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Properties

Label 32.0.59816430901...0000.3
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $66560$ (GRH)
Class group $[2, 2, 2, 8, 1040]$ (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![6561, 0, 63423, 0, 562059, 0, 4951611, 0, 43606026, 0, 36535239, 0, 22756410, 0, 12787305, 0, 6722737, 0, 1926010, 0, 471045, 0, 104078, 0, 19451, 0, 1943, 0, 186, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 186*x^28 + 1943*x^26 + 19451*x^24 + 104078*x^22 + 471045*x^20 + 1926010*x^18 + 6722737*x^16 + 12787305*x^14 + 22756410*x^12 + 36535239*x^10 + 43606026*x^8 + 4951611*x^6 + 562059*x^4 + 63423*x^2 + 6561)
 
gp: K = bnfinit(x^32 + 16*x^30 + 186*x^28 + 1943*x^26 + 19451*x^24 + 104078*x^22 + 471045*x^20 + 1926010*x^18 + 6722737*x^16 + 12787305*x^14 + 22756410*x^12 + 36535239*x^10 + 43606026*x^8 + 4951611*x^6 + 562059*x^4 + 63423*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} + 16 x^{30} + 186 x^{28} + 1943 x^{26} + 19451 x^{24} + 104078 x^{22} + 471045 x^{20} + 1926010 x^{18} + 6722737 x^{16} + 12787305 x^{14} + 22756410 x^{12} + 36535239 x^{10} + 43606026 x^{8} + 4951611 x^{6} + 562059 x^{4} + 63423 x^{2} + 6561 \)

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:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(649,·)$, $\chi_{780}(779,·)$, $\chi_{780}(493,·)$, $\chi_{780}(151,·)$, $\chi_{780}(281,·)$, $\chi_{780}(157,·)$, $\chi_{780}(31,·)$, $\chi_{780}(161,·)$, $\chi_{780}(307,·)$, $\chi_{780}(437,·)$, $\chi_{780}(311,·)$, $\chi_{780}(313,·)$, $\chi_{780}(287,·)$, $\chi_{780}(317,·)$, $\chi_{780}(181,·)$, $\chi_{780}(343,·)$, $\chi_{780}(463,·)$, $\chi_{780}(337,·)$, $\chi_{780}(467,·)$, $\chi_{780}(469,·)$, $\chi_{780}(599,·)$, $\chi_{780}(473,·)$, $\chi_{780}(187,·)$, $\chi_{780}(593,·)$, $\chi_{780}(619,·)$, $\chi_{780}(749,·)$, $\chi_{780}(623,·)$, $\chi_{780}(499,·)$, $\chi_{780}(629,·)$, $\chi_{780}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{16} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{17} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{36} a^{20} + \frac{1}{9} a^{18} + \frac{1}{3} a^{16} + \frac{2}{9} a^{14} - \frac{1}{9} a^{12} - \frac{13}{36} a^{10} - \frac{1}{3} a^{8} - \frac{2}{9} a^{6} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{4}$, $\frac{1}{36} a^{21} + \frac{1}{9} a^{19} + \frac{1}{3} a^{17} + \frac{2}{9} a^{15} - \frac{1}{9} a^{13} - \frac{13}{36} a^{11} - \frac{1}{3} a^{9} - \frac{2}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{4} a$, $\frac{1}{108} a^{22} + \frac{1}{108} a^{20} + \frac{11}{27} a^{16} - \frac{7}{27} a^{14} - \frac{37}{108} a^{12} + \frac{1}{4} a^{10} + \frac{7}{27} a^{8} - \frac{2}{27} a^{6} + \frac{5}{12} a^{2} - \frac{1}{4}$, $\frac{1}{108} a^{23} + \frac{1}{108} a^{21} + \frac{11}{27} a^{17} - \frac{7}{27} a^{15} - \frac{37}{108} a^{13} + \frac{1}{4} a^{11} + \frac{7}{27} a^{9} - \frac{2}{27} a^{7} + \frac{5}{12} a^{3} - \frac{1}{4} a$, $\frac{1}{5508} a^{24} + \frac{11}{2754} a^{22} - \frac{2}{459} a^{20} - \frac{115}{1377} a^{18} + \frac{89}{1377} a^{16} - \frac{121}{5508} a^{14} - \frac{23}{918} a^{12} - \frac{596}{1377} a^{10} + \frac{118}{1377} a^{8} - \frac{29}{459} a^{6} + \frac{47}{204} a^{4} + \frac{49}{102} a^{2} + \frac{4}{17}$, $\frac{1}{16524} a^{25} + \frac{11}{8262} a^{23} - \frac{59}{5508} a^{21} + \frac{650}{4131} a^{19} + \frac{548}{4131} a^{17} - \frac{1345}{16524} a^{15} + \frac{1303}{2754} a^{13} + \frac{6949}{16524} a^{11} - \frac{341}{4131} a^{9} + \frac{532}{1377} a^{7} + \frac{481}{1836} a^{5} + \frac{83}{306} a^{3} + \frac{67}{204} a$, $\frac{1}{56690951690981855556} a^{26} - \frac{1371104173421111}{56690951690981855556} a^{24} - \frac{54314061282042643}{18896983896993951852} a^{22} + \frac{638399666767089635}{56690951690981855556} a^{20} + \frac{651019394355200168}{14172737922745463889} a^{18} - \frac{916009662288218641}{56690951690981855556} a^{16} + \frac{9329828198409621505}{18896983896993951852} a^{14} + \frac{13164924082861117981}{56690951690981855556} a^{12} + \frac{15897233330130718537}{56690951690981855556} a^{10} - \frac{1589431332258917287}{4724245974248487963} a^{8} + \frac{2366176116900695113}{6298994632331317284} a^{6} + \frac{337232700984856591}{2099664877443772428} a^{4} - \frac{133976497319065549}{699888292481257476} a^{2} + \frac{107039711258708329}{233296097493752492}$, $\frac{1}{170072855072945566668} a^{27} - \frac{1371104173421111}{170072855072945566668} a^{25} - \frac{57321533600589253}{14172737922745463889} a^{23} - \frac{1461265210676682793}{170072855072945566668} a^{21} - \frac{923729263727629153}{42518213768236391667} a^{19} + \frac{70472596170800043911}{170072855072945566668} a^{17} - \frac{8867267406103072871}{56690951690981855556} a^{15} - \frac{18624021137604095222}{42518213768236391667} a^{13} + \frac{22196227962462035821}{170072855072945566668} a^{11} - \frac{5963733160266776512}{14172737922745463889} a^{9} + \frac{4232544896850715049}{18896983896993951852} a^{7} - \frac{1995728273952668329}{6298994632331317284} a^{5} + \frac{185220967070626574}{524916219360943107} a^{3} + \frac{107039711258708329}{699888292481257476} a$, $\frac{1}{170072855072945566668} a^{28} + \frac{1}{170072855072945566668} a^{26} + \frac{1611242132380501}{18896983896993951852} a^{24} + \frac{763336954641972545}{170072855072945566668} a^{22} + \frac{874908391187434621}{85036427536472783334} a^{20} + \frac{20440915957932274655}{170072855072945566668} a^{18} + \frac{2976883061765618351}{18896983896993951852} a^{16} - \frac{66085420074038696249}{170072855072945566668} a^{14} + \frac{21715583754120411223}{170072855072945566668} a^{12} - \frac{736193451416267233}{9448491948496975926} a^{10} - \frac{2301093108124494353}{6298994632331317284} a^{8} + \frac{466209788260023239}{6298994632331317284} a^{6} + \frac{68720207785115657}{233296097493752492} a^{4} + \frac{34242858477300665}{233296097493752492} a^{2} + \frac{46718874903870247}{116648048746876246}$, $\frac{1}{510218565218836700004} a^{29} + \frac{1}{510218565218836700004} a^{27} + \frac{1611242132380501}{56690951690981855556} a^{25} - \frac{202852925860214194}{127554641304709175001} a^{23} + \frac{175068124292039921}{510218565218836700004} a^{21} + \frac{20440915957932274655}{510218565218836700004} a^{19} - \frac{4721888155528213885}{56690951690981855556} a^{17} - \frac{21992457647719475261}{510218565218836700004} a^{15} - \frac{22522892742440117642}{127554641304709175001} a^{13} - \frac{25093616774074974281}{56690951690981855556} a^{11} + \frac{8663823474081872771}{18896983896993951852} a^{9} + \frac{2410598871859615169}{6298994632331317284} a^{7} + \frac{68720207785115657}{699888292481257476} a^{5} - \frac{222194959729136524}{524916219360943107} a^{3} - \frac{104943473602108789}{233296097493752492} a$, $\frac{1}{510218565218836700004} a^{30} + \frac{1}{510218565218836700004} a^{28} - \frac{6324404048218852}{127554641304709175001} a^{24} + \frac{1060103112831005143}{255109282609418350002} a^{22} - \frac{1181615779290206887}{127554641304709175001} a^{20} - \frac{1670812520120147327}{18896983896993951852} a^{18} + \frac{40531919818534485055}{127554641304709175001} a^{16} + \frac{40191693398256400639}{127554641304709175001} a^{14} + \frac{3490820197001072345}{9448491948496975926} a^{12} + \frac{232939982648991256}{488715100784326341} a^{10} + \frac{917966637073474835}{2099664877443772428} a^{8} - \frac{462209866215548983}{1574748658082829321} a^{6} - \frac{12499599629394742}{174972073120314369} a^{4} + \frac{53535978778476179}{349944146240628738} a^{2} + \frac{104156796350895583}{233296097493752492}$, $\frac{1}{1530655695656510100012} a^{31} + \frac{1}{1530655695656510100012} a^{29} - \frac{6324404048218852}{382663923914127525003} a^{25} - \frac{2604039748586477677}{1530655695656510100012} a^{23} + \frac{2361014415668074189}{765327847828255050006} a^{21} - \frac{5870142275007692183}{56690951690981855556} a^{19} + \frac{116119855406510292463}{382663923914127525003} a^{17} + \frac{101606891063486744158}{382663923914127525003} a^{15} + \frac{17654936854341321199}{56690951690981855556} a^{13} + \frac{194371575973356767}{2932290604705958046} a^{11} + \frac{7419821861095474345}{18896983896993951852} a^{9} - \frac{231835321236433825}{1574748658082829321} a^{7} - \frac{154146847635060472}{1574748658082829321} a^{5} + \frac{171780042723673073}{699888292481257476} a^{3} - \frac{245787349889733155}{699888292481257476} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{1040}$, which has order $66560$ (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:  \( -\frac{2184250988569}{114991788419841492} a^{30} - \frac{1006939705730309}{3334761864175403268} a^{28} - \frac{1297445485922635}{370529096019489252} a^{26} - \frac{121857178401275941}{3334761864175403268} a^{24} - \frac{21023415764976625}{57495894209920746} a^{22} - \frac{6465184159324280221}{3334761864175403268} a^{20} - \frac{359944904657274079}{41169899557721028} a^{18} - \frac{118912833470298387689}{3334761864175403268} a^{16} - \frac{413219535466561068167}{3334761864175403268} a^{14} - \frac{14184025726290703699}{61754849336581542} a^{12} - \frac{150852618818056987975}{370529096019489252} a^{10} - \frac{80190544151145976847}{123509698673163084} a^{8} - \frac{280358616720669408103}{370529096019489252} a^{6} - \frac{114863206735878003}{13723299852573676} a^{4} - \frac{6493778189015637}{6861649926286838} a^{2} - \frac{688039061399235}{6861649926286838} \) (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:  \( 17345214197601.506 \) (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

\(\Q(\sqrt{195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{65})\), 4.0.4394000.2, 4.0.2471625.2, \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{15}, \sqrt{39})\), 4.0.4394000.1, 4.0.2471625.1, \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{13})\), 4.0.21125.1, 4.0.18000.1, \(\Q(\zeta_{5})\), 4.0.3042000.1, 4.4.35152.1, 4.4.494325.1, 4.4.878800.1, 4.4.19773.1, 8.0.1563886116000000.2, 8.8.370150560000.1, 8.0.1563886116000000.62, 8.0.19307236000000.5, 8.0.1563886116000000.26, 8.0.6108930140625.4, 8.0.1563886116000000.24, 8.0.9253764000000.6, 8.0.9253764000000.10, 8.8.62555444640000.3, 8.8.62555444640000.2, 8.0.446265625.1, 8.0.9253764000000.3, 8.8.772289440000.1, 8.8.244357205625.1, 8.0.9253764000000.4, 8.0.324000000.3, 8.8.100088711424.1, 8.8.62555444640000.4, 16.0.2445739783817565456000000000000.3, 16.0.85632148167696000000000000.6, 16.16.3913183654108104729600000000.1, 16.0.372769361959696000000000000.3, 16.0.2445739783817565456000000000000.16, 16.0.37319027463036582275390625.3, 16.0.2445739783817565456000000000000.4

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed