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Properties

Label 32.0.87309122741...000.10
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $92.66$
Ramified primes $2, 3, 5$
Class number $4352$ (GRH)
Class group $[2, 8, 272]$ (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![779470561, 0, -832092288, 0, 1668661744, 0, -1275041376, 0, 479321832, 0, -99845280, 0, 14823352, 0, -5233056, 0, 4129467, 0, -2670144, 0, 1217920, 0, -382560, 0, 81260, 0, -11328, 0, 984, 0, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 48*x^30 + 984*x^28 - 11328*x^26 + 81260*x^24 - 382560*x^22 + 1217920*x^20 - 2670144*x^18 + 4129467*x^16 - 5233056*x^14 + 14823352*x^12 - 99845280*x^10 + 479321832*x^8 - 1275041376*x^6 + 1668661744*x^4 - 832092288*x^2 + 779470561)
 
gp: K = bnfinit(x^32 - 48*x^30 + 984*x^28 - 11328*x^26 + 81260*x^24 - 382560*x^22 + 1217920*x^20 - 2670144*x^18 + 4129467*x^16 - 5233056*x^14 + 14823352*x^12 - 99845280*x^10 + 479321832*x^8 - 1275041376*x^6 + 1668661744*x^4 - 832092288*x^2 + 779470561, 1)
 

Normalized defining polynomial

\( x^{32} - 48 x^{30} + 984 x^{28} - 11328 x^{26} + 81260 x^{24} - 382560 x^{22} + 1217920 x^{20} - 2670144 x^{18} + 4129467 x^{16} - 5233056 x^{14} + 14823352 x^{12} - 99845280 x^{10} + 479321832 x^{8} - 1275041376 x^{6} + 1668661744 x^{4} - 832092288 x^{2} + 779470561 \)

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:  \(873091227416114037923609044826021953536000000000000000000000000=2^{128}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.66$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(259,·)$, $\chi_{480}(263,·)$, $\chi_{480}(139,·)$, $\chi_{480}(143,·)$, $\chi_{480}(19,·)$, $\chi_{480}(407,·)$, $\chi_{480}(409,·)$, $\chi_{480}(413,·)$, $\chi_{480}(287,·)$, $\chi_{480}(289,·)$, $\chi_{480}(293,·)$, $\chi_{480}(167,·)$, $\chi_{480}(169,·)$, $\chi_{480}(173,·)$, $\chi_{480}(47,·)$, $\chi_{480}(49,·)$, $\chi_{480}(53,·)$, $\chi_{480}(23,·)$, $\chi_{480}(317,·)$, $\chi_{480}(437,·)$, $\chi_{480}(451,·)$, $\chi_{480}(197,·)$, $\chi_{480}(331,·)$, $\chi_{480}(77,·)$, $\chi_{480}(211,·)$, $\chi_{480}(91,·)$, $\chi_{480}(361,·)$, $\chi_{480}(241,·)$, $\chi_{480}(121,·)$, $\chi_{480}(379,·)$, $\chi_{480}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{27919} a^{23} + \frac{5460}{27919} a^{21} - \frac{8793}{27919} a^{19} - \frac{11780}{27919} a^{17} + \frac{12285}{27919} a^{15} - \frac{13594}{27919} a^{13} + \frac{12363}{27919} a^{11} + \frac{272}{27919} a^{9} - \frac{9398}{27919} a^{7} - \frac{10222}{27919} a^{5} - \frac{12825}{27919} a^{3} - \frac{12292}{27919} a$, $\frac{1}{105561739} a^{24} + \frac{17036050}{105561739} a^{22} - \frac{19300822}{105561739} a^{20} + \frac{38767711}{105561739} a^{18} - \frac{17213738}{105561739} a^{16} - \frac{45074860}{105561739} a^{14} - \frac{37510773}{105561739} a^{12} - \frac{22223252}{105561739} a^{10} + \frac{28300468}{105561739} a^{8} + \frac{52337903}{105561739} a^{6} + \frac{2779075}{105561739} a^{4} - \frac{47725863}{105561739} a^{2} - \frac{193}{3781}$, $\frac{1}{105561739} a^{25} - \frac{1136}{105561739} a^{23} - \frac{42444323}{105561739} a^{21} - \frac{50925171}{105561739} a^{19} + \frac{7971503}{105561739} a^{17} - \frac{17976433}{105561739} a^{15} - \frac{36459655}{105561739} a^{13} + \frac{48277274}{105561739} a^{11} + \frac{38902392}{105561739} a^{9} + \frac{30653868}{105561739} a^{7} - \frac{19974983}{105561739} a^{5} + \frac{46946596}{105561739} a^{3} - \frac{18788231}{105561739} a$, $\frac{1}{105561739} a^{26} - \frac{7289760}{105561739} a^{22} - \frac{19817251}{105561739} a^{20} + \frac{28846036}{105561739} a^{18} - \frac{43861086}{105561739} a^{16} - \frac{2318800}{5555881} a^{14} - \frac{1188423}{5555881} a^{12} + \frac{22543741}{105561739} a^{10} - \frac{16344879}{105561739} a^{8} + \frac{4623768}{105561739} a^{6} + \frac{37123626}{105561739} a^{4} + \frac{23365247}{105561739} a^{2} + \frac{50}{3781}$, $\frac{1}{105561739} a^{27} + \frac{8}{105561739} a^{23} - \frac{14459574}{105561739} a^{21} + \frac{5891585}{105561739} a^{19} + \frac{9927420}{105561739} a^{17} - \frac{295368}{5555881} a^{15} + \frac{146668}{5555881} a^{13} - \frac{3779581}{105561739} a^{11} - \frac{39201024}{105561739} a^{9} + \frac{4952715}{105561739} a^{7} + \frac{47702864}{105561739} a^{5} - \frac{45770338}{105561739} a^{3} + \frac{17484105}{105561739} a$, $\frac{1}{105561739} a^{28} - \frac{45186235}{105561739} a^{22} - \frac{50825317}{105561739} a^{20} + \frac{16470949}{105561739} a^{18} + \frac{26536173}{105561739} a^{16} + \frac{46700355}{105561739} a^{14} - \frac{20378614}{105561739} a^{12} + \frac{33023253}{105561739} a^{10} - \frac{10327551}{105561739} a^{8} + \frac{51246596}{105561739} a^{6} + \frac{1976779}{5555881} a^{4} - \frac{22955947}{105561739} a^{2} + \frac{1544}{3781}$, $\frac{1}{105561739} a^{29} + \frac{496}{105561739} a^{23} - \frac{29058100}{105561739} a^{21} + \frac{23930862}{105561739} a^{19} - \frac{30866969}{105561739} a^{17} + \frac{16505289}{105561739} a^{15} - \frac{25040587}{105561739} a^{13} + \frac{43855818}{105561739} a^{11} + \frac{35301557}{105561739} a^{9} - \frac{44337084}{105561739} a^{7} - \frac{1505124}{5555881} a^{5} - \frac{8833912}{105561739} a^{3} - \frac{31881637}{105561739} a$, $\frac{1}{87608143099560270780671492581339936893288887941} a^{30} + \frac{82833024836585603610468802394819364478}{87608143099560270780671492581339936893288887941} a^{28} + \frac{356614013880202484805857312184168688089}{87608143099560270780671492581339936893288887941} a^{26} - \frac{337391149296355167915496768508468445987}{87608143099560270780671492581339936893288887941} a^{24} + \frac{29578187866430266654985279959295684034047459555}{87608143099560270780671492581339936893288887941} a^{22} - \frac{6195705823590708389795242935405589577755453744}{87608143099560270780671492581339936893288887941} a^{20} - \frac{14897925101704433598363483282059870628700666011}{87608143099560270780671492581339936893288887941} a^{18} + \frac{22232667717803984905173725470069473554664420437}{87608143099560270780671492581339936893288887941} a^{16} - \frac{29771275178328897114538014864672123775925535405}{87608143099560270780671492581339936893288887941} a^{14} - \frac{41832636572947555290510422115264467572580151156}{87608143099560270780671492581339936893288887941} a^{12} + \frac{10520184872084513076779868869375594136741279849}{87608143099560270780671492581339936893288887941} a^{10} + \frac{38761975818298766972631673118904432185896553876}{87608143099560270780671492581339936893288887941} a^{8} + \frac{33919532686583560095722554385580953769224448295}{87608143099560270780671492581339936893288887941} a^{6} - \frac{553701516135920555882197181161202587387927366}{87608143099560270780671492581339936893288887941} a^{4} + \frac{38103965876367668785416020565129643659716511299}{87608143099560270780671492581339936893288887941} a^{2} + \frac{14694098124371319166305652675501923770}{112394421910130703166699187990680172581}$, $\frac{1}{87608143099560270780671492581339936893288887941} a^{31} + \frac{82833024836585603610468802394819364478}{87608143099560270780671492581339936893288887941} a^{29} + \frac{356614013880202484805857312184168688089}{87608143099560270780671492581339936893288887941} a^{27} - \frac{337391149296355167915496768508468445987}{87608143099560270780671492581339936893288887941} a^{25} - \frac{33303971793317743309498482540299064079459}{87608143099560270780671492581339936893288887941} a^{23} - \frac{41475563729259110710332854995006754035337994921}{87608143099560270780671492581339936893288887941} a^{21} - \frac{42176036350835041209537855036405945753646412738}{87608143099560270780671492581339936893288887941} a^{19} + \frac{36092948102873568917431542108623092998686664000}{87608143099560270780671492581339936893288887941} a^{17} + \frac{359223408367536140091723727900177311124856873}{87608143099560270780671492581339936893288887941} a^{15} + \frac{10737269990573101480455211152946713442974443911}{87608143099560270780671492581339936893288887941} a^{13} + \frac{11361152755987308756038436870084756466602715501}{87608143099560270780671492581339936893288887941} a^{11} - \frac{34173160471076904568438834494838329131153255301}{87608143099560270780671492581339936893288887941} a^{9} + \frac{29404037220403996728360317993713473945826665074}{87608143099560270780671492581339936893288887941} a^{7} + \frac{12321265751226656578438342023725711738811589351}{87608143099560270780671492581339936893288887941} a^{5} + \frac{35530855186814338722012939368929967874319581319}{87608143099560270780671492581339936893288887941} a^{3} + \frac{471162302209613700920438476938286863273532}{3137939865308939101711074629511799738288939} a$

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

Class group and class number

$C_{2}\times C_{8}\times C_{272}$, which has order $4352$ (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:  \( 459974601348790.56 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.72000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.18000.1, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.2304000.2, 4.0.2304000.1, 8.0.82944000000.6, 8.8.2621440000.1, 8.0.5308416000000.7, 8.8.2717908992000000.1, 8.8.2717908992000000.2, 8.0.2147483648.1, 8.0.1342177280000.1, 16.0.450868486864896000000000000.13, 16.16.7387029288794456064000000000000.1, 16.0.1801439850948198400000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed
5Data not computed