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Properties

Label 30.0.40311179820...6875.1
Degree $30$
Signature $[0, 15]$
Discriminant $-\,5^{15}\cdot 7^{25}\cdot 11^{24}$
Root discriminant $77.06$
Ramified primes $5, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_{30}$ (as 30T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2988813059, -688273342, 5106469195, -338196921, 5155475097, -288473284, 3146885812, 268246414, 1402022003, 296908461, 495709581, 127074731, 146815832, 35846744, 35140797, 7266688, 6916917, 1313102, 1070782, 216461, 136385, 28037, 15662, 2795, 1643, 129, 173, 6, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 12*x^28 + 6*x^27 + 173*x^26 + 129*x^25 + 1643*x^24 + 2795*x^23 + 15662*x^22 + 28037*x^21 + 136385*x^20 + 216461*x^19 + 1070782*x^18 + 1313102*x^17 + 6916917*x^16 + 7266688*x^15 + 35140797*x^14 + 35846744*x^13 + 146815832*x^12 + 127074731*x^11 + 495709581*x^10 + 296908461*x^9 + 1402022003*x^8 + 268246414*x^7 + 3146885812*x^6 - 288473284*x^5 + 5155475097*x^4 - 338196921*x^3 + 5106469195*x^2 - 688273342*x + 2988813059)
 
gp: K = bnfinit(x^30 - x^29 + 12*x^28 + 6*x^27 + 173*x^26 + 129*x^25 + 1643*x^24 + 2795*x^23 + 15662*x^22 + 28037*x^21 + 136385*x^20 + 216461*x^19 + 1070782*x^18 + 1313102*x^17 + 6916917*x^16 + 7266688*x^15 + 35140797*x^14 + 35846744*x^13 + 146815832*x^12 + 127074731*x^11 + 495709581*x^10 + 296908461*x^9 + 1402022003*x^8 + 268246414*x^7 + 3146885812*x^6 - 288473284*x^5 + 5155475097*x^4 - 338196921*x^3 + 5106469195*x^2 - 688273342*x + 2988813059, 1)
 

Normalized defining polynomial

\( x^{30} - x^{29} + 12 x^{28} + 6 x^{27} + 173 x^{26} + 129 x^{25} + 1643 x^{24} + 2795 x^{23} + 15662 x^{22} + 28037 x^{21} + 136385 x^{20} + 216461 x^{19} + 1070782 x^{18} + 1313102 x^{17} + 6916917 x^{16} + 7266688 x^{15} + 35140797 x^{14} + 35846744 x^{13} + 146815832 x^{12} + 127074731 x^{11} + 495709581 x^{10} + 296908461 x^{9} + 1402022003 x^{8} + 268246414 x^{7} + 3146885812 x^{6} - 288473284 x^{5} + 5155475097 x^{4} - 338196921 x^{3} + 5106469195 x^{2} - 688273342 x + 2988813059 \)

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

Invariants

Degree:  $30$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 15]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-403111798205699600205760315000413793695475942596435546875=-\,5^{15}\cdot 7^{25}\cdot 11^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.06$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(256,·)$, $\chi_{385}(1,·)$, $\chi_{385}(69,·)$, $\chi_{385}(199,·)$, $\chi_{385}(331,·)$, $\chi_{385}(141,·)$, $\chi_{385}(334,·)$, $\chi_{385}(269,·)$, $\chi_{385}(16,·)$, $\chi_{385}(81,·)$, $\chi_{385}(339,·)$, $\chi_{385}(366,·)$, $\chi_{385}(86,·)$, $\chi_{385}(279,·)$, $\chi_{385}(89,·)$, $\chi_{385}(221,·)$, $\chi_{385}(159,·)$, $\chi_{385}(34,·)$, $\chi_{385}(291,·)$, $\chi_{385}(36,·)$, $\chi_{385}(229,·)$, $\chi_{385}(104,·)$, $\chi_{385}(361,·)$, $\chi_{385}(234,·)$, $\chi_{385}(71,·)$, $\chi_{385}(174,·)$, $\chi_{385}(246,·)$, $\chi_{385}(59,·)$, $\chi_{385}(124,·)$, $\chi_{385}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13} a^{20} - \frac{3}{13} a^{19} + \frac{4}{13} a^{18} + \frac{2}{13} a^{17} + \frac{1}{13} a^{16} - \frac{3}{13} a^{15} + \frac{5}{13} a^{14} + \frac{3}{13} a^{13} + \frac{5}{13} a^{12} + \frac{5}{13} a^{11} - \frac{5}{13} a^{10} - \frac{5}{13} a^{7} + \frac{6}{13} a^{6} - \frac{5}{13} a^{5} - \frac{3}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a^{2} + \frac{1}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{21} - \frac{5}{13} a^{19} + \frac{1}{13} a^{18} - \frac{6}{13} a^{17} - \frac{4}{13} a^{15} + \frac{5}{13} a^{14} + \frac{1}{13} a^{13} - \frac{6}{13} a^{12} - \frac{3}{13} a^{11} - \frac{2}{13} a^{10} - \frac{5}{13} a^{8} + \frac{4}{13} a^{7} - \frac{5}{13} a^{5} + \frac{2}{13} a^{4} + \frac{6}{13} a^{3} - \frac{2}{13} a^{2} + \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{22} - \frac{1}{13} a^{19} + \frac{1}{13} a^{18} - \frac{3}{13} a^{17} + \frac{1}{13} a^{16} + \frac{3}{13} a^{15} - \frac{4}{13} a^{13} - \frac{4}{13} a^{12} - \frac{3}{13} a^{11} + \frac{1}{13} a^{10} - \frac{5}{13} a^{9} + \frac{4}{13} a^{8} + \frac{1}{13} a^{7} - \frac{1}{13} a^{6} + \frac{3}{13} a^{5} + \frac{4}{13} a^{4} + \frac{1}{13} a^{3} - \frac{2}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{23} - \frac{2}{13} a^{19} + \frac{1}{13} a^{18} + \frac{3}{13} a^{17} + \frac{4}{13} a^{16} - \frac{3}{13} a^{15} + \frac{1}{13} a^{14} - \frac{1}{13} a^{13} + \frac{2}{13} a^{12} + \frac{6}{13} a^{11} + \frac{3}{13} a^{10} + \frac{4}{13} a^{9} + \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{4}{13} a^{6} - \frac{1}{13} a^{5} - \frac{2}{13} a^{4} - \frac{2}{13} a^{3} - \frac{3}{13} a^{2} - \frac{2}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{24} - \frac{5}{13} a^{19} - \frac{2}{13} a^{18} - \frac{5}{13} a^{17} - \frac{1}{13} a^{16} - \frac{5}{13} a^{15} - \frac{4}{13} a^{14} - \frac{5}{13} a^{13} + \frac{3}{13} a^{12} - \frac{6}{13} a^{10} + \frac{1}{13} a^{9} - \frac{6}{13} a^{8} - \frac{1}{13} a^{7} - \frac{2}{13} a^{6} + \frac{1}{13} a^{5} + \frac{5}{13} a^{4} + \frac{6}{13} a^{3} - \frac{4}{13} a^{2} + \frac{4}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{25} - \frac{4}{13} a^{19} + \frac{2}{13} a^{18} - \frac{4}{13} a^{17} - \frac{6}{13} a^{15} - \frac{6}{13} a^{14} + \frac{5}{13} a^{13} - \frac{1}{13} a^{12} + \frac{6}{13} a^{11} + \frac{2}{13} a^{10} - \frac{6}{13} a^{9} - \frac{1}{13} a^{8} - \frac{1}{13} a^{7} + \frac{5}{13} a^{6} + \frac{6}{13} a^{5} + \frac{4}{13} a^{4} - \frac{1}{13} a^{3} - \frac{1}{13} a^{2} - \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{26} + \frac{3}{13} a^{19} - \frac{1}{13} a^{18} - \frac{5}{13} a^{17} - \frac{2}{13} a^{16} - \frac{5}{13} a^{15} - \frac{1}{13} a^{14} - \frac{2}{13} a^{13} - \frac{4}{13} a^{11} - \frac{1}{13} a^{9} - \frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{4}{13} a^{6} - \frac{3}{13} a^{5} + \frac{4}{13} a^{3} + \frac{5}{13} a^{2} + \frac{1}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{27} - \frac{5}{13} a^{19} - \frac{4}{13} a^{18} + \frac{5}{13} a^{17} + \frac{5}{13} a^{16} - \frac{5}{13} a^{15} - \frac{4}{13} a^{14} + \frac{4}{13} a^{13} - \frac{6}{13} a^{12} - \frac{2}{13} a^{11} + \frac{1}{13} a^{10} - \frac{1}{13} a^{9} - \frac{2}{13} a^{8} + \frac{6}{13} a^{7} + \frac{5}{13} a^{6} + \frac{2}{13} a^{5} - \frac{2}{13} a^{3} + \frac{4}{13} a^{2} + \frac{5}{13} a - \frac{6}{13}$, $\frac{1}{13} a^{28} - \frac{6}{13} a^{19} - \frac{1}{13} a^{18} + \frac{2}{13} a^{17} - \frac{6}{13} a^{15} + \frac{3}{13} a^{14} - \frac{4}{13} a^{13} - \frac{3}{13} a^{12} - \frac{2}{13} a^{9} + \frac{6}{13} a^{8} + \frac{6}{13} a^{7} + \frac{6}{13} a^{6} + \frac{1}{13} a^{5} - \frac{4}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a - \frac{3}{13}$, $\frac{1}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{29} + \frac{1986852555114428944534677031217758279412681156770413160255148798773926209622360305527905128290864126447758571067113822462465851916619560334}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{28} + \frac{935042451021110179562257178271255218014418270324332349602348893678370021759478826422655435781590434142842907897586649560172645593564023489}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{27} + \frac{1926183670878701954311961648031307728722311595594450891176368457139668337833019088059502702579891139747335390146244806840626239709082559949}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{26} - \frac{1058799702436412621918332805784855220166708000644173849753186019184981244429459523843944970902299707579609134170740321000410172322830687620}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{25} - \frac{1389995951983779398046093260990381286905598041585855765873654122766655445750961279786687767372702025880563640731150582487590703402361801366}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{24} - \frac{1841445628923700550646746789755817421357780641734081793043763870537879873098051322527187773540257479453436663525355408627824486314723752510}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{23} + \frac{1910886813849565451938187605004117788427001519389119374322991267345263099690682354108975038820813224666982861758785384634583696555344600167}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{22} - \frac{2025987221817317973623615324958323217364103681268631078464769998743770012946894009158804435226962425156527247165161281918827303631628676371}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{21} - \frac{496925175790220861373813902716928941970338671792626481013668498176550209758447463501362204344207897420385734437242555487736221084382174714}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{20} + \frac{11046160828375133522675611608793303638632999562870225435204100463449238160618633436569828919371924022456449304882442586051910925329679591921}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{19} + \frac{20334911141733927272256597074504231939987898769976084598274341569377148099153092549512867672591687351334887221070859184519130015637133557118}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{18} + \frac{23517156385403012073378822259723052068076540927148919018028673211120731514539917766521590597446593711862592290053059344943545163368304154643}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{17} - \frac{26983182748835618605829706602548603997122798616412184954563574610936476415252654235008519276490545906296242946527750338606328699072049607207}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{16} + \frac{19139653931889264720824112868622386522439171550971775361556718908077576284343680440880623565006672482705428756388566867446188934710600979196}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{15} + \frac{14229264619569673865756565396114382523963782236032020889912385295392639076953723059086968851958268651724525509284768129307594188188770690635}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{14} + \frac{7023214958673140020838867382927589734862461237649784896458044177715774951266422641733216069178792877121857232485245229113332849093037993993}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{13} - \frac{22044082115913115733350394666223383727006682082866364805615891045284022012342894802465859279551400701965307634780769187185112999671251695912}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{12} + \frac{3872722228363545425374228850305843277891828829351427809804701182979490923083570739914752689707631168809852080672274197265717367048516762700}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{11} - \frac{12662380448660046806016560888218340620229830217192260528191287442923463101517135762716950814168757662116478565024723778459458499388345567651}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{10} - \frac{13876085715878215256717965257027108825882969363965251068895898782823842272500178651153220958583611508946431386170753964198776004463610760558}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{9} - \frac{14022886539613028361829096574647867392853837379915533951679009718266933983492954618750175274715797347020731617383494658265584190294773670113}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{8} + \frac{1326405675380919464407023434673316970690024456268363826673846713870995695097321411059666685421330425699056224900839835322564403744487074332}{4552744283894509125258787270355746599714378758007306229867151108339699897057296015852261527665153568941139665906129371636996009784747550611} a^{7} - \frac{4305925408653307135233855942671410166929709712023265031011925656960526366942316458446209226026021559810521661670025731623602582221440623782}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{6} + \frac{21682810988431193799833587759508715050560522237025856473497042332003208943649146428688359187356028246602249481284233296439849511291365117182}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{5} + \frac{27357704240322631699982766029247539061945826924227182478858604540892352944835546591661020563431526108813860368187279747231903466033028349834}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{4} - \frac{6405703238936745428032954890290672566699863498501066498820274889211173601785856803496028834287890543037081284071484126222830090315615012356}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{3} + \frac{28510890979546160868811135329533425357235955760289377211670209542302615708853066386985031399318220265218457476527999403764867586006980098498}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a^{2} + \frac{4965803226637890800118270462431212567220127473587431636380330641565594057456626909164814688260622669522138809837261028824621075648076853842}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943} a + \frac{5266877853613371540430952828533820103928395279375193810759694290013087797932136439506205225354758360064026708334990134930144039503397383706}{59185675690628618628364234514624705796286923854094980988272964408416098661744848206079399859646996396234815656779681831280948127201718157943}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $14$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{30}$ (as 30T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.2100875.1, 10.0.11258530353021875.4, 15.15.886528337182930278529.1

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 $30$ $15^{2}$ R R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ $15^{2}$ $30$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $30$ $30$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $15^{2}$ $30$ $30$

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
5Data not computed
7Data not computed
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$