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Properties

Label 32.32.5456820171...0000.2
Degree $32$
Signature $[32, 0]$
Discriminant $2^{124}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $84.97$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -192, 376, 16600, -38980, -427104, 803092, 4705064, -6034148, -22254968, 25065416, 52549560, -61202386, -65186904, 87563076, 40994896, -73953556, -9662688, 37734684, -2751040, -11792220, 2495960, 2231260, -732960, -240255, 113616, 11652, -9792, 134, 440, -36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 - 36*x^30 + 440*x^29 + 134*x^28 - 9792*x^27 + 11652*x^26 + 113616*x^25 - 240255*x^24 - 732960*x^23 + 2231260*x^22 + 2495960*x^21 - 11792220*x^20 - 2751040*x^19 + 37734684*x^18 - 9662688*x^17 - 73953556*x^16 + 40994896*x^15 + 87563076*x^14 - 65186904*x^13 - 61202386*x^12 + 52549560*x^11 + 25065416*x^10 - 22254968*x^9 - 6034148*x^8 + 4705064*x^7 + 803092*x^6 - 427104*x^5 - 38980*x^4 + 16600*x^3 + 376*x^2 - 192*x + 1)
 
gp: K = bnfinit(x^32 - 8*x^31 - 36*x^30 + 440*x^29 + 134*x^28 - 9792*x^27 + 11652*x^26 + 113616*x^25 - 240255*x^24 - 732960*x^23 + 2231260*x^22 + 2495960*x^21 - 11792220*x^20 - 2751040*x^19 + 37734684*x^18 - 9662688*x^17 - 73953556*x^16 + 40994896*x^15 + 87563076*x^14 - 65186904*x^13 - 61202386*x^12 + 52549560*x^11 + 25065416*x^10 - 22254968*x^9 - 6034148*x^8 + 4705064*x^7 + 803092*x^6 - 427104*x^5 - 38980*x^4 + 16600*x^3 + 376*x^2 - 192*x + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} - 36 x^{30} + 440 x^{29} + 134 x^{28} - 9792 x^{27} + 11652 x^{26} + 113616 x^{25} - 240255 x^{24} - 732960 x^{23} + 2231260 x^{22} + 2495960 x^{21} - 11792220 x^{20} - 2751040 x^{19} + 37734684 x^{18} - 9662688 x^{17} - 73953556 x^{16} + 40994896 x^{15} + 87563076 x^{14} - 65186904 x^{13} - 61202386 x^{12} + 52549560 x^{11} + 25065416 x^{10} - 22254968 x^{9} - 6034148 x^{8} + 4705064 x^{7} + 803092 x^{6} - 427104 x^{5} - 38980 x^{4} + 16600 x^{3} + 376 x^{2} - 192 x + 1 \)

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:  \(54568201713507127370225565301626372096000000000000000000000000=2^{124}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.97$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(257,·)$, $\chi_{480}(137,·)$, $\chi_{480}(301,·)$, $\chi_{480}(17,·)$, $\chi_{480}(409,·)$, $\chi_{480}(413,·)$, $\chi_{480}(289,·)$, $\chi_{480}(293,·)$, $\chi_{480}(113,·)$, $\chi_{480}(169,·)$, $\chi_{480}(173,·)$, $\chi_{480}(49,·)$, $\chi_{480}(53,·)$, $\chi_{480}(317,·)$, $\chi_{480}(437,·)$, $\chi_{480}(181,·)$, $\chi_{480}(197,·)$, $\chi_{480}(77,·)$, $\chi_{480}(469,·)$, $\chi_{480}(377,·)$, $\chi_{480}(473,·)$, $\chi_{480}(349,·)$, $\chi_{480}(421,·)$, $\chi_{480}(353,·)$, $\chi_{480}(229,·)$, $\chi_{480}(361,·)$, $\chi_{480}(109,·)$, $\chi_{480}(61,·)$, $\chi_{480}(241,·)$, $\chi_{480}(233,·)$, $\chi_{480}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{31} a^{23} - \frac{9}{31} a^{22} + \frac{5}{31} a^{21} + \frac{8}{31} a^{20} + \frac{7}{31} a^{19} + \frac{15}{31} a^{18} - \frac{13}{31} a^{17} + \frac{5}{31} a^{16} - \frac{3}{31} a^{15} - \frac{7}{31} a^{14} + \frac{2}{31} a^{13} + \frac{3}{31} a^{12} + \frac{11}{31} a^{11} - \frac{14}{31} a^{10} - \frac{11}{31} a^{9} + \frac{13}{31} a^{8} + \frac{9}{31} a^{7} + \frac{9}{31} a^{6} - \frac{14}{31} a^{5} + \frac{3}{31} a^{4} - \frac{6}{31} a^{3} + \frac{2}{31} a^{2} - \frac{2}{31} a - \frac{11}{31}$, $\frac{1}{31} a^{24} - \frac{14}{31} a^{22} - \frac{9}{31} a^{21} - \frac{14}{31} a^{20} - \frac{15}{31} a^{19} - \frac{2}{31} a^{18} + \frac{12}{31} a^{17} + \frac{11}{31} a^{16} - \frac{3}{31} a^{15} + \frac{1}{31} a^{14} - \frac{10}{31} a^{13} + \frac{7}{31} a^{12} - \frac{8}{31} a^{11} - \frac{13}{31} a^{10} + \frac{7}{31} a^{9} + \frac{2}{31} a^{8} - \frac{3}{31} a^{7} + \frac{5}{31} a^{6} + \frac{1}{31} a^{5} - \frac{10}{31} a^{4} + \frac{10}{31} a^{3} - \frac{15}{31} a^{2} + \frac{2}{31} a - \frac{6}{31}$, $\frac{1}{31} a^{25} - \frac{11}{31} a^{22} - \frac{6}{31} a^{21} + \frac{4}{31} a^{20} + \frac{3}{31} a^{19} + \frac{5}{31} a^{18} + \frac{15}{31} a^{17} + \frac{5}{31} a^{16} - \frac{10}{31} a^{15} - \frac{15}{31} a^{14} + \frac{4}{31} a^{13} + \frac{3}{31} a^{12} - \frac{14}{31} a^{11} - \frac{3}{31} a^{10} + \frac{3}{31} a^{9} - \frac{7}{31} a^{8} + \frac{7}{31} a^{7} + \frac{3}{31} a^{6} + \frac{11}{31} a^{5} - \frac{10}{31} a^{4} - \frac{6}{31} a^{3} - \frac{1}{31} a^{2} - \frac{3}{31} a + \frac{1}{31}$, $\frac{1}{31} a^{26} - \frac{12}{31} a^{22} - \frac{3}{31} a^{21} - \frac{2}{31} a^{20} - \frac{11}{31} a^{19} - \frac{6}{31} a^{18} - \frac{14}{31} a^{17} + \frac{14}{31} a^{16} + \frac{14}{31} a^{15} - \frac{11}{31} a^{14} - \frac{6}{31} a^{13} - \frac{12}{31} a^{12} - \frac{6}{31} a^{11} + \frac{4}{31} a^{10} - \frac{4}{31} a^{9} - \frac{5}{31} a^{8} + \frac{9}{31} a^{7} - \frac{14}{31} a^{6} - \frac{9}{31} a^{5} - \frac{4}{31} a^{4} - \frac{5}{31} a^{3} - \frac{12}{31} a^{2} + \frac{10}{31} a + \frac{3}{31}$, $\frac{1}{31} a^{27} + \frac{13}{31} a^{22} - \frac{4}{31} a^{21} - \frac{8}{31} a^{20} - \frac{15}{31} a^{19} + \frac{11}{31} a^{18} + \frac{13}{31} a^{17} + \frac{12}{31} a^{16} + \frac{15}{31} a^{15} + \frac{3}{31} a^{14} + \frac{12}{31} a^{13} - \frac{1}{31} a^{12} + \frac{12}{31} a^{11} + \frac{14}{31} a^{10} - \frac{13}{31} a^{9} + \frac{10}{31} a^{8} + \frac{1}{31} a^{7} + \frac{6}{31} a^{6} + \frac{14}{31} a^{5} + \frac{9}{31} a^{3} + \frac{3}{31} a^{2} + \frac{10}{31} a - \frac{8}{31}$, $\frac{1}{217} a^{28} + \frac{3}{217} a^{26} + \frac{1}{217} a^{24} + \frac{3}{217} a^{23} - \frac{88}{217} a^{22} - \frac{107}{217} a^{21} + \frac{40}{217} a^{20} - \frac{2}{31} a^{19} - \frac{2}{217} a^{18} - \frac{74}{217} a^{17} - \frac{106}{217} a^{16} + \frac{10}{217} a^{15} + \frac{81}{217} a^{14} - \frac{18}{217} a^{13} - \frac{78}{217} a^{12} - \frac{13}{31} a^{11} + \frac{95}{217} a^{10} - \frac{71}{217} a^{9} + \frac{13}{217} a^{8} + \frac{64}{217} a^{7} - \frac{51}{217} a^{6} - \frac{41}{217} a^{5} - \frac{12}{217} a^{4} + \frac{89}{217} a^{3} + \frac{94}{217} a^{2} + \frac{106}{217} a - \frac{11}{217}$, $\frac{1}{217} a^{29} + \frac{3}{217} a^{27} + \frac{1}{217} a^{25} + \frac{3}{217} a^{24} + \frac{3}{217} a^{23} - \frac{58}{217} a^{22} + \frac{61}{217} a^{21} + \frac{9}{31} a^{20} - \frac{16}{217} a^{19} - \frac{11}{217} a^{18} + \frac{13}{217} a^{17} + \frac{1}{7} a^{16} + \frac{25}{217} a^{15} - \frac{4}{217} a^{14} + \frac{104}{217} a^{13} - \frac{5}{31} a^{12} + \frac{11}{217} a^{11} - \frac{43}{217} a^{10} + \frac{97}{217} a^{9} - \frac{55}{217} a^{8} - \frac{100}{217} a^{7} - \frac{90}{217} a^{6} + \frac{16}{217} a^{5} - \frac{72}{217} a^{4} - \frac{18}{217} a^{3} + \frac{71}{217} a^{2} + \frac{24}{217} a + \frac{12}{31}$, $\frac{1}{94405722113007991940106044857} a^{30} + \frac{167928544597000598410455262}{94405722113007991940106044857} a^{29} + \frac{24879488696186367930513908}{13486531730429713134300863551} a^{28} + \frac{1088213928393912380425808764}{94405722113007991940106044857} a^{27} + \frac{1437021569558145011405695975}{94405722113007991940106044857} a^{26} + \frac{302276769759941779309571034}{94405722113007991940106044857} a^{25} - \frac{99109529567735983109327410}{94405722113007991940106044857} a^{24} - \frac{902455885712922219579138166}{94405722113007991940106044857} a^{23} + \frac{39980354964134815736717547895}{94405722113007991940106044857} a^{22} + \frac{14884403946853952558121034466}{94405722113007991940106044857} a^{21} - \frac{37304076253212431486790346180}{94405722113007991940106044857} a^{20} + \frac{1140493685424623332376447974}{3045345874613161030326001447} a^{19} + \frac{2300575755969470869478385010}{94405722113007991940106044857} a^{18} - \frac{18263991507676352652269030424}{94405722113007991940106044857} a^{17} + \frac{10473523628400285296894931082}{94405722113007991940106044857} a^{16} - \frac{14175460460923704119464508737}{94405722113007991940106044857} a^{15} + \frac{4247590103590862865352678387}{13486531730429713134300863551} a^{14} - \frac{7468956298074166209703936964}{94405722113007991940106044857} a^{13} + \frac{6405560142095155556252991948}{13486531730429713134300863551} a^{12} + \frac{4468377678439853585608805437}{13486531730429713134300863551} a^{11} + \frac{27498850327245157288578127368}{94405722113007991940106044857} a^{10} + \frac{33713557224558971759958666717}{94405722113007991940106044857} a^{9} + \frac{5084433403573806021101173941}{94405722113007991940106044857} a^{8} - \frac{35059353743267557519464144306}{94405722113007991940106044857} a^{7} + \frac{39995602290441469332319232731}{94405722113007991940106044857} a^{6} + \frac{8687416462506586003300150048}{94405722113007991940106044857} a^{5} - \frac{6663551261578988709343032762}{13486531730429713134300863551} a^{4} - \frac{46383611537938325174170905210}{94405722113007991940106044857} a^{3} - \frac{20193818173257086236191977171}{94405722113007991940106044857} a^{2} - \frac{13118487669209348688918244180}{94405722113007991940106044857} a + \frac{26042479338862772218728823687}{94405722113007991940106044857}$, $\frac{1}{1132105189960412363031518246101768300076983} a^{31} - \frac{1846924389546}{1132105189960412363031518246101768300076983} a^{30} - \frac{633893511854603443399740028218037955435}{1132105189960412363031518246101768300076983} a^{29} + \frac{362010940265112526780694861973592515606}{161729312851487480433074035157395471439569} a^{28} + \frac{9162351897136968767448071420406217297285}{1132105189960412363031518246101768300076983} a^{27} + \frac{7301084734699109865502659138545716811562}{1132105189960412363031518246101768300076983} a^{26} - \frac{144300979470707250095510227735802942172}{161729312851487480433074035157395471439569} a^{25} - \frac{8993724559955311869790749898502566560650}{1132105189960412363031518246101768300076983} a^{24} - \frac{2087235168320040761644876345679183153441}{1132105189960412363031518246101768300076983} a^{23} - \frac{39561076703011714353532934786146026091852}{1132105189960412363031518246101768300076983} a^{22} - \frac{44768598033211530911912899148320521053897}{161729312851487480433074035157395471439569} a^{21} + \frac{253793293089683404115230207068699240275943}{1132105189960412363031518246101768300076983} a^{20} + \frac{111754037613234135987677071280752492324974}{1132105189960412363031518246101768300076983} a^{19} - \frac{47061900321567950965443399561051151589652}{161729312851487480433074035157395471439569} a^{18} + \frac{238944576357431763399864492768674936276611}{1132105189960412363031518246101768300076983} a^{17} + \frac{299966061575937529749247304334325798910952}{1132105189960412363031518246101768300076983} a^{16} - \frac{10716966458879239482816067096722991659224}{161729312851487480433074035157395471439569} a^{15} + \frac{111636078034436996792428220798154884801261}{1132105189960412363031518246101768300076983} a^{14} - \frac{147707803445779802331318930238642876288413}{1132105189960412363031518246101768300076983} a^{13} + \frac{477327427558094299022211978841611708581014}{1132105189960412363031518246101768300076983} a^{12} - \frac{183866769512811096895409080393392518074582}{1132105189960412363031518246101768300076983} a^{11} - \frac{543598844619363398731995552689776162245294}{1132105189960412363031518246101768300076983} a^{10} - \frac{211340446125100203796649927001152851832181}{1132105189960412363031518246101768300076983} a^{9} - \frac{28374061284856874052972892124861918055952}{1132105189960412363031518246101768300076983} a^{8} + \frac{262711749844118777854576396923436866936967}{1132105189960412363031518246101768300076983} a^{7} - \frac{508887770411727936723616776460568139050122}{1132105189960412363031518246101768300076983} a^{6} + \frac{403273159782972715551002342746631406237822}{1132105189960412363031518246101768300076983} a^{5} + \frac{188627696017584487715398021753757231153292}{1132105189960412363031518246101768300076983} a^{4} - \frac{468879044433409768712204256314524204055859}{1132105189960412363031518246101768300076983} a^{3} - \frac{34108200662315085290552550030150111574344}{161729312851487480433074035157395471439569} a^{2} - \frac{337009439832795844575723878543412685490974}{1132105189960412363031518246101768300076983} a - \frac{318499428050886374824028488516972339045934}{1132105189960412363031518246101768300076983}$

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:  \( 1022818136839327000000 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.72000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.4.2304000.2, 4.4.2304000.1, 8.8.5184000000.1, 8.8.2621440000.1, 8.8.5308416000000.1, 8.8.2717908992000000.2, 8.8.2717908992000000.1, 8.8.1342177280000.1, \(\Q(\zeta_{32})^+\), 16.16.28179280429056000000000000.2, 16.16.7387029288794456064000000000000.1, 16.16.1801439850948198400000000.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed