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Properties

Label 32.0.18622155271...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{16}\cdot 17^{28}$
Root discriminant $46.20$
Ramified primes $3, 5, 17$
Class number $2040$ (GRH)
Class group $[2040]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 62, 184, 1791, 2674, 13558, 9995, 67299, 28296, 187873, 46478, 369102, 64009, 494181, 64136, 490499, 50946, 346044, 24855, 180555, 7650, 64543, -338, 16520, -327, 2779, -122, 329, -12, 23, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 23*x^30 - 12*x^29 + 329*x^28 - 122*x^27 + 2779*x^26 - 327*x^25 + 16520*x^24 - 338*x^23 + 64543*x^22 + 7650*x^21 + 180555*x^20 + 24855*x^19 + 346044*x^18 + 50946*x^17 + 490499*x^16 + 64136*x^15 + 494181*x^14 + 64009*x^13 + 369102*x^12 + 46478*x^11 + 187873*x^10 + 28296*x^9 + 67299*x^8 + 9995*x^7 + 13558*x^6 + 2674*x^5 + 1791*x^4 + 184*x^3 + 62*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^32 - x^31 + 23*x^30 - 12*x^29 + 329*x^28 - 122*x^27 + 2779*x^26 - 327*x^25 + 16520*x^24 - 338*x^23 + 64543*x^22 + 7650*x^21 + 180555*x^20 + 24855*x^19 + 346044*x^18 + 50946*x^17 + 490499*x^16 + 64136*x^15 + 494181*x^14 + 64009*x^13 + 369102*x^12 + 46478*x^11 + 187873*x^10 + 28296*x^9 + 67299*x^8 + 9995*x^7 + 13558*x^6 + 2674*x^5 + 1791*x^4 + 184*x^3 + 62*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 23 x^{30} - 12 x^{29} + 329 x^{28} - 122 x^{27} + 2779 x^{26} - 327 x^{25} + 16520 x^{24} - 338 x^{23} + 64543 x^{22} + 7650 x^{21} + 180555 x^{20} + 24855 x^{19} + 346044 x^{18} + 50946 x^{17} + 490499 x^{16} + 64136 x^{15} + 494181 x^{14} + 64009 x^{13} + 369102 x^{12} + 46478 x^{11} + 187873 x^{10} + 28296 x^{9} + 67299 x^{8} + 9995 x^{7} + 13558 x^{6} + 2674 x^{5} + 1791 x^{4} + 184 x^{3} + 62 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(186221552718646678775874367536462493032617340087890625=3^{16}\cdot 5^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.20$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(4,·)$, $\chi_{255}(134,·)$, $\chi_{255}(16,·)$, $\chi_{255}(19,·)$, $\chi_{255}(149,·)$, $\chi_{255}(151,·)$, $\chi_{255}(26,·)$, $\chi_{255}(154,·)$, $\chi_{255}(161,·)$, $\chi_{255}(166,·)$, $\chi_{255}(169,·)$, $\chi_{255}(49,·)$, $\chi_{255}(179,·)$, $\chi_{255}(59,·)$, $\chi_{255}(191,·)$, $\chi_{255}(64,·)$, $\chi_{255}(196,·)$, $\chi_{255}(76,·)$, $\chi_{255}(206,·)$, $\chi_{255}(86,·)$, $\chi_{255}(89,·)$, $\chi_{255}(94,·)$, $\chi_{255}(101,·)$, $\chi_{255}(229,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(236,·)$, $\chi_{255}(239,·)$, $\chi_{255}(121,·)$, $\chi_{255}(251,·)$, $\chi_{255}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{18} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{19} - \frac{1}{4} a^{13} - \frac{1}{2} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{20} - \frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} - \frac{1}{4} a^{15} + \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{188} a^{28} + \frac{5}{188} a^{27} - \frac{15}{188} a^{26} - \frac{3}{47} a^{24} - \frac{1}{47} a^{23} + \frac{31}{188} a^{22} - \frac{3}{188} a^{21} - \frac{29}{188} a^{20} + \frac{21}{94} a^{19} - \frac{23}{94} a^{18} + \frac{4}{47} a^{17} - \frac{21}{188} a^{16} - \frac{33}{188} a^{15} - \frac{33}{188} a^{14} + \frac{4}{47} a^{13} - \frac{10}{47} a^{12} + \frac{7}{94} a^{11} - \frac{51}{188} a^{10} + \frac{61}{188} a^{9} - \frac{65}{188} a^{8} - \frac{23}{94} a^{7} - \frac{23}{47} a^{6} + \frac{22}{47} a^{5} + \frac{59}{188} a^{4} - \frac{61}{188} a^{3} + \frac{91}{188} a^{2} + \frac{35}{94} a + \frac{23}{94}$, $\frac{1}{188} a^{29} + \frac{7}{188} a^{27} - \frac{19}{188} a^{26} - \frac{3}{47} a^{25} + \frac{9}{188} a^{24} - \frac{43}{188} a^{23} + \frac{15}{94} a^{22} + \frac{33}{188} a^{21} - \frac{1}{188} a^{20} + \frac{13}{94} a^{19} + \frac{11}{188} a^{18} - \frac{7}{188} a^{17} - \frac{11}{94} a^{16} - \frac{9}{188} a^{15} - \frac{7}{188} a^{14} - \frac{13}{94} a^{13} - \frac{21}{188} a^{12} - \frac{27}{188} a^{11} + \frac{17}{94} a^{10} + \frac{53}{188} a^{9} - \frac{3}{188} a^{8} + \frac{11}{47} a^{7} - \frac{63}{188} a^{6} - \frac{5}{188} a^{5} + \frac{5}{47} a^{4} + \frac{67}{188} a^{3} + \frac{85}{188} a^{2} + \frac{18}{47} a - \frac{89}{188}$, $\frac{1}{8917714962141037156} a^{30} - \frac{13684232734587351}{8917714962141037156} a^{29} - \frac{187919616363481}{2229428740535259289} a^{28} + \frac{462134890013706903}{8917714962141037156} a^{27} - \frac{104551188913173075}{8917714962141037156} a^{26} - \frac{77561511962948555}{8917714962141037156} a^{25} - \frac{775120587387181167}{8917714962141037156} a^{24} - \frac{559559220237530639}{8917714962141037156} a^{23} - \frac{5164869785921548}{47434654053941687} a^{22} - \frac{1585611064367090937}{8917714962141037156} a^{21} + \frac{951156303944434161}{8917714962141037156} a^{20} - \frac{12636102215634753}{8917714962141037156} a^{19} + \frac{1992322241138876609}{8917714962141037156} a^{18} + \frac{298576119357280377}{8917714962141037156} a^{17} + \frac{4841162222930637}{94869308107883374} a^{16} + \frac{28994870494930979}{8917714962141037156} a^{15} + \frac{1304165479883795253}{8917714962141037156} a^{14} + \frac{1916152699242410821}{8917714962141037156} a^{13} + \frac{1096991895101777065}{8917714962141037156} a^{12} - \frac{3160697357333510827}{8917714962141037156} a^{11} - \frac{926121646083071467}{2229428740535259289} a^{10} + \frac{3797681093834689823}{8917714962141037156} a^{9} - \frac{4237380666717256443}{8917714962141037156} a^{8} + \frac{2212223650356178869}{8917714962141037156} a^{7} + \frac{3151963145991898861}{8917714962141037156} a^{6} - \frac{2518355984190439859}{8917714962141037156} a^{5} + \frac{1246749091243979897}{4458857481070518578} a^{4} - \frac{2658093013936379969}{8917714962141037156} a^{3} + \frac{725213211001218293}{8917714962141037156} a^{2} - \frac{1594306239932223369}{8917714962141037156} a + \frac{202397835938063495}{2229428740535259289}$, $\frac{1}{27844693362534749437634760664619689943375785856596} a^{31} + \frac{818476907587350441926692763527}{27844693362534749437634760664619689943375785856596} a^{30} + \frac{1425155984811305331819482245617835746198655102}{6961173340633687359408690166154922485843946464149} a^{29} + \frac{10678518230629474269038331968247856887548467105}{27844693362534749437634760664619689943375785856596} a^{28} - \frac{426379954280061094140813921849513332147629930395}{13922346681267374718817380332309844971687892928298} a^{27} - \frac{1734450349921338780240702850512376616995717438399}{13922346681267374718817380332309844971687892928298} a^{26} - \frac{3387785943268954200977253515941647619816662880453}{27844693362534749437634760664619689943375785856596} a^{25} - \frac{2905159913779165341746174862547804709525443382441}{27844693362534749437634760664619689943375785856596} a^{24} - \frac{65176991531816180685861049794265932387624113139}{6961173340633687359408690166154922485843946464149} a^{23} + \frac{1241796409084552158993860835265798524245690527955}{27844693362534749437634760664619689943375785856596} a^{22} + \frac{1638952373987559082053340572638919187914629400099}{6961173340633687359408690166154922485843946464149} a^{21} - \frac{3161576273800712948692340948567233995452353581951}{13922346681267374718817380332309844971687892928298} a^{20} - \frac{1174187353864229455753407094483976397270327775905}{27844693362534749437634760664619689943375785856596} a^{19} + \frac{1066001918716406373060341908365700228760196256061}{27844693362534749437634760664619689943375785856596} a^{18} - \frac{21832809847360281736445547339919363524019066319}{135168414381236647755508546915629562831921290566} a^{17} - \frac{6646354417017005612333573650437422258385469208091}{27844693362534749437634760664619689943375785856596} a^{16} + \frac{409759915085617353641977923624777319464594239906}{6961173340633687359408690166154922485843946464149} a^{15} + \frac{1174760153857376495408422145678018407275368942696}{6961173340633687359408690166154922485843946464149} a^{14} - \frac{6029494922322361101292820768730971483564196270075}{27844693362534749437634760664619689943375785856596} a^{13} - \frac{2826932069213476714433424354748896570626437953451}{27844693362534749437634760664619689943375785856596} a^{12} + \frac{6502734319407035724982308781231566398212262603261}{13922346681267374718817380332309844971687892928298} a^{11} + \frac{7750957978499176535260471035044289169229067434787}{27844693362534749437634760664619689943375785856596} a^{10} + \frac{3294463635329185005537416920990163267016501708625}{6961173340633687359408690166154922485843946464149} a^{9} + \frac{648115107890303777960854800882250647099802151891}{6961173340633687359408690166154922485843946464149} a^{8} + \frac{12103034179236845559907395783431442189470035707087}{27844693362534749437634760664619689943375785856596} a^{7} + \frac{41428380773579717946718601816303720207245175021}{90699326913793972109559480992246547046826664028} a^{6} + \frac{792168491353952763387572874468564591894930128953}{13922346681267374718817380332309844971687892928298} a^{5} - \frac{7053027669535474055101089793502449039120497595737}{27844693362534749437634760664619689943375785856596} a^{4} + \frac{4021078362971770089921086834913941434106123590659}{13922346681267374718817380332309844971687892928298} a^{3} + \frac{3250734369971017115670091329254693145990301487889}{13922346681267374718817380332309844971687892928298} a^{2} + \frac{1878552601739869158048778956642092926826159890752}{6961173340633687359408690166154922485843946464149} a + \frac{3329818865389917415371254552636308323719878660}{535474872356437489185283858934994037372611266473}$

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

Class group and class number

$C_{2040}$, which has order $2040$ (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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{298645634274941971178703072668392056786645}{3568955800412968998594032627962117247016731} a^{31} + \frac{905224901404256777243461065177306532946023}{14275823201651875994376130511848468988066924} a^{30} - \frac{27161252639963989134722848181898232487959609}{14275823201651875994376130511848468988066924} a^{29} + \frac{7668622559643726792438903963277065504911607}{14275823201651875994376130511848468988066924} a^{28} - \frac{97246981692799824761374878886238579608107236}{3568955800412968998594032627962117247016731} a^{27} + \frac{25276398177358375400255780203680537362008607}{7137911600825937997188065255924234494033462} a^{26} - \frac{3276427262323453479991819833227753239797826653}{14275823201651875994376130511848468988066924} a^{25} - \frac{4384162715250273465186629841952992102422171}{151870459592041233982724792679239031787946} a^{24} - \frac{19572325318284888547704873773129506689669899509}{14275823201651875994376130511848468988066924} a^{23} - \frac{4347698515337263449619952634491740637918141107}{14275823201651875994376130511848468988066924} a^{22} - \frac{19149067654953303852696086917067746023518104332}{3568955800412968998594032627962117247016731} a^{21} - \frac{13802989248210558440807207984750691390723798057}{7137911600825937997188065255924234494033462} a^{20} - \frac{216291018982861737061446317187260744462724142913}{14275823201651875994376130511848468988066924} a^{19} - \frac{20228592292180471072530715508717038940789496762}{3568955800412968998594032627962117247016731} a^{18} - \frac{415936679302621057161726966965358969220562834611}{14275823201651875994376130511848468988066924} a^{17} - \frac{158086769775180028445477930791123825756843973477}{14275823201651875994376130511848468988066924} a^{16} - \frac{147912803375990889329214496702163210895877930230}{3568955800412968998594032627962117247016731} a^{15} - \frac{106522729932167494985738062234647028461764066261}{7137911600825937997188065255924234494033462} a^{14} - \frac{595874413312939086760418113746796916352716952487}{14275823201651875994376130511848468988066924} a^{13} - \frac{53017585943776490225573982239425339886761931379}{3568955800412968998594032627962117247016731} a^{12} - \frac{446150951181574613211158045930876563182978495093}{14275823201651875994376130511848468988066924} a^{11} - \frac{155230197875011730200997869827426527302625571991}{14275823201651875994376130511848468988066924} a^{10} - \frac{113879849850586919930602648995695502322491532699}{7137911600825937997188065255924234494033462} a^{9} - \frac{41576176734028776209941354620981659460988489229}{7137911600825937997188065255924234494033462} a^{8} - \frac{83287392282502206752983751232121184722856507185}{14275823201651875994376130511848468988066924} a^{7} - \frac{7247693422549006291272788558351400261831263196}{3568955800412968998594032627962117247016731} a^{6} - \frac{17087742441795037244298011782202805432040679877}{14275823201651875994376130511848468988066924} a^{5} - \frac{6334869749614727178432626128466430863603579463}{14275823201651875994376130511848468988066924} a^{4} - \frac{47887102825267382777354846455089761071380172}{274535061570228384507233279074009019001287} a^{3} - \frac{157518929273454653292089707381838564218533437}{3568955800412968998594032627962117247016731} a^{2} - \frac{65396898248455572973505462341992025793093495}{14275823201651875994376130511848468988066924} a + \frac{2776519760477847458865363517840746723725577}{14275823201651875994376130511848468988066924} \) (order $6$)
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:  \( 42597284566.379654 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{85})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-15}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-51})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-15}, \sqrt{-51})\), 4.4.4913.1, 4.0.44217.1, 4.0.1105425.1, 4.4.122825.1, 8.0.4228250625.1, 8.0.1955143089.1, 8.0.1221964430625.4, 8.0.1221964430625.2, 8.0.1221964430625.1, 8.8.15085980625.1, 8.0.1221964430625.3, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 8.0.20773395320625.1, 8.8.256461670625.1, 16.0.1493197069712680437890625.1, 16.0.1104726920056229495169.1, 16.0.431533953146964646550390625.3, 16.0.431533953146964646550390625.1, 16.0.431533953146964646550390625.4, 16.16.65772588499765987890625.1, 16.0.431533953146964646550390625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
3Data not computed
5Data not computed
17Data not computed