↑

Properties

Label 32.32.5615889357...0625.1
Degree $32$
Signature $[32, 0]$
Discriminant $5^{24}\cdot 7^{16}\cdot 17^{28}$
Root discriminant $105.54$
Ramified primes $5, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times C_8$ (as 32T43)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3700304, 63812896, -384738348, 828328332, 731994753, -5889335682, 5947593437, 9449532252, -20290513343, -1494801150, 26279937053, -9777588040, -17900177751, 11614969466, 6946567999, -6592149900, -1474815293, 2251615110, 105769163, -497884694, 26561229, 73242962, -8540945, -7186022, 1146281, 461322, -87837, -18474, 3953, 416, -97, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 97*x^30 + 416*x^29 + 3953*x^28 - 18474*x^27 - 87837*x^26 + 461322*x^25 + 1146281*x^24 - 7186022*x^23 - 8540945*x^22 + 73242962*x^21 + 26561229*x^20 - 497884694*x^19 + 105769163*x^18 + 2251615110*x^17 - 1474815293*x^16 - 6592149900*x^15 + 6946567999*x^14 + 11614969466*x^13 - 17900177751*x^12 - 9777588040*x^11 + 26279937053*x^10 - 1494801150*x^9 - 20290513343*x^8 + 9449532252*x^7 + 5947593437*x^6 - 5889335682*x^5 + 731994753*x^4 + 828328332*x^3 - 384738348*x^2 + 63812896*x - 3700304)
 
gp: K = bnfinit(x^32 - 4*x^31 - 97*x^30 + 416*x^29 + 3953*x^28 - 18474*x^27 - 87837*x^26 + 461322*x^25 + 1146281*x^24 - 7186022*x^23 - 8540945*x^22 + 73242962*x^21 + 26561229*x^20 - 497884694*x^19 + 105769163*x^18 + 2251615110*x^17 - 1474815293*x^16 - 6592149900*x^15 + 6946567999*x^14 + 11614969466*x^13 - 17900177751*x^12 - 9777588040*x^11 + 26279937053*x^10 - 1494801150*x^9 - 20290513343*x^8 + 9449532252*x^7 + 5947593437*x^6 - 5889335682*x^5 + 731994753*x^4 + 828328332*x^3 - 384738348*x^2 + 63812896*x - 3700304, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 97 x^{30} + 416 x^{29} + 3953 x^{28} - 18474 x^{27} - 87837 x^{26} + 461322 x^{25} + 1146281 x^{24} - 7186022 x^{23} - 8540945 x^{22} + 73242962 x^{21} + 26561229 x^{20} - 497884694 x^{19} + 105769163 x^{18} + 2251615110 x^{17} - 1474815293 x^{16} - 6592149900 x^{15} + 6946567999 x^{14} + 11614969466 x^{13} - 17900177751 x^{12} - 9777588040 x^{11} + 26279937053 x^{10} - 1494801150 x^{9} - 20290513343 x^{8} + 9449532252 x^{7} + 5947593437 x^{6} - 5889335682 x^{5} + 731994753 x^{4} + 828328332 x^{3} - 384738348 x^{2} + 63812896 x - 3700304 \)

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:  \(56158893576626869963359344621945280012905849018156528472900390625=5^{24}\cdot 7^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.54$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(1,·)$, $\chi_{595}(132,·)$, $\chi_{595}(134,·)$, $\chi_{595}(519,·)$, $\chi_{595}(13,·)$, $\chi_{595}(526,·)$, $\chi_{595}(274,·)$, $\chi_{595}(281,·)$, $\chi_{595}(412,·)$, $\chi_{595}(36,·)$, $\chi_{595}(293,·)$, $\chi_{595}(552,·)$, $\chi_{595}(169,·)$, $\chi_{595}(433,·)$, $\chi_{595}(307,·)$, $\chi_{595}(188,·)$, $\chi_{595}(64,·)$, $\chi_{595}(118,·)$, $\chi_{595}(202,·)$, $\chi_{595}(587,·)$, $\chi_{595}(83,·)$, $\chi_{595}(468,·)$, $\chi_{595}(342,·)$, $\chi_{595}(344,·)$, $\chi_{595}(223,·)$, $\chi_{595}(484,·)$, $\chi_{595}(421,·)$, $\chi_{595}(106,·)$, $\chi_{595}(491,·)$, $\chi_{595}(237,·)$, $\chi_{595}(239,·)$, $\chi_{595}(246,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{14}$, $\frac{1}{7790632} a^{30} + \frac{449137}{3895316} a^{29} + \frac{1865067}{7790632} a^{28} - \frac{627963}{3895316} a^{27} + \frac{1858301}{7790632} a^{26} + \frac{158188}{973829} a^{25} - \frac{392225}{7790632} a^{24} + \frac{150660}{973829} a^{23} + \frac{1278745}{7790632} a^{22} + \frac{399023}{1947658} a^{21} + \frac{95163}{7790632} a^{20} - \frac{132890}{973829} a^{19} - \frac{1205831}{7790632} a^{18} + \frac{215260}{973829} a^{17} + \frac{1478375}{7790632} a^{16} - \frac{402763}{1947658} a^{15} - \frac{1828445}{7790632} a^{14} - \frac{1757777}{3895316} a^{13} - \frac{646581}{7790632} a^{12} - \frac{627941}{1947658} a^{11} + \frac{3780229}{7790632} a^{10} - \frac{311935}{3895316} a^{9} + \frac{118941}{7790632} a^{8} - \frac{29679}{973829} a^{7} - \frac{1979691}{7790632} a^{6} - \frac{3597}{9524} a^{5} - \frac{2842547}{7790632} a^{4} - \frac{233790}{973829} a^{3} + \frac{93033}{7790632} a^{2} - \frac{215053}{3895316} a + \frac{342003}{1947658}$, $\frac{1}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{31} + \frac{6283778163080785264478022797902462716841173960412781498585044222990695329915795950350854475821413060133364409511}{123340072463385473511668325254408694919121267132095215551328211854821281760684621695826020442033900710683880527569400116} a^{30} - \frac{75604199201885410869401158488251212910733420657121308149788938072078941344846214466712778579059167571866694172096613037}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{29} + \frac{5183630159580280565190255210813249690463490871918381703155550794509593046617572195155713094479565728981061268132655325}{30835018115846368377917081313602173729780316783023803887832052963705320440171155423956505110508475177670970131892350029} a^{28} - \frac{105661990148937625342460749709468859122064089168278838201856808083375297009211304314274810656338807264799805395652212675}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{27} - \frac{61563423255250121023482868282959469228931398188580606871118867408648443379386245425935902721579212362574929456807465329}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{26} + \frac{61202794546247642761754272311183640252702442026541222973899718931503172007350175201736357341735068512459591533288183903}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{25} - \frac{60323332768048888990387420897310311433551494252771520666510382355850498612562898629455161215333554340464267843287266271}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{24} - \frac{19656051079176817493142390819621607000417940076947872057900385649163448682467556964500310591573580277068055760359327211}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{23} + \frac{32596172585822268442426238146938968973474410783260611117678944074427928485351566487841909611965909929991665834099041381}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{22} + \frac{79205137257227392411598412471844116144883562321823098404844032152199899082674322068837137176185196008826818831804349723}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{21} - \frac{53765027774253024524256650933904538204226365570486809870181043796504520828896060216639626496231754120690828325158863519}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{20} + \frac{15671821520505786603050031845911267482817042904965645906372604044215649460946340521813952529186182797867615501044262233}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{19} - \frac{56812209829898093995399708280972447756716688945856030482005859159779775547967688320032067620757230586267190755771171335}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{18} + \frac{35269850659856162060225175454931878656250336105384458757642171717334189018443099941057759456273994543808127003071038359}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{17} - \frac{54584045198322884019747070736269149566281385998516101657509479095134718249618339582443756360852038144336235120137413}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{16} + \frac{34842865978401568044549199569222568273412605510986746025800319026674388808655249542059934714514896124075147805403848127}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{15} + \frac{39257253526465135853273122823406882061831382183085597291958713060997761511284222802939739702441773857601426618885928685}{123340072463385473511668325254408694919121267132095215551328211854821281760684621695826020442033900710683880527569400116} a^{14} - \frac{160828123026577929065885104306974772934539463802688114139688328159686518274898964312817013884048137773106547983817230749}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{13} + \frac{102615559805862323741165490568273389399144212987134893852139335301890375751469992469274514085182755265431630878414048657}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{12} - \frac{191442064441433070442266838343749300506630170240333041100861185511693315292339215867224493715254736668640047578054783707}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{11} + \frac{5444456978472482335723116272373397313922354454036185192612573789557680714686007068803886092845777945352195805798160187}{30835018115846368377917081313602173729780316783023803887832052963705320440171155423956505110508475177670970131892350029} a^{10} - \frac{136487483010284970748007866515341451278870062278174662818798840990964668022765916335380271328528451910168514993830010167}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{9} - \frac{21385953372910060477810400934635329241523977803339936112909429349066350321475007274620452511213360702616907072893599619}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{8} + \frac{87086133280257967902794822765601646455511709958725634397671835176961476153069295759789717110515818864378042178165445253}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{7} - \frac{34859382502897515985453814064600951238722675557666596369756360808719771702870706082876086365143163999880874540453224099}{123340072463385473511668325254408694919121267132095215551328211854821281760684621695826020442033900710683880527569400116} a^{6} + \frac{84119170584168570997862489518033909237256253338049251581713645611410586871406178416264195086260784466791450334756500305}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{5} - \frac{86757593963770070970749126317881367475334247890784115477040676663102200619285548348939189422972019113335841241784598181}{246680144926770947023336650508817389838242534264190431102656423709642563521369243391652040884067801421367761055138800232} a^{4} - \frac{106209952926302475310261701695741982303154177489880474111973888349851962993305100944152356852938731248300870526725143427}{493360289853541894046673301017634779676485068528380862205312847419285127042738486783304081768135602842735522110277600464} a^{3} + \frac{50750750235907636856220718511505260861829047840986180647794461445947742095713760837220738512958178175211499992183372341}{123340072463385473511668325254408694919121267132095215551328211854821281760684621695826020442033900710683880527569400116} a^{2} - \frac{2516664454140010929146374713129938755961856719292710286907155196879183675111189450734187833906420953955361252605351263}{61670036231692736755834162627204347459560633566047607775664105927410640880342310847913010221016950355341940263784700058} a + \frac{26734125895490406043244594920683713469022222166939528665521556805821006800116178650609951150315703805206623257078035085}{61670036231692736755834162627204347459560633566047607775664105927410640880342310847913010221016950355341940263784700058}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $31$
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:  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:  \( 16668045472879155000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.30092125.1, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.30092125.2, 4.4.122825.1, 4.4.4913.1, 4.4.1770125.1, 4.4.6125.1, 8.8.905535987015625.1, 8.8.15085980625.1, 8.8.3133342515625.1, 8.8.15394111779265625.1, 8.8.15394111779265625.2, 8.8.256461670625.1, \(\Q(\zeta_{17})^+\), 16.16.819995423780362168593994140625.1, 16.16.236978677472524666723664306640625.1, 16.16.65772588499765987890625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
17Data not computed