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Properties

Label 32.0.84762290704...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 5^{24}\cdot 11^{16}$
Root discriminant $74.60$
Ramified primes $2, 5, 11$
Class number $3480$ (GRH)
Class group $[3480]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279841, -1314036, 3615186, -7703344, 13978351, -24740140, 41285982, -58142980, 55508750, 12703888, -94903292, 112078936, -58075371, -11220832, 74184448, -86346268, 39761917, 9556052, -11212032, 1567840, 271699, 434440, -78600, -119600, 43750, 5068, -2622, -260, 33, 80, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 10*x^30 + 80*x^29 + 33*x^28 - 260*x^27 - 2622*x^26 + 5068*x^25 + 43750*x^24 - 119600*x^23 - 78600*x^22 + 434440*x^21 + 271699*x^20 + 1567840*x^19 - 11212032*x^18 + 9556052*x^17 + 39761917*x^16 - 86346268*x^15 + 74184448*x^14 - 11220832*x^13 - 58075371*x^12 + 112078936*x^11 - 94903292*x^10 + 12703888*x^9 + 55508750*x^8 - 58142980*x^7 + 41285982*x^6 - 24740140*x^5 + 13978351*x^4 - 7703344*x^3 + 3615186*x^2 - 1314036*x + 279841)
 
gp: K = bnfinit(x^32 - 4*x^31 - 10*x^30 + 80*x^29 + 33*x^28 - 260*x^27 - 2622*x^26 + 5068*x^25 + 43750*x^24 - 119600*x^23 - 78600*x^22 + 434440*x^21 + 271699*x^20 + 1567840*x^19 - 11212032*x^18 + 9556052*x^17 + 39761917*x^16 - 86346268*x^15 + 74184448*x^14 - 11220832*x^13 - 58075371*x^12 + 112078936*x^11 - 94903292*x^10 + 12703888*x^9 + 55508750*x^8 - 58142980*x^7 + 41285982*x^6 - 24740140*x^5 + 13978351*x^4 - 7703344*x^3 + 3615186*x^2 - 1314036*x + 279841, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 10 x^{30} + 80 x^{29} + 33 x^{28} - 260 x^{27} - 2622 x^{26} + 5068 x^{25} + 43750 x^{24} - 119600 x^{23} - 78600 x^{22} + 434440 x^{21} + 271699 x^{20} + 1567840 x^{19} - 11212032 x^{18} + 9556052 x^{17} + 39761917 x^{16} - 86346268 x^{15} + 74184448 x^{14} - 11220832 x^{13} - 58075371 x^{12} + 112078936 x^{11} - 94903292 x^{10} + 12703888 x^{9} + 55508750 x^{8} - 58142980 x^{7} + 41285982 x^{6} - 24740140 x^{5} + 13978351 x^{4} - 7703344 x^{3} + 3615186 x^{2} - 1314036 x + 279841 \)

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:  \(847622907049404564614012839370162176000000000000000000000000=2^{88}\cdot 5^{24}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.60$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(880=2^{4}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{880}(1,·)$, $\chi_{880}(131,·)$, $\chi_{880}(769,·)$, $\chi_{880}(529,·)$, $\chi_{880}(659,·)$, $\chi_{880}(793,·)$, $\chi_{880}(153,·)$, $\chi_{880}(417,·)$, $\chi_{880}(419,·)$, $\chi_{880}(683,·)$, $\chi_{880}(681,·)$, $\chi_{880}(43,·)$, $\chi_{880}(177,·)$, $\chi_{880}(307,·)$, $\chi_{880}(219,·)$, $\chi_{880}(441,·)$, $\chi_{880}(571,·)$, $\chi_{880}(67,·)$, $\chi_{880}(329,·)$, $\chi_{880}(331,·)$, $\chi_{880}(593,·)$, $\chi_{880}(857,·)$, $\chi_{880}(859,·)$, $\chi_{880}(353,·)$, $\chi_{880}(483,·)$, $\chi_{880}(617,·)$, $\chi_{880}(747,·)$, $\chi_{880}(241,·)$, $\chi_{880}(243,·)$, $\chi_{880}(89,·)$, $\chi_{880}(771,·)$, $\chi_{880}(507,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{41} a^{24} - \frac{3}{41} a^{23} + \frac{6}{41} a^{22} - \frac{10}{41} a^{21} - \frac{5}{41} a^{20} + \frac{5}{41} a^{19} - \frac{4}{41} a^{18} - \frac{15}{41} a^{17} + \frac{13}{41} a^{16} + \frac{1}{41} a^{15} - \frac{12}{41} a^{14} + \frac{15}{41} a^{13} + \frac{10}{41} a^{12} - \frac{8}{41} a^{11} + \frac{18}{41} a^{10} - \frac{11}{41} a^{9} + \frac{1}{41} a^{8} + \frac{6}{41} a^{7} + \frac{4}{41} a^{6} - \frac{6}{41} a^{5} - \frac{12}{41} a^{4} + \frac{19}{41} a^{3} - \frac{5}{41} a^{2} - \frac{1}{41} a + \frac{10}{41}$, $\frac{1}{943} a^{25} + \frac{8}{943} a^{24} - \frac{150}{943} a^{23} + \frac{466}{943} a^{22} + \frac{418}{943} a^{21} - \frac{296}{943} a^{20} - \frac{113}{943} a^{19} - \frac{305}{943} a^{18} + \frac{176}{943} a^{17} + \frac{390}{943} a^{16} - \frac{247}{943} a^{15} - \frac{404}{943} a^{14} + \frac{93}{943} a^{13} + \frac{143}{943} a^{12} - \frac{193}{943} a^{11} + \frac{1}{41} a^{10} - \frac{243}{943} a^{9} + \frac{222}{943} a^{8} + \frac{439}{943} a^{7} - \frac{249}{943} a^{6} + \frac{86}{943} a^{5} - \frac{441}{943} a^{4} + \frac{409}{943} a^{3} - \frac{15}{943} a^{2} - \frac{247}{943} a + \frac{3}{41}$, $\frac{1}{80544459} a^{26} + \frac{8955}{26848153} a^{25} - \frac{487231}{80544459} a^{24} - \frac{1342664}{26848153} a^{23} - \frac{39211726}{80544459} a^{22} - \frac{34018589}{80544459} a^{21} + \frac{10676494}{26848153} a^{20} - \frac{34201135}{80544459} a^{19} - \frac{4909132}{80544459} a^{18} + \frac{10376252}{80544459} a^{17} + \frac{22662488}{80544459} a^{16} + \frac{7116680}{26848153} a^{15} - \frac{159047}{1167311} a^{14} + \frac{12660095}{80544459} a^{13} - \frac{23411077}{80544459} a^{12} + \frac{11966116}{26848153} a^{11} - \frac{27835280}{80544459} a^{10} + \frac{25996178}{80544459} a^{9} - \frac{29872595}{80544459} a^{8} - \frac{14776406}{80544459} a^{7} + \frac{2721502}{26848153} a^{6} + \frac{5512816}{80544459} a^{5} + \frac{1267883}{3501933} a^{4} + \frac{6803905}{26848153} a^{3} - \frac{38225107}{80544459} a^{2} + \frac{115200}{378143} a + \frac{1610869}{3501933}$, $\frac{1}{80544459} a^{27} + \frac{36872}{80544459} a^{25} + \frac{110683}{26848153} a^{24} - \frac{37363411}{80544459} a^{23} - \frac{10895207}{80544459} a^{22} + \frac{165516}{654833} a^{21} + \frac{7105490}{80544459} a^{20} + \frac{16927232}{80544459} a^{19} + \frac{27382883}{80544459} a^{18} - \frac{40159423}{80544459} a^{17} + \frac{130440}{26848153} a^{16} - \frac{9590209}{26848153} a^{15} + \frac{12571322}{80544459} a^{14} + \frac{3668300}{80544459} a^{13} + \frac{10272362}{26848153} a^{12} - \frac{5203991}{80544459} a^{11} + \frac{24426533}{80544459} a^{10} - \frac{5343125}{80544459} a^{9} + \frac{362009}{3501933} a^{8} + \frac{1567720}{26848153} a^{7} + \frac{21248749}{80544459} a^{6} - \frac{4251077}{80544459} a^{5} - \frac{11909704}{26848153} a^{4} + \frac{15973277}{80544459} a^{3} + \frac{12645817}{26848153} a^{2} + \frac{35646104}{80544459} a + \frac{41204}{1167311}$, $\frac{1}{80544459} a^{28} - \frac{194}{378143} a^{25} + \frac{483451}{80544459} a^{24} - \frac{37391024}{80544459} a^{23} + \frac{7586996}{80544459} a^{22} + \frac{8643247}{26848153} a^{21} + \frac{23986997}{80544459} a^{20} + \frac{20960359}{80544459} a^{19} - \frac{4733675}{80544459} a^{18} - \frac{38707462}{80544459} a^{17} - \frac{30641431}{80544459} a^{16} - \frac{81434}{1964499} a^{15} + \frac{26809292}{80544459} a^{14} + \frac{16416047}{80544459} a^{13} - \frac{2707458}{26848153} a^{12} + \frac{25800077}{80544459} a^{11} - \frac{35046505}{80544459} a^{10} + \frac{7331896}{26848153} a^{9} - \frac{880600}{1964499} a^{8} + \frac{5228456}{80544459} a^{7} - \frac{39144380}{80544459} a^{6} - \frac{24330119}{80544459} a^{5} + \frac{4992448}{26848153} a^{4} - \frac{4381893}{26848153} a^{3} - \frac{1646839}{3501933} a^{2} + \frac{12781262}{26848153} a + \frac{323845}{3501933}$, $\frac{1}{81385013542542534813603231581738297816109} a^{29} + \frac{98073192936873031215416147116611}{27128337847514178271201077193912765938703} a^{28} - \frac{65063341837623199403782537724528}{81385013542542534813603231581738297816109} a^{27} - \frac{316404327593375327165126293792153}{81385013542542534813603231581738297816109} a^{26} - \frac{2111376173300424716431916328226571646}{27128337847514178271201077193912765938703} a^{25} - \frac{14478497568368320501158180752590590782}{3538478849675762383200140503553839035483} a^{24} - \frac{19931988286047718151125118583466246909985}{81385013542542534813603231581738297816109} a^{23} + \frac{40451298539461371875102564352006826311641}{81385013542542534813603231581738297816109} a^{22} - \frac{29702799571052674756141439780990386449734}{81385013542542534813603231581738297816109} a^{21} - \frac{2753450604099084049574976694701590658007}{27128337847514178271201077193912765938703} a^{20} + \frac{11984972574520887960163528474255189813966}{81385013542542534813603231581738297816109} a^{19} + \frac{196612430428977456069566015773647308314}{3538478849675762383200140503553839035483} a^{18} + \frac{21671390562021594866654562839856949781978}{81385013542542534813603231581738297816109} a^{17} - \frac{4608729076816194686259906906818416338202}{27128337847514178271201077193912765938703} a^{16} + \frac{33624133698638388024548280448149250898972}{81385013542542534813603231581738297816109} a^{15} - \frac{12190571762925987972601287257964842023508}{81385013542542534813603231581738297816109} a^{14} + \frac{9334050868348608035520148245032478778580}{27128337847514178271201077193912765938703} a^{13} + \frac{6307837136083916615636300875936063800642}{27128337847514178271201077193912765938703} a^{12} + \frac{6076094534031881211949845223050606785007}{27128337847514178271201077193912765938703} a^{11} - \frac{27334129362836632801504524216951165951056}{81385013542542534813603231581738297816109} a^{10} + \frac{10331909779299155578392693805130218732831}{27128337847514178271201077193912765938703} a^{9} - \frac{4740121709214969978609958330024860208600}{81385013542542534813603231581738297816109} a^{8} - \frac{12393332955085065307673323396017941066909}{27128337847514178271201077193912765938703} a^{7} - \frac{14506323042514745033667457059193929264679}{81385013542542534813603231581738297816109} a^{6} + \frac{1035143708904301208757519434200194737969}{27128337847514178271201077193912765938703} a^{5} - \frac{9898624946596122796103672501146425393565}{81385013542542534813603231581738297816109} a^{4} - \frac{4239297294405299367787816319889608829160}{27128337847514178271201077193912765938703} a^{3} + \frac{21498223924215193802338635999047143409782}{81385013542542534813603231581738297816109} a^{2} + \frac{5156319840047429140058404631471452110880}{81385013542542534813603231581738297816109} a + \frac{1015698892829110900530549458965160646611}{3538478849675762383200140503553839035483}$, $\frac{1}{1871855311478478300712874326379980849770507} a^{30} - \frac{4}{1871855311478478300712874326379980849770507} a^{29} - \frac{4042539394439986495830825601130762}{1871855311478478300712874326379980849770507} a^{28} + \frac{3816071576008445529419612915662049}{623951770492826100237624775459993616590169} a^{27} - \frac{10406912290250805088795604943969902}{1871855311478478300712874326379980849770507} a^{26} - \frac{263338825887426340910497409820284504559}{623951770492826100237624775459993616590169} a^{25} - \frac{357112688968949128425297016142556263180}{81385013542542534813603231581738297816109} a^{24} + \frac{654614795850615440518664349251980781983412}{1871855311478478300712874326379980849770507} a^{23} + \frac{319028750040704335410368404369395535633066}{1871855311478478300712874326379980849770507} a^{22} - \frac{35948604742077176253746618796603396230}{86304362187213716663418061062288756963} a^{21} - \frac{388085144674631285057809292323092664275847}{1871855311478478300712874326379980849770507} a^{20} - \frac{744030162399921310550983130291533573317163}{1871855311478478300712874326379980849770507} a^{19} - \frac{40415150442357744131264613896656939478830}{81385013542542534813603231581738297816109} a^{18} - \frac{237231786320365804646712467574385793492653}{1871855311478478300712874326379980849770507} a^{17} + \frac{150560236910721214415485564244640321697211}{1871855311478478300712874326379980849770507} a^{16} - \frac{535653827415213956256412699185630768779284}{1871855311478478300712874326379980849770507} a^{15} + \frac{8342167043947232535334664633453221679817}{27128337847514178271201077193912765938703} a^{14} + \frac{637461218100777725840166515226179250351892}{1871855311478478300712874326379980849770507} a^{13} - \frac{3542262088215803033702053121373228361381}{8788053105532761975177813738873149529439} a^{12} + \frac{207298187998794417986979391011545665943484}{623951770492826100237624775459993616590169} a^{11} - \frac{274065702601972623932104372375944168609117}{623951770492826100237624775459993616590169} a^{10} + \frac{316204731750617163315338657445485698968808}{1871855311478478300712874326379980849770507} a^{9} - \frac{208744578518036963347631858119766602915509}{623951770492826100237624775459993616590169} a^{8} + \frac{261190264851577874545242520359380726298641}{1871855311478478300712874326379980849770507} a^{7} + \frac{144703622302548480258364739554996763983626}{623951770492826100237624775459993616590169} a^{6} + \frac{883372231327040589534033162314969820203197}{1871855311478478300712874326379980849770507} a^{5} - \frac{144242393508553200931146039772376615711423}{1871855311478478300712874326379980849770507} a^{4} + \frac{303581875826969182730892633779691040204301}{1871855311478478300712874326379980849770507} a^{3} - \frac{223208504869476522273901426750989715866327}{623951770492826100237624775459993616590169} a^{2} - \frac{32255799909777595202274206995648116481645}{81385013542542534813603231581738297816109} a + \frac{1528228730532133606023171799238054821547}{3538478849675762383200140503553839035483}$, $\frac{1}{43052672164005000916396109506739559544721661} a^{31} - \frac{4}{43052672164005000916396109506739559544721661} a^{30} - \frac{10}{43052672164005000916396109506739559544721661} a^{29} - \frac{202592128453774491177876917666785498}{43052672164005000916396109506739559544721661} a^{28} - \frac{93852566024152190501142406536714793}{43052672164005000916396109506739559544721661} a^{27} + \frac{160424939039584291169112818064674863}{43052672164005000916396109506739559544721661} a^{26} - \frac{458534502460839115563206981713942224536}{1871855311478478300712874326379980849770507} a^{25} + \frac{472831427306679872107260727001878436279209}{43052672164005000916396109506739559544721661} a^{24} + \frac{16455335251085005566809294325595882320560185}{43052672164005000916396109506739559544721661} a^{23} + \frac{258338232634559508333596297049005499791439}{623951770492826100237624775459993616590169} a^{22} + \frac{2681916373150198138457678171093432468670446}{43052672164005000916396109506739559544721661} a^{21} - \frac{6970990080501035070226587232322992288304202}{14350890721335000305465369835579853181573887} a^{20} - \frac{852070785187157845159046070508205229804923}{1871855311478478300712874326379980849770507} a^{19} - \frac{15506565287062265918589780895527730703136668}{43052672164005000916396109506739559544721661} a^{18} - \frac{6511462352686649565980708144205879247452581}{14350890721335000305465369835579853181573887} a^{17} + \frac{10054619901283384598262575130949394967529861}{43052672164005000916396109506739559544721661} a^{16} - \frac{107046180291196914384963945215031699089084}{623951770492826100237624775459993616590169} a^{15} + \frac{2664771430541857925659764633637296694211658}{14350890721335000305465369835579853181573887} a^{14} + \frac{19574393285385527581508453964422319242241569}{43052672164005000916396109506739559544721661} a^{13} + \frac{5075530418050695371177773760179638950249422}{43052672164005000916396109506739559544721661} a^{12} - \frac{3956653522035493096872533880124888736796074}{43052672164005000916396109506739559544721661} a^{11} + \frac{20928893437630236706806107962296653987060963}{43052672164005000916396109506739559544721661} a^{10} + \frac{18667387962529415791845331451669976416215267}{43052672164005000916396109506739559544721661} a^{9} + \frac{10388604672786230799317916292712898494278142}{43052672164005000916396109506739559544721661} a^{8} - \frac{4860121767785052268738797286407365433239459}{14350890721335000305465369835579853181573887} a^{7} - \frac{5628490609114148275894059379116277053792442}{43052672164005000916396109506739559544721661} a^{6} - \frac{584269201825625474300905371164639643856657}{43052672164005000916396109506739559544721661} a^{5} - \frac{6099653885418577528059781070416365758673767}{14350890721335000305465369835579853181573887} a^{4} - \frac{6596822530487605887658309062599940521191304}{14350890721335000305465369835579853181573887} a^{3} + \frac{708345096290835038162725016028229921548064}{1871855311478478300712874326379980849770507} a^{2} + \frac{473738408198940971157569122544112250220}{1179492949891920794400046834517946345161} a + \frac{527569471114996773926437477335651821015}{1179492949891920794400046834517946345161}$

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

Class group and class number

$C_{3480}$, which has order $3480$ (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{2106666720628878475868526987876034520}{1050065174731829290643807548944867305968821} a^{31} + \frac{2728491587602468632036463321901350932}{350021724910609763547935849648289101989607} a^{30} + \frac{21699768149597426199704412020109739328}{1050065174731829290643807548944867305968821} a^{29} - \frac{165106337961696915216373650716506089712}{1050065174731829290643807548944867305968821} a^{28} - \frac{28139228121644107852443327911698258328}{350021724910609763547935849648289101989607} a^{27} + \frac{516970254303805548062550225898175275480}{1050065174731829290643807548944867305968821} a^{26} + \frac{241405532808813770954955245517598437040}{45655007597036056114948154301950752433427} a^{25} - \frac{9980890024985761138135567878264266387789}{1050065174731829290643807548944867305968821} a^{24} - \frac{30815720715125886676714701479916636681360}{350021724910609763547935849648289101989607} a^{23} + \frac{10460024242214767581837776403303372699644}{45655007597036056114948154301950752433427} a^{22} + \frac{178782504619034375802645965202328469748368}{1050065174731829290643807548944867305968821} a^{21} - \frac{290312047141602922699950585536914419752328}{350021724910609763547935849648289101989607} a^{20} - \frac{27008809480489140300617052242410515192400}{45655007597036056114948154301950752433427} a^{19} - \frac{1156798941805965512289534080528004100704332}{350021724910609763547935849648289101989607} a^{18} + \frac{23033442410434154571577217402859027609130112}{1050065174731829290643807548944867305968821} a^{17} - \frac{18098009503076946141524941094205931845453512}{1050065174731829290643807548944867305968821} a^{16} - \frac{3604307379456181258826677676131528982028200}{45655007597036056114948154301950752433427} a^{15} + \frac{172357619804392149018444747209778908070608000}{1050065174731829290643807548944867305968821} a^{14} - \frac{149866137489322176584999170291430541568638704}{1050065174731829290643807548944867305968821} a^{13} + \frac{58118598814395862144847973826031728979576}{2924972631564984096500856682297680518019} a^{12} + \frac{120614667542462230718797212777194691509660936}{1050065174731829290643807548944867305968821} a^{11} - \frac{228992779910557696826949681604820936885661368}{1050065174731829290643807548944867305968821} a^{10} + \frac{186152442826307463125116089722434743591202952}{1050065174731829290643807548944867305968821} a^{9} - \frac{7902426289288953868575919521826951029922456}{350021724910609763547935849648289101989607} a^{8} - \frac{37903610840383315830554824062268654453177952}{350021724910609763547935849648289101989607} a^{7} + \frac{39481430523476047520635625026804655106072268}{350021724910609763547935849648289101989607} a^{6} - \frac{83946447164556606944696023638176602775557216}{1050065174731829290643807548944867305968821} a^{5} + \frac{15232033821296197304215626091374949453112872}{350021724910609763547935849648289101989607} a^{4} - \frac{28400872819925371569882096583727082010933816}{1050065174731829290643807548944867305968821} a^{3} + \frac{679590599948461212499255810187637742381732}{45655007597036056114948154301950752433427} a^{2} - \frac{13842206655799060063440779136860535408736}{1985000330305915483258615404432641410149} a + \frac{218177140528305415525697667068883271208}{86304362187213716663418061062288756963} \) (order $10$)
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:  \( 93528182489607.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{2}, \sqrt{-11})\), 4.4.247808.1, 4.0.2048.2, \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-55})\), 4.4.6195200.2, 4.0.51200.2, \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.4.256000.2, 4.0.30976000.2, 4.4.256000.1, 4.0.30976000.4, 4.4.968000.2, \(\Q(\zeta_{5})\), 4.4.15125.1, 4.0.8000.2, 8.0.61408804864.1, 8.0.37480960000.9, 8.0.38380503040000.15, 8.8.38380503040000.3, 8.0.38380503040000.20, 8.0.2621440000.1, 8.0.38380503040000.50, 8.0.959512576000000.64, 8.0.959512576000000.63, 8.0.937024000000.5, 8.0.937024000000.1, 8.8.65536000000.1, 8.0.959512576000000.56, 8.8.937024000000.1, 8.0.64000000.2, 8.0.959512576000000.41, 8.0.959512576000000.31, 8.0.937024000000.2, 8.0.228765625.1, 16.0.1473063013603449241600000000.2, 16.0.920664383502155776000000000000.3, 16.0.878013976576000000000000.1, 16.16.920664383502155776000000000000.3, 16.0.920664383502155776000000000000.7, 16.0.4294967296000000000000.2, 16.0.920664383502155776000000000000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
2Data not computed
5Data not computed
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$