↑

Properties

Label 32.0.11544095681...0976.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 13^{24}$
Root discriminant $100.45$
Ramified primes $2, 13$
Class number $2025$ (GRH)
Class group $[45, 45]$ (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![19747393, 26719104, -26861080, -60652600, -17251700, 54746128, 45725380, -19426712, -15710452, -34775816, 45640456, -37727976, 48961614, -51444648, 45545668, -38582432, 30299804, -22097552, 14525436, -8981104, 5132100, -2640872, 1299116, -549536, 230321, -78416, 27220, -7200, 1990, -376, 76, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 76*x^30 - 376*x^29 + 1990*x^28 - 7200*x^27 + 27220*x^26 - 78416*x^25 + 230321*x^24 - 549536*x^23 + 1299116*x^22 - 2640872*x^21 + 5132100*x^20 - 8981104*x^19 + 14525436*x^18 - 22097552*x^17 + 30299804*x^16 - 38582432*x^15 + 45545668*x^14 - 51444648*x^13 + 48961614*x^12 - 37727976*x^11 + 45640456*x^10 - 34775816*x^9 - 15710452*x^8 - 19426712*x^7 + 45725380*x^6 + 54746128*x^5 - 17251700*x^4 - 60652600*x^3 - 26861080*x^2 + 26719104*x + 19747393)
 
gp: K = bnfinit(x^32 - 8*x^31 + 76*x^30 - 376*x^29 + 1990*x^28 - 7200*x^27 + 27220*x^26 - 78416*x^25 + 230321*x^24 - 549536*x^23 + 1299116*x^22 - 2640872*x^21 + 5132100*x^20 - 8981104*x^19 + 14525436*x^18 - 22097552*x^17 + 30299804*x^16 - 38582432*x^15 + 45545668*x^14 - 51444648*x^13 + 48961614*x^12 - 37727976*x^11 + 45640456*x^10 - 34775816*x^9 - 15710452*x^8 - 19426712*x^7 + 45725380*x^6 + 54746128*x^5 - 17251700*x^4 - 60652600*x^3 - 26861080*x^2 + 26719104*x + 19747393, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 76 x^{30} - 376 x^{29} + 1990 x^{28} - 7200 x^{27} + 27220 x^{26} - 78416 x^{25} + 230321 x^{24} - 549536 x^{23} + 1299116 x^{22} - 2640872 x^{21} + 5132100 x^{20} - 8981104 x^{19} + 14525436 x^{18} - 22097552 x^{17} + 30299804 x^{16} - 38582432 x^{15} + 45545668 x^{14} - 51444648 x^{13} + 48961614 x^{12} - 37727976 x^{11} + 45640456 x^{10} - 34775816 x^{9} - 15710452 x^{8} - 19426712 x^{7} + 45725380 x^{6} + 54746128 x^{5} - 17251700 x^{4} - 60652600 x^{3} - 26861080 x^{2} + 26719104 x + 19747393 \)

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:  \(11544095681843725705116280971624020302917345750478466341169790976=2^{124}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.45$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(416=2^{5}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(259,·)$, $\chi_{416}(385,·)$, $\chi_{416}(363,·)$, $\chi_{416}(265,·)$, $\chi_{416}(395,·)$, $\chi_{416}(131,·)$, $\chi_{416}(281,·)$, $\chi_{416}(25,·)$, $\chi_{416}(27,·)$, $\chi_{416}(161,·)$, $\chi_{416}(291,·)$, $\chi_{416}(129,·)$, $\chi_{416}(307,·)$, $\chi_{416}(177,·)$, $\chi_{416}(51,·)$, $\chi_{416}(155,·)$, $\chi_{416}(57,·)$, $\chi_{416}(187,·)$, $\chi_{416}(411,·)$, $\chi_{416}(73,·)$, $\chi_{416}(203,·)$, $\chi_{416}(337,·)$, $\chi_{416}(83,·)$, $\chi_{416}(313,·)$, $\chi_{416}(99,·)$, $\chi_{416}(209,·)$, $\chi_{416}(105,·)$, $\chi_{416}(235,·)$, $\chi_{416}(369,·)$, $\chi_{416}(339,·)$, $\chi_{416}(233,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3} a^{24} + \frac{1}{3} a^{22} + \frac{1}{3} a^{20} - \frac{1}{3} a^{18} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{51} a^{25} - \frac{4}{51} a^{24} - \frac{20}{51} a^{23} + \frac{1}{3} a^{22} + \frac{10}{51} a^{21} - \frac{4}{51} a^{20} + \frac{2}{51} a^{19} - \frac{8}{51} a^{18} + \frac{4}{51} a^{17} - \frac{22}{51} a^{16} - \frac{11}{51} a^{15} - \frac{25}{51} a^{14} - \frac{1}{17} a^{13} + \frac{8}{17} a^{12} + \frac{7}{17} a^{11} + \frac{5}{17} a^{10} + \frac{4}{51} a^{9} + \frac{23}{51} a^{8} + \frac{2}{51} a^{7} + \frac{4}{51} a^{6} - \frac{1}{17} a^{5} - \frac{1}{17} a^{4} - \frac{1}{51} a^{3} + \frac{16}{51} a^{2} + \frac{23}{51} a + \frac{19}{51}$, $\frac{1}{51} a^{26} - \frac{2}{51} a^{24} - \frac{4}{17} a^{23} + \frac{10}{51} a^{22} - \frac{5}{17} a^{21} + \frac{20}{51} a^{20} - \frac{11}{51} a^{18} - \frac{2}{17} a^{17} - \frac{14}{51} a^{16} - \frac{6}{17} a^{15} - \frac{6}{17} a^{14} + \frac{4}{17} a^{13} + \frac{5}{17} a^{12} - \frac{1}{17} a^{11} + \frac{13}{51} a^{10} - \frac{4}{17} a^{9} - \frac{25}{51} a^{8} + \frac{4}{17} a^{7} - \frac{7}{17} a^{6} - \frac{5}{17} a^{5} - \frac{13}{51} a^{4} + \frac{4}{17} a^{3} + \frac{2}{51} a^{2} + \frac{3}{17} a - \frac{3}{17}$, $\frac{1}{51} a^{27} - \frac{1}{17} a^{24} + \frac{7}{17} a^{23} - \frac{5}{17} a^{22} - \frac{11}{51} a^{21} + \frac{3}{17} a^{20} - \frac{7}{51} a^{19} + \frac{4}{17} a^{18} - \frac{2}{17} a^{17} + \frac{2}{17} a^{16} + \frac{11}{51} a^{15} - \frac{7}{17} a^{14} + \frac{3}{17} a^{13} - \frac{2}{17} a^{12} + \frac{4}{51} a^{11} + \frac{6}{17} a^{10} - \frac{1}{3} a^{9} + \frac{8}{17} a^{8} - \frac{1}{3} a^{7} - \frac{8}{17} a^{6} - \frac{19}{51} a^{5} + \frac{2}{17} a^{4} + \frac{8}{17} a^{2} - \frac{14}{51} a + \frac{7}{17}$, $\frac{1}{28203} a^{28} + \frac{13}{4029} a^{27} - \frac{200}{28203} a^{26} + \frac{130}{28203} a^{25} + \frac{210}{1343} a^{24} - \frac{3617}{28203} a^{23} - \frac{13457}{28203} a^{22} - \frac{3394}{28203} a^{21} - \frac{254}{9401} a^{20} - \frac{512}{28203} a^{19} - \frac{10}{51} a^{18} + \frac{4762}{28203} a^{17} + \frac{1271}{4029} a^{16} + \frac{2144}{9401} a^{15} + \frac{11990}{28203} a^{14} - \frac{403}{1343} a^{13} + \frac{1786}{4029} a^{12} - \frac{11678}{28203} a^{11} + \frac{8717}{28203} a^{10} - \frac{1906}{28203} a^{9} - \frac{6808}{28203} a^{8} - \frac{3343}{9401} a^{7} - \frac{347}{9401} a^{6} - \frac{1621}{28203} a^{5} + \frac{7184}{28203} a^{4} - \frac{1699}{4029} a^{3} + \frac{838}{28203} a^{2} + \frac{614}{9401} a - \frac{167}{357}$, $\frac{1}{28203} a^{29} - \frac{62}{9401} a^{27} + \frac{27}{9401} a^{26} - \frac{11}{1343} a^{25} + \frac{325}{9401} a^{24} + \frac{1033}{28203} a^{23} - \frac{2340}{9401} a^{22} - \frac{1588}{28203} a^{21} + \frac{1192}{9401} a^{20} - \frac{125}{1343} a^{19} - \frac{3574}{9401} a^{18} + \frac{353}{4029} a^{17} - \frac{2476}{9401} a^{16} - \frac{2350}{9401} a^{15} + \frac{149}{1343} a^{14} - \frac{533}{4029} a^{13} - \frac{997}{9401} a^{12} + \frac{12973}{28203} a^{11} - \frac{4588}{9401} a^{10} - \frac{8663}{28203} a^{9} - \frac{2734}{9401} a^{8} + \frac{1913}{28203} a^{7} - \frac{1406}{9401} a^{6} - \frac{1523}{9401} a^{5} - \frac{15}{79} a^{4} + \frac{10834}{28203} a^{3} + \frac{1923}{9401} a^{2} - \frac{4603}{9401} a + \frac{7}{17}$, $\frac{1}{4003937047403466493509450716734928319700290011483397} a^{30} + \frac{7175734665789664988183894769007691953144458008}{571991006771923784787064388104989759957184287354771} a^{29} - \frac{45199078499152477844169801575122801188913958041}{4003937047403466493509450716734928319700290011483397} a^{28} - \frac{6823076937505130211188006069444408552395551150668}{4003937047403466493509450716734928319700290011483397} a^{27} + \frac{5309193997271399386151844334514988719228505393076}{1334645682467822164503150238911642773233430003827799} a^{26} + \frac{53771521169641338578501957005976688540085213881}{10509021121793875311048427078044431285302598455337} a^{25} - \frac{2660183861948217330946163803798345546946329069824}{16894249145162305879786711884957503458651012706681} a^{24} + \frac{20970626921102244686250537502152345037838337315269}{50682747435486917639360135654872510375953038120043} a^{23} - \frac{417057780962144959709475921754589905607509255551473}{1334645682467822164503150238911642773233430003827799} a^{22} - \frac{4300306066770256531911454574979319140890991586090}{571991006771923784787064388104989759957184287354771} a^{21} - \frac{256076490430686271016142456698609498067370073229329}{1334645682467822164503150238911642773233430003827799} a^{20} - \frac{1461938600669080115461728425679242115648141433976731}{4003937047403466493509450716734928319700290011483397} a^{19} + \frac{90050208634702314771333983904016775224954086879006}{190663668923974594929021462701663253319061429118257} a^{18} - \frac{615178354384607976968266803758562225143829917830211}{4003937047403466493509450716734928319700290011483397} a^{17} - \frac{4204741703191146381415496144648273354282545411015}{16894249145162305879786711884957503458651012706681} a^{16} - \frac{1148181644636668581229180295060750663995526237507056}{4003937047403466493509450716734928319700290011483397} a^{15} - \frac{613326249301284503602240639686995252210878497104956}{1334645682467822164503150238911642773233430003827799} a^{14} - \frac{721352062598308615984526787134715600151139474456389}{4003937047403466493509450716734928319700290011483397} a^{13} + \frac{549277194228797145452690086854160061624704735733378}{1334645682467822164503150238911642773233430003827799} a^{12} - \frac{77314570154311508721865782424283591379551030802565}{1334645682467822164503150238911642773233430003827799} a^{11} - \frac{444602416167879841057865430948079480387915620872064}{1334645682467822164503150238911642773233430003827799} a^{10} + \frac{3058314574530229024714825365031203038369094511319}{190663668923974594929021462701663253319061429118257} a^{9} + \frac{542772757874082930059730266814792334610966015309237}{1334645682467822164503150238911642773233430003827799} a^{8} + \frac{1954014745492707900292238131570935896519769182092037}{4003937047403466493509450716734928319700290011483397} a^{7} + \frac{18537855081672594951870737882446578417084328824401}{4003937047403466493509450716734928319700290011483397} a^{6} + \frac{1090746559455965497821378718431813606491442352101062}{4003937047403466493509450716734928319700290011483397} a^{5} - \frac{124391024111375834192841363113638930931141944763387}{571991006771923784787064388104989759957184287354771} a^{4} + \frac{826078385547918230685287728933804252350298014909239}{4003937047403466493509450716734928319700290011483397} a^{3} + \frac{51533153355319464053167990506132598646320208715416}{190663668923974594929021462701663253319061429118257} a^{2} - \frac{1141300997825189802723610197020718351636956474227814}{4003937047403466493509450716734928319700290011483397} a - \frac{3196349887994107248754661690898817276781001823301}{50682747435486917639360135654872510375953038120043}$, $\frac{1}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{31} + \frac{12746965520116815041}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{30} - \frac{304420953938540733882529982000105684304626746647837981555603261027}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{29} + \frac{677790649506476766437529520756780995603422547235041393504062369861}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{28} - \frac{301089662535232624135245647717594488059938579670556364306221451462241}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{27} + \frac{142985385568451133897701235715522918324348496852255014231917176362109}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{26} - \frac{915433993731173510232971785851962414979164296397938944721527169658594}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{25} + \frac{1289640512426129328867951088004539615359615154662817500070511660965439}{12150843490535885184354339881417755624595694555856934982883432790558833} a^{24} + \frac{32883952518092579720743684006879474295657117863881595125945738389172188}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{23} - \frac{94930225866770242985717459740805868102551318254845239676794757461493103}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{22} + \frac{17881005099466909572170385122136261972569921219378498373234808617210318}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{21} + \frac{72417495079828218924812642344672129121454027457880303377961986352536303}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{20} - \frac{10083782189074477255841813949922637958809041475219471235412808437605421}{36452530471607655553063019644253266873787083667570804948650298371676499} a^{19} + \frac{16316171650627271179919358333968283193747428180277408144041294731182927}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{18} + \frac{2052169089273001210919433768989610881758692573653872081939140734191755}{12150843490535885184354339881417755624595694555856934982883432790558833} a^{17} + \frac{110848778479334704329310394424825811282624349040020291016913309562428417}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{16} + \frac{118258889074852109330551800708048652655884174282255091963850961391571328}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{15} + \frac{16520371464172619605108342675497179336824603184948295454212539477795823}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{14} + \frac{100246076767147787194998796009014537125339308888666694632486234778403448}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{13} + \frac{55072518913274218063211438356998040185888543609811466868668777707986977}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{12} - \frac{34723747928939608651410073149156622118843548056000565177049226965786447}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{11} + \frac{54463531709313147595993569949699521082917947498945141870087105748656972}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{10} - \frac{5039772785457243520186909133083379995030254482449590391211223228683140}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{9} - \frac{104621565907831237471154258488939720914572075121217326841659834992883217}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{8} - \frac{28603389267914288095231399902402398350890826284728390578393277166635197}{85055904433751196290480379169924289372169861890998544880184029533911831} a^{7} - \frac{104361930442057764178334093262386954731898324065894727912275406671691182}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{6} + \frac{2661089801072861046358453397382263542524245818627234571224071517689400}{36452530471607655553063019644253266873787083667570804948650298371676499} a^{5} + \frac{1336132024924706757840845120861304844482655134716834241809601093548417}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{4} - \frac{1683975641124837723368126368689400076470796266663676259971891495570048}{12150843490535885184354339881417755624595694555856934982883432790558833} a^{3} - \frac{32478421419357808606049756226792571747847198067135398002452179483726842}{255167713301253588871441137509772868116509585672995634640552088601735493} a^{2} + \frac{35557663664718564434842618480091934038858877081499484844974606810905138}{85055904433751196290480379169924289372169861890998544880184029533911831} a + \frac{234710046602145132116741377942275793863220090874197661315354859873854}{3229971054446247960397989082402188204006450451556906767601925172173867}$

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

Class group and class number

$C_{45}\times C_{45}$, which has order $2025$ (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:  \( 2849230308504732.5 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.0.140608.2, \(\Q(\sqrt{2}, \sqrt{13})\), 4.0.2197.1, \(\Q(\zeta_{16})^+\), 4.4.346112.1, 4.0.4499456.2, 4.0.4499456.1, 8.0.19770609664.2, 8.8.119793516544.1, 8.0.20245104295936.1, 8.0.2147483648.1, 8.0.61334280470528.11, 8.8.10365493399519232.2, 8.8.10365493399519232.1, 16.0.409864247953326282266116096.2, 16.0.3761893960837392421076598784.1, 16.16.107443453415476764938368737869824.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
13Data not computed