} + 384577 x^{9} - 1489368 x^{8} + 2068027 x^{7} - 1731374 x^{6} - 444980 x^{5} + 2213182 x^{4} - 1190713 x^{3} - 21588 x^{2} + 127306 x + 52081 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1097586964004374803146658147777020931243896484375\)\(\medspace = -\,3^{15}\cdot 5^{25}\cdot 47^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.93$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 47$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{11} a^{27} + \frac{5}{11} a^{26} - \frac{4}{11} a^{25} + \frac{1}{11} a^{23} + \frac{1}{11} a^{22} + \frac{1}{11} a^{21} + \frac{1}{11} a^{20} - \frac{2}{11} a^{19} + \frac{1}{11} a^{18} - \frac{5}{11} a^{17} - \frac{4}{11} a^{16} - \frac{3}{11} a^{15} + \frac{5}{11} a^{14} - \frac{5}{11} a^{13} + \frac{5}{11} a^{12} - \frac{3}{11} a^{10} - \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{11} a^{4} - \frac{1}{11} a^{2} + \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{111397} a^{28} + \frac{750}{111397} a^{27} + \frac{27393}{111397} a^{26} + \frac{45827}{111397} a^{25} - \frac{33076}{111397} a^{24} + \frac{45043}{111397} a^{23} + \frac{30314}{111397} a^{22} + \frac{43151}{111397} a^{21} + \frac{51068}{111397} a^{20} - \frac{40176}{111397} a^{19} + \frac{5}{209} a^{18} + \frac{3565}{10127} a^{17} + \frac{44537}{111397} a^{16} + \frac{14809}{111397} a^{15} - \frac{55339}{111397} a^{14} - \frac{13554}{111397} a^{13} + \frac{35053}{111397} a^{12} - \frac{49910}{111397} a^{11} + \frac{1048}{8569} a^{10} - \frac{930}{5863} a^{9} - \frac{2220}{5863} a^{8} - \frac{4695}{111397} a^{7} - \frac{5003}{10127} a^{6} + \frac{17892}{111397} a^{5} + \frac{50983}{111397} a^{4} - \frac{50744}{111397} a^{3} + \frac{31830}{111397} a^{2} + \frac{2836}{111397} a - \frac{8357}{111397}$, $\frac{1}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{29} - \frac{4318115161972396767393357695084714578598772675094230685505374353148075570051852919966501845183519377727}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{28} - \frac{69686083801754455144855092485800993930545951717701278816823901520777337964324927205892977636874839933276247}{2230396901117535591816136095534108533165030827690851258162451939215337974100456001950068782593372101813085451} a^{27} + \frac{20210410949686267916254230488074300636206203392176787113282664195262400991765492514872711831898110161989294}{208598271327539299954027116848513747706082019855978894648286872012945278153280057736337368156214657004101517} a^{26} - \frac{1908947282810239070725915482148148100482818061741796164529390364944977849400544331347255359543056297012103724}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{25} + \frac{1489118883964017178758861868740641626008066060340847547440400917396483785126256891693075392419806323875939942}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{24} - \frac{237427492545542861458766088686605421514313701570543553501102319846872142798895542106088439286523214124570240}{1526061037606734878611040486418074259533968461051635071374309221568389140173996211860573377563886174924742677} a^{23} - \frac{9361227503587612168477142777769618934564582758920975466231776793090635990804537237343588818374948705473524949}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{22} + \frac{4346203041679468164415901638730060990550335839625850636298619960428723692596294026848595714344031095249067543}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{21} + \frac{10899401680191692441879514276701585468496965189494205942744024211921979446221767357303602492038948190483140974}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{20} + \frac{10986339744871078170175467712276972279898380025405423817067566027148937241030310451909150709340352584194156621}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{19} + \frac{936399554572860093503726754037501704854888529606411295741446196214090715711490596375775640355887859413362274}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{18} + \frac{7230082206020785766002590288099572607648068882935582919854489392240432184675226214452584820829337250960947826}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{17} + \frac{520949959995590691615238611812293431292046507438363833845837815707512897648542212272061451585808406998577426}{1526061037606734878611040486418074259533968461051635071374309221568389140173996211860573377563886174924742677} a^{16} - \frac{1417551589199051343581939474036900117365386040088536563026035967329846320112813613217266802470951970222223712}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{15} - \frac{2742204977774280363722358964464045364878240377815000686767653296252027361816716471168739118162281046210279552}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{14} - \frac{4342489797155302096895391049623317395759081135368042189770684003743262034166594720737578943205542551754379034}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{13} - \frac{18743826273976223188167328872064785699448481272236747655904980089574309031267277938875353353175442301161820}{80319001979301835716370551916740750501787813739559740598647853766757323167052432203188072503362430259196983} a^{12} - \frac{4496111277867511946773711451678372567469258341550930659219501738636031980526950750345517718628969793461649427}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{11} - \frac{13935768049519357612590982865966554015229323957946204047318104283492504260722309371402934038105333731909200964}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{10} - \frac{304532541400587425451410474054225212222582078328819486588203394666285966124423177001245478445325234635570489}{1526061037606734878611040486418074259533968461051635071374309221568389140173996211860573377563886174924742677} a^{9} + \frac{11726654685860068069644325872146257954166483856825830125904232526712574546504691288964932564509568724383088632}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{8} - \frac{574765073627184611387605367941835944100379271352044148524885239140107799982693152310872093918567059282881526}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{7} + \frac{580018871544782295418783928964703428501745405672845708471816399963207299879927421044256098809652638163718245}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{6} - \frac{5004995648413774578758802240260270662416586272110267603618210851445209870942940869234578965918758650584372443}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{5} + \frac{70684623891962971162435255123208832462905297062371537164613241683818030748984522716128605171245716548333909}{202763354647048690165103281412191684833184620699168296196586539928667088554586909268188071144852009255735041} a^{4} + \frac{1382882982324247378185936826110947778947333287811274614573790777334206374774559248173342011846100294492858549}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{3} - \frac{13387938109889961256477589301291404339943193790683199336856079573551609507288768672260249080987107720819239982}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a^{2} - \frac{2619270016103084494890277517692580434759835368079089411906432853442626467164425454876192812213657428788040911}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863} a + \frac{8560461315690175384369144837593853626066635814767123791305410460373145570035939635262230415927793811643657049}{28995159714527962693609769241943410931145400759981066356111875209799393663305928025350894173713837323570110863}$

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

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.1175.1, 5.1.2209.1, 6.0.186384375.1, 10.0.3705507759375.1, 15.1.4947491410771484375.1

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

Sibling fields

Degree 30 sibling: 30.2.51586587308205615747892932945519983768463134765625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ R R $30$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.705.2t1.a.a$1$ $ 3 \cdot 5 \cdot 47 $ \(\Q(\sqrt{705}) \) $C_2$ (as 2T1) $1$ $1$
1.47.2t1.a.a$1$ $ 47 $ \(\Q(\sqrt{-47}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.15.2t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\sqrt{-15}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.10575.6t3.b.a$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 6.2.8760065625.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.1175.3t2.a.a$2$ $ 5^{2} \cdot 47 $ 3.1.1175.1 $S_3$ (as 3T2) $1$ $0$
* 2.47.5t2.a.a$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.47.5t2.a.b$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.10575.10t3.b.b$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 10.2.174158864690625.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.10575.10t3.b.a$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 10.2.174158864690625.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.1175.15t2.a.b$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10575.30t14.b.c$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.0.1097586964004374803146658147777020931243896484375.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.1175.15t2.a.d$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10575.30t14.b.a$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.0.1097586964004374803146658147777020931243896484375.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.1175.15t2.a.c$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10575.30t14.b.d$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.0.1097586964004374803146658147777020931243896484375.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.1175.15t2.a.a$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10575.30t14.b.b$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.0.1097586964004374803146658147777020931243896484375.1 $D_{30}$ (as 30T14) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.