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Properties

Label 32.0.72742794030...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $69.09$
Ramified primes $3, 5, 17$
Class number $4964$ (GRH)
Class group $[2, 2482]$ (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![330991, 211141, 1319440, -102743, 7220745, 9007615, 28309424, 19525837, 31270930, 3190803, 10963461, -11329204, -48880, -6589011, -197086, 27606, 580886, 488309, 259309, -60297, -43829, -28328, -15683, -784, 6029, 123, 194, 136, -157, -2, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + x^30 - 2*x^29 - 157*x^28 + 136*x^27 + 194*x^26 + 123*x^25 + 6029*x^24 - 784*x^23 - 15683*x^22 - 28328*x^21 - 43829*x^20 - 60297*x^19 + 259309*x^18 + 488309*x^17 + 580886*x^16 + 27606*x^15 - 197086*x^14 - 6589011*x^13 - 48880*x^12 - 11329204*x^11 + 10963461*x^10 + 3190803*x^9 + 31270930*x^8 + 19525837*x^7 + 28309424*x^6 + 9007615*x^5 + 7220745*x^4 - 102743*x^3 + 1319440*x^2 + 211141*x + 330991)
 
gp: K = bnfinit(x^32 - x^31 + x^30 - 2*x^29 - 157*x^28 + 136*x^27 + 194*x^26 + 123*x^25 + 6029*x^24 - 784*x^23 - 15683*x^22 - 28328*x^21 - 43829*x^20 - 60297*x^19 + 259309*x^18 + 488309*x^17 + 580886*x^16 + 27606*x^15 - 197086*x^14 - 6589011*x^13 - 48880*x^12 - 11329204*x^11 + 10963461*x^10 + 3190803*x^9 + 31270930*x^8 + 19525837*x^7 + 28309424*x^6 + 9007615*x^5 + 7220745*x^4 - 102743*x^3 + 1319440*x^2 + 211141*x + 330991, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + x^{30} - 2 x^{29} - 157 x^{28} + 136 x^{27} + 194 x^{26} + 123 x^{25} + 6029 x^{24} - 784 x^{23} - 15683 x^{22} - 28328 x^{21} - 43829 x^{20} - 60297 x^{19} + 259309 x^{18} + 488309 x^{17} + 580886 x^{16} + 27606 x^{15} - 197086 x^{14} - 6589011 x^{13} - 48880 x^{12} - 11329204 x^{11} + 10963461 x^{10} + 3190803 x^{9} + 31270930 x^{8} + 19525837 x^{7} + 28309424 x^{6} + 9007615 x^{5} + 7220745 x^{4} - 102743 x^{3} + 1319440 x^{2} + 211141 x + 330991 \)

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:  \(72742794030721358896825924818930661340866148471832275390625=3^{16}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.09$
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$$\chi_{255}(1,·)$, $\chi_{255}(4,·)$, $\chi_{255}(134,·)$, $\chi_{255}(137,·)$, $\chi_{255}(16,·)$, $\chi_{255}(152,·)$, $\chi_{255}(26,·)$, $\chi_{255}(154,·)$, $\chi_{255}(161,·)$, $\chi_{255}(38,·)$, $\chi_{255}(169,·)$, $\chi_{255}(43,·)$, $\chi_{255}(172,·)$, $\chi_{255}(47,·)$, $\chi_{255}(178,·)$, $\chi_{255}(179,·)$, $\chi_{255}(59,·)$, $\chi_{255}(188,·)$, $\chi_{255}(64,·)$, $\chi_{255}(202,·)$, $\chi_{255}(203,·)$, $\chi_{255}(206,·)$, $\chi_{255}(223,·)$, $\chi_{255}(98,·)$, $\chi_{255}(166,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(236,·)$, $\chi_{255}(242,·)$, $\chi_{255}(247,·)$, $\chi_{255}(253,·)$, $\chi_{255}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{10}$, $\frac{1}{356} a^{26} - \frac{25}{178} a^{25} - \frac{25}{178} a^{24} + \frac{7}{356} a^{23} - \frac{5}{356} a^{22} + \frac{1}{178} a^{21} - \frac{1}{356} a^{20} - \frac{13}{89} a^{19} - \frac{5}{356} a^{18} + \frac{1}{356} a^{17} + \frac{85}{356} a^{16} + \frac{53}{356} a^{15} - \frac{83}{178} a^{14} - \frac{75}{178} a^{13} - \frac{7}{89} a^{12} - \frac{33}{356} a^{11} + \frac{71}{178} a^{10} + \frac{15}{89} a^{9} + \frac{117}{356} a^{8} - \frac{139}{356} a^{7} - \frac{24}{89} a^{6} - \frac{25}{356} a^{5} - \frac{14}{89} a^{4} - \frac{125}{356} a^{3} + \frac{15}{356} a^{2} - \frac{51}{356} a + \frac{1}{4}$, $\frac{1}{53044} a^{27} - \frac{19}{26522} a^{26} - \frac{1542}{13261} a^{25} - \frac{237}{53044} a^{24} + \frac{613}{53044} a^{23} + \frac{297}{13261} a^{22} + \frac{1981}{53044} a^{21} - \frac{3309}{13261} a^{20} - \frac{6859}{53044} a^{19} - \frac{3085}{53044} a^{18} - \frac{8625}{53044} a^{17} - \frac{6047}{53044} a^{16} + \frac{1942}{13261} a^{15} + \frac{531}{26522} a^{14} + \frac{611}{13261} a^{13} + \frac{23839}{53044} a^{12} + \frac{1119}{26522} a^{11} + \frac{10761}{26522} a^{10} + \frac{25045}{53044} a^{9} + \frac{22091}{53044} a^{8} + \frac{3301}{26522} a^{7} - \frac{10611}{53044} a^{6} + \frac{4}{149} a^{5} + \frac{16113}{53044} a^{4} - \frac{25871}{53044} a^{3} - \frac{14289}{53044} a^{2} - \frac{7643}{53044} a + \frac{109}{298}$, $\frac{1}{53044} a^{28} - \frac{13}{53044} a^{26} - \frac{4565}{53044} a^{25} + \frac{9487}{53044} a^{24} - \frac{1891}{53044} a^{23} + \frac{4565}{26522} a^{22} - \frac{1163}{26522} a^{21} + \frac{6507}{26522} a^{20} + \frac{4175}{53044} a^{19} - \frac{2359}{26522} a^{18} - \frac{3967}{26522} a^{17} - \frac{455}{53044} a^{16} + \frac{9421}{53044} a^{15} + \frac{689}{26522} a^{14} - \frac{15303}{53044} a^{13} + \frac{1444}{13261} a^{12} + \frac{14931}{53044} a^{11} - \frac{14167}{53044} a^{10} + \frac{24075}{53044} a^{9} + \frac{11213}{53044} a^{8} - \frac{5084}{13261} a^{7} + \frac{2287}{13261} a^{6} - \frac{6831}{26522} a^{5} - \frac{24775}{53044} a^{4} + \frac{7683}{26522} a^{3} - \frac{3072}{13261} a^{2} - \frac{22053}{53044} a - \frac{209}{596}$, $\frac{1}{53044} a^{29} + \frac{7}{53044} a^{26} - \frac{5733}{53044} a^{25} + \frac{1737}{13261} a^{24} - \frac{483}{53044} a^{23} - \frac{3053}{13261} a^{22} - \frac{4145}{53044} a^{21} + \frac{12695}{53044} a^{20} - \frac{12531}{53044} a^{19} + \frac{6197}{53044} a^{18} - \frac{713}{26522} a^{17} - \frac{2472}{13261} a^{16} - \frac{112}{13261} a^{15} + \frac{6251}{53044} a^{14} + \frac{34}{89} a^{13} + \frac{11929}{26522} a^{12} + \frac{6881}{53044} a^{11} + \frac{173}{596} a^{10} + \frac{2115}{26522} a^{9} + \frac{10865}{53044} a^{8} - \frac{6429}{13261} a^{7} - \frac{1413}{53044} a^{6} - \frac{303}{53044} a^{5} + \frac{20705}{53044} a^{4} - \frac{547}{53044} a^{3} - \frac{6433}{13261} a^{2} - \frac{2509}{26522} a + \frac{38}{149}$, $\frac{1}{1806686441794958212} a^{30} - \frac{1935227505002}{451671610448739553} a^{29} - \frac{8528333360397}{1806686441794958212} a^{28} + \frac{3489384232013}{451671610448739553} a^{27} - \frac{1005807844263085}{903343220897479106} a^{26} - \frac{369644412111958465}{1806686441794958212} a^{25} - \frac{205264245526533841}{1806686441794958212} a^{24} + \frac{9680753198062031}{451671610448739553} a^{23} - \frac{24406342252978461}{1806686441794958212} a^{22} - \frac{92290245594451281}{451671610448739553} a^{21} - \frac{22777760838043857}{1806686441794958212} a^{20} - \frac{226121302123993437}{1806686441794958212} a^{19} + \frac{5428399385772135}{1806686441794958212} a^{18} - \frac{201481233982356419}{1806686441794958212} a^{17} + \frac{179682496624894047}{903343220897479106} a^{16} + \frac{94924448639798105}{451671610448739553} a^{15} + \frac{90643138670613016}{451671610448739553} a^{14} - \frac{631755205449818095}{1806686441794958212} a^{13} + \frac{56976556568756798}{451671610448739553} a^{12} - \frac{202657886876987861}{451671610448739553} a^{11} - \frac{529217179598552527}{1806686441794958212} a^{10} - \frac{869149988775897383}{1806686441794958212} a^{9} + \frac{4616337908121155}{19220068529733598} a^{8} - \frac{550457846949470049}{1806686441794958212} a^{7} - \frac{191882912724175754}{451671610448739553} a^{6} + \frac{578542849902525591}{1806686441794958212} a^{5} + \frac{368385938815307701}{1806686441794958212} a^{4} - \frac{481432461069809257}{1806686441794958212} a^{3} - \frac{66417105967377579}{1806686441794958212} a^{2} - \frac{32681574259667008}{451671610448739553} a + \frac{441540156602833}{20299847660617508}$, $\frac{1}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{31} + \frac{100675209409174417584515757678282010139748390898877218000945516198420072}{499181900196211880734003480682617783216280031497173910352484669394868058222597694097086893} a^{30} + \frac{24253428412302461152299754819172248165105091890784531103852363281906531129773372090767}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{29} + \frac{1171419423059172880068808744503197226985342673562481425928623018046110931111250037595}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{28} - \frac{12360505830545710674101799547248343453729522087428661771526912898265715069832838238551}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{27} - \frac{342548994533981123164519609601690923363175753187209791720760773751316069873689062096931}{499181900196211880734003480682617783216280031497173910352484669394868058222597694097086893} a^{26} - \frac{284292758458743769225129241768223597492367966854858227090442314913030387210586204866615929}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{25} + \frac{750729113846159457728965211026648735482643761457616893721585404007864870127568613449105603}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{24} + \frac{181243749732858417335557339011048327816336348622027829331024180461902339506474381926896911}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{23} + \frac{215340504077012786147400369090206604460959244101531740652236831135831762845564397667669545}{998363800392423761468006961365235566432560062994347820704969338789736116445195388194173786} a^{22} + \frac{293827822719545983014692666014651849886553543589818026742382818725317984788716447771523647}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{21} + \frac{569252301961048478116433929252932316250276151356460675683236965920929693378543902163400975}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{20} - \frac{40268032429122144795044602221421572284270903833728285139419024216521336185847304282660569}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{19} + \frac{141610315782843512726268972292065136159658494602203107905523567930929703412062652842325005}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{18} - \frac{50334875275244326889251002591387729491364583180696161628769416464632123777878564578189789}{998363800392423761468006961365235566432560062994347820704969338789736116445195388194173786} a^{17} + \frac{813803476547564516593127284270525599786956680420570940085491173374233866444913816614721617}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{16} + \frac{777112190127594589582631744723492940573224685736120637138790870639387216080642906363002803}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{15} + \frac{86511192544315517122926526939625610746724255815974730991954879770323604325515639209977371}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{14} - \frac{893962596357384276509669608976830655564393756016923212342622471189564353096857359499230313}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{13} - \frac{462812630760784725335564342315609999318901879790569429705228823893408441597620051782044911}{998363800392423761468006961365235566432560062994347820704969338789736116445195388194173786} a^{12} - \frac{299395189897279462999583014969327576027361608107909136102003617721796129192362253010409575}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{11} - \frac{215506710312172185147743999285997151932487809486333875865318895893591821153380450542249455}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{10} + \frac{679019618347417112070611846476072594899580095659717653386342711267719185955859652896339097}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{9} + \frac{540198372415907824751218863417942427491423615722935659459932477867026931844666171121355307}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{8} + \frac{60183582657625161845885924530258090793444685325833258089667330482131801425121381408533883}{1996727600784847522936013922730471132865120125988695641409938677579472232890390776388347572} a^{7} - \frac{834420812150721423671871065859591280262628755593267790347449173257994997147174440153719359}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{6} + \frac{257322799625304982936919899014773478907184106281452367548382683804331353040526467950558061}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{5} - \frac{401039904761310690564211899682732686345570336478828720289267867943158456773614484348702159}{998363800392423761468006961365235566432560062994347820704969338789736116445195388194173786} a^{4} - \frac{1069960530555482092526773434243599128926016242716893297939316455875175837822255454522928275}{3993455201569695045872027845460942265730240251977391282819877355158944465780781552776695144} a^{3} - \frac{203582017676512633938247282482107866747107516967867511223294651795247559377771777652715379}{998363800392423761468006961365235566432560062994347820704969338789736116445195388194173786} a^{2} + \frac{269121384875705080552030136539589181864245002633053494608008263733424679312843193766506501}{998363800392423761468006961365235566432560062994347820704969338789736116445195388194173786} a - \frac{2224107345445790998645515368156930420786198242116031612976768221579297317618864331997695}{44870283163704438717663234218662272648654384853678553739549183765830836694166084862659496}$

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

Class group and class number

$C_{2}\times C_{2482}$, which has order $4964$ (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:  \( -1 \) (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:  \( 5880820493631.187 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.5527125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.5527125.1, 4.4.4913.1, 4.4.122825.1, \(\Q(\zeta_{15})^+\), 4.4.325125.1, 8.8.30549110765625.1, 8.8.15085980625.1, 8.8.105706265625.1, 8.0.33237432513.1, 8.0.20773395320625.1, 8.0.6411541765625.1, 8.0.6411541765625.2, 16.16.933248168570425273681640625.1, 16.0.431533953146964646550390625.2, 16.0.41107867812353742431640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ 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
3Data not computed
5Data not computed
17Data not computed