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Properties

Label 32.32.4212774438...0000.2
Degree $32$
Signature $[32, 0]$
Discriminant $2^{124}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $129.80$
Ramified primes $2, 5, 7$
Class number Not computed
Class group Not computed
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![-9793409, 103141288, -286773844, -498020920, 3842975350, -4658045344, -7738765748, 20765809224, -1328020573, -31998194928, 18807108516, 23589054160, -23836615436, -8043244424, 14927443376, 176836216, -5510095721, 888389032, 1267393024, -357343720, -182544660, 72455680, 15632420, -8827760, -634585, 666456, -7628, -30352, 1954, 760, -76, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 - 76*x^30 + 760*x^29 + 1954*x^28 - 30352*x^27 - 7628*x^26 + 666456*x^25 - 634585*x^24 - 8827760*x^23 + 15632420*x^22 + 72455680*x^21 - 182544660*x^20 - 357343720*x^19 + 1267393024*x^18 + 888389032*x^17 - 5510095721*x^16 + 176836216*x^15 + 14927443376*x^14 - 8043244424*x^13 - 23836615436*x^12 + 23589054160*x^11 + 18807108516*x^10 - 31998194928*x^9 - 1328020573*x^8 + 20765809224*x^7 - 7738765748*x^6 - 4658045344*x^5 + 3842975350*x^4 - 498020920*x^3 - 286773844*x^2 + 103141288*x - 9793409)
 
gp: K = bnfinit(x^32 - 8*x^31 - 76*x^30 + 760*x^29 + 1954*x^28 - 30352*x^27 - 7628*x^26 + 666456*x^25 - 634585*x^24 - 8827760*x^23 + 15632420*x^22 + 72455680*x^21 - 182544660*x^20 - 357343720*x^19 + 1267393024*x^18 + 888389032*x^17 - 5510095721*x^16 + 176836216*x^15 + 14927443376*x^14 - 8043244424*x^13 - 23836615436*x^12 + 23589054160*x^11 + 18807108516*x^10 - 31998194928*x^9 - 1328020573*x^8 + 20765809224*x^7 - 7738765748*x^6 - 4658045344*x^5 + 3842975350*x^4 - 498020920*x^3 - 286773844*x^2 + 103141288*x - 9793409, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} - 76 x^{30} + 760 x^{29} + 1954 x^{28} - 30352 x^{27} - 7628 x^{26} + 666456 x^{25} - 634585 x^{24} - 8827760 x^{23} + 15632420 x^{22} + 72455680 x^{21} - 182544660 x^{20} - 357343720 x^{19} + 1267393024 x^{18} + 888389032 x^{17} - 5510095721 x^{16} + 176836216 x^{15} + 14927443376 x^{14} - 8043244424 x^{13} - 23836615436 x^{12} + 23589054160 x^{11} + 18807108516 x^{10} - 31998194928 x^{9} - 1328020573 x^{8} + 20765809224 x^{7} - 7738765748 x^{6} - 4658045344 x^{5} + 3842975350 x^{4} - 498020920 x^{3} - 286773844 x^{2} + 103141288 x - 9793409 \)

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:  \(42127744383897781264481528176959874165374976000000000000000000000000=2^{124}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.80$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(1,·)$, $\chi_{1120}(517,·)$, $\chi_{1120}(13,·)$, $\chi_{1120}(657,·)$, $\chi_{1120}(153,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(29,·)$, $\chi_{1120}(293,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(169,·)$, $\chi_{1120}(797,·)$, $\chi_{1120}(433,·)$, $\chi_{1120}(1077,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(573,·)$, $\chi_{1120}(309,·)$, $\chi_{1120}(449,·)$, $\chi_{1120}(97,·)$, $\chi_{1120}(713,·)$, $\chi_{1120}(589,·)$, $\chi_{1120}(141,·)$, $\chi_{1120}(853,·)$, $\chi_{1120}(729,·)$, $\chi_{1120}(421,·)$, $\chi_{1120}(993,·)$, $\chi_{1120}(869,·)$, $\chi_{1120}(237,·)$, $\chi_{1120}(701,·)$, $\chi_{1120}(1009,·)$, $\chi_{1120}(937,·)$, $\chi_{1120}(377,·)$, $\chi_{1120}(981,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{46031} a^{28} + \frac{8187}{46031} a^{27} - \frac{10459}{46031} a^{26} + \frac{17439}{46031} a^{25} + \frac{713}{46031} a^{24} + \frac{6520}{46031} a^{23} + \frac{617}{46031} a^{22} + \frac{1449}{46031} a^{21} + \frac{9514}{46031} a^{20} + \frac{8972}{46031} a^{19} - \frac{18991}{46031} a^{18} + \frac{10406}{46031} a^{17} + \frac{22876}{46031} a^{16} - \frac{19157}{46031} a^{15} - \frac{2607}{46031} a^{14} - \frac{16188}{46031} a^{13} + \frac{2781}{46031} a^{12} - \frac{12581}{46031} a^{11} - \frac{607}{46031} a^{10} - \frac{16787}{46031} a^{9} + \frac{21460}{46031} a^{8} + \frac{12121}{46031} a^{7} + \frac{10056}{46031} a^{6} - \frac{9214}{46031} a^{5} - \frac{3924}{46031} a^{4} + \frac{168}{46031} a^{3} + \frac{4920}{46031} a^{2} + \frac{22164}{46031} a + \frac{19986}{46031}$, $\frac{1}{3636449} a^{29} - \frac{32}{3636449} a^{28} + \frac{1701057}{3636449} a^{27} + \frac{1374982}{3636449} a^{26} + \frac{654540}{3636449} a^{25} - \frac{1158465}{3636449} a^{24} + \frac{1649937}{3636449} a^{23} - \frac{6264}{3636449} a^{22} - \frac{852377}{3636449} a^{21} + \frac{664509}{3636449} a^{20} - \frac{1445158}{3636449} a^{19} + \frac{190438}{3636449} a^{18} - \frac{70502}{3636449} a^{17} + \frac{874223}{3636449} a^{16} + \frac{483066}{3636449} a^{15} - \frac{269856}{3636449} a^{14} + \frac{712828}{3636449} a^{13} + \frac{1802996}{3636449} a^{12} - \frac{121087}{3636449} a^{11} - \frac{91264}{3636449} a^{10} - \frac{421404}{3636449} a^{9} - \frac{989509}{3636449} a^{8} + \frac{1333540}{3636449} a^{7} - \frac{310019}{3636449} a^{6} - \frac{777580}{3636449} a^{5} - \frac{154300}{3636449} a^{4} + \frac{511399}{3636449} a^{3} - \frac{460408}{3636449} a^{2} + \frac{597140}{3636449} a + \frac{986356}{3636449}$, $\frac{1}{1056545045456183297852734113343707267195473665677761} a^{30} - \frac{73801689683345086075020431510268837575371493}{1056545045456183297852734113343707267195473665677761} a^{29} - \frac{4399111741260534122860741022437245456094658866}{1056545045456183297852734113343707267195473665677761} a^{28} - \frac{374673685046529783240815287956227542459129045634137}{1056545045456183297852734113343707267195473665677761} a^{27} - \frac{68393462758954120197630755365807621394438631482855}{1056545045456183297852734113343707267195473665677761} a^{26} + \frac{356205567647093571874129913146031843236900839254595}{1056545045456183297852734113343707267195473665677761} a^{25} - \frac{147376391788469510435626194673413517814008351245580}{1056545045456183297852734113343707267195473665677761} a^{24} - \frac{238352858899603559539730495012656517332073157162056}{1056545045456183297852734113343707267195473665677761} a^{23} + \frac{124569417357417963779598460583517736727494332675174}{1056545045456183297852734113343707267195473665677761} a^{22} + \frac{481671191181615228310656714394415866274766627407250}{1056545045456183297852734113343707267195473665677761} a^{21} - \frac{195525249542557816462780289773683766578948384818694}{1056545045456183297852734113343707267195473665677761} a^{20} - \frac{186797070486379689588734178886259459568471940848419}{1056545045456183297852734113343707267195473665677761} a^{19} + \frac{115151189577034338672237780353035203960711274964072}{1056545045456183297852734113343707267195473665677761} a^{18} - \frac{250172547712771162533362683199826575278945262188946}{1056545045456183297852734113343707267195473665677761} a^{17} - \frac{376889142158115450505034218448890255692782905447937}{1056545045456183297852734113343707267195473665677761} a^{16} - \frac{6972533389741003098392372185292180109269572268772}{1056545045456183297852734113343707267195473665677761} a^{15} + \frac{393141409594938983955415313104400707134139659112287}{1056545045456183297852734113343707267195473665677761} a^{14} - \frac{189900489570324643167231593824227772137649920256790}{1056545045456183297852734113343707267195473665677761} a^{13} - \frac{65114142888986096056969881336354586321178291368186}{1056545045456183297852734113343707267195473665677761} a^{12} - \frac{336648913090274527052603327833966533665251001002239}{1056545045456183297852734113343707267195473665677761} a^{11} + \frac{246795746283417773702963188161432001075858596793929}{1056545045456183297852734113343707267195473665677761} a^{10} + \frac{480136146682619242660488813089966786107273249501826}{1056545045456183297852734113343707267195473665677761} a^{9} + \frac{516038673957672706418523355865996598114389015067712}{1056545045456183297852734113343707267195473665677761} a^{8} - \frac{280706144004860537276084645867649612967992896706032}{1056545045456183297852734113343707267195473665677761} a^{7} + \frac{401889847816023271731698013275647889477368614678000}{1056545045456183297852734113343707267195473665677761} a^{6} + \frac{504872528531603639173939708050365999338925825713665}{1056545045456183297852734113343707267195473665677761} a^{5} - \frac{459000659225448771364788705610114362209793319081203}{1056545045456183297852734113343707267195473665677761} a^{4} + \frac{501554080015816616226684702707965057013958914371630}{1056545045456183297852734113343707267195473665677761} a^{3} + \frac{368413909693379186734691617923470569843417685939631}{1056545045456183297852734113343707267195473665677761} a^{2} - \frac{174293626482326778605156243821510612028735092561296}{1056545045456183297852734113343707267195473665677761} a + \frac{253188020534429715211958901814938439834970850955477}{1056545045456183297852734113343707267195473665677761}$, $\frac{1}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{31} - \frac{458740422476100362}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{30} + \frac{41946247069102130740254086713353769093217716243348876696471826}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{29} - \frac{9424508011089967498157601952590629553367695814406885802617106591}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{28} - \frac{618712569432979252334803909552165250568324319257995037315631606328551}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{27} + \frac{669676498178733184386330123895967123382174063049095768326939525441747}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{26} + \frac{288777503535403696473029111386858445749078607866650768322018561719730}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{25} + \frac{7291360587446540488457976262340237737031089454973273593211837850106}{17355640664822967280248307783597865610084087953446466762990354398351} a^{24} + \frac{54839531819467213711960938095845281538044579139397940847591717486481}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{23} + \frac{2611759351308845712249843759534192148268578214887199107854182806664}{5689193412950267282736997157278968395006817212955480806125468869169} a^{22} - \frac{132825410054210682841824565404489605810299456734084192841250472974761}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{21} + \frac{178475735508264830497284256962302089023361787670563631520336099118128}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{20} + \frac{554445453134294004063209492899185370705268011982957556450257923039159}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{19} - \frac{501588978951497371533956782686845659964845298034728059397405742959488}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{18} + \frac{168573502842300456701476922204083540408371019089942890491991870456466}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{17} + \frac{61520753467972065160620142537119645874092215334478531000692236290206}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{16} + \frac{224572609862962339767928246050709269153645299619099554040638316026829}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{15} - \frac{322233956883897122571132467749810743208668397258289447261633712515830}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{14} + \frac{496241342712244015132202924245977477267161153219132251013339777899591}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{13} - \frac{360349670748564851385344080382427130430211350933570620945345439922989}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{12} + \frac{313444807507450609648444977574431282991466231646881778758985497778928}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{11} + \frac{35699147348153221085311026363274748760743340932698969768301345100634}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{10} - \frac{114247575213858902567551697526428594717010523046977180495862077500590}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{9} + \frac{58844778966708780082408962093764692955417223845895685067311332010443}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{8} + \frac{468294122071353580761358620203243873308023730435505337848204551810330}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{7} - \frac{318260498190587244478954827872319263124339604144571346311251763082763}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{6} - \frac{579125298177774345808957206564660809302307271010528658840246955295025}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{5} - \frac{488978351249384038229833558927162533440682383715420058449182263181016}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{4} + \frac{288610844368299119221834720222065109235244900270251583092087951720575}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{3} + \frac{357392472588333392243266676287600277704443735096612562402140336424868}{1371095612521014415139616314904231383196642948322270874276237997469729} a^{2} - \frac{449618778159640056694467971880235384733411397869307632348225847075884}{1371095612521014415139616314904231383196642948322270874276237997469729} a - \frac{504606974280744494634234672942711072321139915088462105807143563571136}{1371095612521014415139616314904231383196642948322270874276237997469729}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.392000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.6125.1, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.4.12544000.1, 4.4.12544000.2, 8.8.153664000000.1, 8.8.2621440000.1, 8.8.157351936000000.4, 8.8.80564191232000000.4, 8.8.80564191232000000.5, 8.8.1342177280000.1, \(\Q(\zeta_{32})^+\), 16.16.24759631762948096000000000000.2, 16.16.6490588908866265677824000000000000.1, 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$