Properties

Label 32.32.1292684086...6753.1
Degree $32$
Signature $[32, 0]$
Discriminant $7^{16}\cdot 97^{31}$
Root discriminant $222.45$
Ramified primes $7, 97$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14559956239, 190504344411, 2097291371636, 5776725400421, 1413749921964, -18650067236811, -29363649724089, -1991967071353, 33098017056437, 26831580182190, -5531030792826, -17489038973421, -5872939632229, 3668327522541, 2957237171335, 51715582409, -559716324137, -139397040512, 47464737498, 24362757568, -767939030, -2067532985, -188031041, 98480308, 17183464, -2636561, -704581, 35407, 15855, -134, -192, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 192*x^30 - 134*x^29 + 15855*x^28 + 35407*x^27 - 704581*x^26 - 2636561*x^25 + 17183464*x^24 + 98480308*x^23 - 188031041*x^22 - 2067532985*x^21 - 767939030*x^20 + 24362757568*x^19 + 47464737498*x^18 - 139397040512*x^17 - 559716324137*x^16 + 51715582409*x^15 + 2957237171335*x^14 + 3668327522541*x^13 - 5872939632229*x^12 - 17489038973421*x^11 - 5531030792826*x^10 + 26831580182190*x^9 + 33098017056437*x^8 - 1991967071353*x^7 - 29363649724089*x^6 - 18650067236811*x^5 + 1413749921964*x^4 + 5776725400421*x^3 + 2097291371636*x^2 + 190504344411*x - 14559956239)
 
gp: K = bnfinit(x^32 - x^31 - 192*x^30 - 134*x^29 + 15855*x^28 + 35407*x^27 - 704581*x^26 - 2636561*x^25 + 17183464*x^24 + 98480308*x^23 - 188031041*x^22 - 2067532985*x^21 - 767939030*x^20 + 24362757568*x^19 + 47464737498*x^18 - 139397040512*x^17 - 559716324137*x^16 + 51715582409*x^15 + 2957237171335*x^14 + 3668327522541*x^13 - 5872939632229*x^12 - 17489038973421*x^11 - 5531030792826*x^10 + 26831580182190*x^9 + 33098017056437*x^8 - 1991967071353*x^7 - 29363649724089*x^6 - 18650067236811*x^5 + 1413749921964*x^4 + 5776725400421*x^3 + 2097291371636*x^2 + 190504344411*x - 14559956239, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 192 x^{30} - 134 x^{29} + 15855 x^{28} + 35407 x^{27} - 704581 x^{26} - 2636561 x^{25} + 17183464 x^{24} + 98480308 x^{23} - 188031041 x^{22} - 2067532985 x^{21} - 767939030 x^{20} + 24362757568 x^{19} + 47464737498 x^{18} - 139397040512 x^{17} - 559716324137 x^{16} + 51715582409 x^{15} + 2957237171335 x^{14} + 3668327522541 x^{13} - 5872939632229 x^{12} - 17489038973421 x^{11} - 5531030792826 x^{10} + 26831580182190 x^{9} + 33098017056437 x^{8} - 1991967071353 x^{7} - 29363649724089 x^{6} - 18650067236811 x^{5} + 1413749921964 x^{4} + 5776725400421 x^{3} + 2097291371636 x^{2} + 190504344411 x - 14559956239 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1292684086480957274390235903780365469718835315185422412744961998012838206753=7^{16}\cdot 97^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $222.45$
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:  $7, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(679=7\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{679}(512,·)$, $\chi_{679}(1,·)$, $\chi_{679}(8,·)$, $\chi_{679}(139,·)$, $\chi_{679}(652,·)$, $\chi_{679}(272,·)$, $\chi_{679}(531,·)$, $\chi_{679}(20,·)$, $\chi_{679}(22,·)$, $\chi_{679}(160,·)$, $\chi_{679}(34,·)$, $\chi_{679}(421,·)$, $\chi_{679}(552,·)$, $\chi_{679}(174,·)$, $\chi_{679}(176,·)$, $\chi_{679}(433,·)$, $\chi_{679}(50,·)$, $\chi_{679}(435,·)$, $\chi_{679}(309,·)$, $\chi_{679}(55,·)$, $\chi_{679}(440,·)$, $\chi_{679}(64,·)$, $\chi_{679}(321,·)$, $\chi_{679}(69,·)$, $\chi_{679}(463,·)$, $\chi_{679}(85,·)$, $\chi_{679}(342,·)$, $\chi_{679}(601,·)$, $\chi_{679}(477,·)$, $\chi_{679}(400,·)$, $\chi_{679}(484,·)$, $\chi_{679}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{1148093} a^{30} - \frac{525410}{1148093} a^{29} - \frac{406840}{1148093} a^{28} + \frac{573750}{1148093} a^{27} + \frac{236401}{1148093} a^{26} + \frac{71821}{1148093} a^{25} + \frac{490193}{1148093} a^{24} + \frac{419555}{1148093} a^{23} + \frac{61774}{1148093} a^{22} - \frac{74194}{1148093} a^{21} - \frac{181886}{1148093} a^{20} - \frac{355383}{1148093} a^{19} - \frac{70821}{1148093} a^{18} - \frac{374034}{1148093} a^{17} + \frac{388122}{1148093} a^{16} + \frac{481466}{1148093} a^{15} + \frac{375937}{1148093} a^{14} - \frac{569579}{1148093} a^{13} - \frac{28253}{1148093} a^{12} - \frac{121823}{1148093} a^{11} - \frac{353149}{1148093} a^{10} + \frac{105409}{1148093} a^{9} - \frac{429360}{1148093} a^{8} + \frac{521986}{1148093} a^{7} + \frac{316113}{1148093} a^{6} - \frac{221742}{1148093} a^{5} + \frac{422688}{1148093} a^{4} - \frac{119012}{1148093} a^{3} + \frac{487482}{1148093} a^{2} - \frac{347704}{1148093} a - \frac{550857}{1148093}$, $\frac{1}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{31} + \frac{100585678903713427158383970605847788526086626257715582860720255820173472151134972211890834329206906729163308540454595556250803707280956463908314605010519200}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{30} - \frac{349851376896528955110081680286870110760286330136919321152739690279375544320494599389516839110100600492343794768576315611387870843785051848256688179266027119359499}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{29} - \frac{38531681967536865226518551639350323673294965583823165947006940533802804970219905837849558529729344254362478182379978955778647688388365522402725344646014590935367}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{28} - \frac{517962333755438916672867094039087041523826343440623623245640431273975467649792562920429068590901301953814243723258902081277217315751475881154982656815356451907330}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{27} - \frac{252681631194095634951169098773311294859588004053334260974101298694399891755192710048750887494889267138335978430890342419357773723338740699056002911643549339249710}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{26} - \frac{399734213621195832788843534548393789354151303081019257333618960488934528164893302044189405071481870116603473004472602708753613553925379477209977730168318152397276}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{25} + \frac{343267078713947902603811390167896872359801239861546855469125478082701480748195271593526180203294203741077354955306140353888957548829606597202964634840478381130501}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{24} - \frac{293421528467421189808561475778449302122885259840735509807773951136284411978888834989076047884502163452627452392029081905112636294724946042388292392168704861342500}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{23} + \frac{339154919796652654797574071988422057276541662170741133325118358225218614376837471000493357970721220783253669712749752775043338736967521552269389932914021088043094}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{22} + \frac{188432207649120674954677490595698887795340312953470121740699551053356755918803660412170205112510528683483051269262434433602915735666907945413150096268643570413658}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{21} - \frac{516028731247626913683406907042285172758556822566305534481412548386931003727848760260427778789399314419635839007038187436005756096037261062712904842104460267847362}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{20} - \frac{274625829969616316678154754944513969881318187234068637925800825153446445542466317334753261917323519869512043365775339501229041240390275432975274411188320327220231}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{19} + \frac{83180275685486268727787290228836557238081927495191690434866864007595340769820866275017349665710251669238942147794476117161654591902645628516598530102190379248361}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{18} + \frac{6477600372895126487172981085926022523139389677707905602082428281986925162324667256316720077860955857977046893035282284938525053364643233035438803064135886538761}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{17} - \frac{599914811061529654674331689133397824171947915857105569754035042734100683099242270864184180068719924943069071031064476220277315867022302929105316445079545875751945}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{16} + \frac{454191306312631342129363223516316010258887506213572629485287916404529192470589239196399151345715856526483544316601449953719862997745904233084453396125792088924015}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{15} - \frac{593623095210932600840928419836526443828708619748305867490084313161620335074736208021805030392997724554269673685189140115614192075957759985946615270271117183892373}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{14} - \frac{315697830552622953471064125062648339433462612320460190504374946572796991931185561338956456595665528462018076175237753916044611354068494086746991649089999858312997}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{13} - \frac{168239948943831373349089397853711719459451146562474409706758410233320466378944568305237448629677328356658597433822817430905378187393730108566658028616655936187563}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{12} - \frac{591685908020140684152676263102645519756945275412484195080135747942974570104285108640141572242467338251661932643163697593479248277125230466789097501721767464116312}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{11} + \frac{49194191353371178780339468879387002408880779257288571398836906303053832220698596203130894593863424417540771770324320832879538620073788781575326290455280449923377}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{10} + \frac{305717891774245331192036186921949603388188176431108255002624473512335076316185919963844524841239954366310678451494485668179890824794479332780176857814194225776323}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{9} + \frac{168668866388096692828357657846883573838838405551839209771120704749893291036391611616184826088327962500636100886633999076786172586099790090334068884105453616339115}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{8} - \frac{237032579606332586193673125486068301812149367084714487797522369844140291537759617266389464105470271743689446914012780781109066645681238616132883090870069315781568}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{7} + \frac{233157063591867981960710798494322015874139873046690286613780387166178008552825366166327749636950057465526330582181217592419398128046827990911082149684487729679821}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{6} - \frac{319284129758305665159832545731096245287923696161136996183791955618261837655305949233418691743079188109266906993373635014037076830378668426473591051923702626434980}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{5} + \frac{323940436269316783911766344201001488937374914995753336519182311747029541942068588547099088881837072023819723416397100424716409699014228435833689417643255908518781}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{4} - \frac{25323844311837575653962412221778708536028445098892535804433350832487373932721353214811284461179394238042229338675548380066149880651866395547228894130109392204894}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{3} + \frac{386766720599078741450578213481388485088940662099079242252440315549085740897804076538695648227784398075676992724981900324417840396324174923266811696702296227053559}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a^{2} + \frac{388932768013682867998158281818879551933366030202309441613634678350636295703221231350892548352697705208115363753209442717215613953941588654634827979506141994465131}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051} a - \frac{438259269279722866119835259416381453792631251538383536652744939128323680497655902369423617661856940716033659749444482950159816109257129976876578550307960779725082}{1223439461167619170118548594397450534969827073547766334966899435089256720100087814296605741558661216030701194779925176486274101220712111324170153813318846974834051}$

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:  $31$
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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1, 16.16.633251189136789386043275954593.1

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 $16^{2}$ $16^{2}$ $32$ R $16^{2}$ $32$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $16^{2}$ $32$

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
7Data not computed
97Data not computed