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Properties

Label 32.0.72578796740...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $79.78$
Ramified primes $2, 5, 17$
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![83521, 0, -417605, 0, 1753941, 0, -7182806, 0, 29310958, 0, -25297037, 0, 16320986, 0, -9305222, 0, 4815029, 0, -1687182, 0, 525402, 0, -145945, 0, 32062, 0, -2686, 0, 221, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 17*x^30 + 221*x^28 - 2686*x^26 + 32062*x^24 - 145945*x^22 + 525402*x^20 - 1687182*x^18 + 4815029*x^16 - 9305222*x^14 + 16320986*x^12 - 25297037*x^10 + 29310958*x^8 - 7182806*x^6 + 1753941*x^4 - 417605*x^2 + 83521)
 
gp: K = bnfinit(x^32 - 17*x^30 + 221*x^28 - 2686*x^26 + 32062*x^24 - 145945*x^22 + 525402*x^20 - 1687182*x^18 + 4815029*x^16 - 9305222*x^14 + 16320986*x^12 - 25297037*x^10 + 29310958*x^8 - 7182806*x^6 + 1753941*x^4 - 417605*x^2 + 83521, 1)
 

Normalized defining polynomial

\( x^{32} - 17 x^{30} + 221 x^{28} - 2686 x^{26} + 32062 x^{24} - 145945 x^{22} + 525402 x^{20} - 1687182 x^{18} + 4815029 x^{16} - 9305222 x^{14} + 16320986 x^{12} - 25297037 x^{10} + 29310958 x^{8} - 7182806 x^{6} + 1753941 x^{4} - 417605 x^{2} + 83521 \)

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:  \(7257879674078131427258907485712138496000000000000000000000000=2^{32}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.78$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(263,·)$, $\chi_{340}(137,·)$, $\chi_{340}(13,·)$, $\chi_{340}(273,·)$, $\chi_{340}(19,·)$, $\chi_{340}(149,·)$, $\chi_{340}(151,·)$, $\chi_{340}(157,·)$, $\chi_{340}(287,·)$, $\chi_{340}(33,·)$, $\chi_{340}(291,·)$, $\chi_{340}(293,·)$, $\chi_{340}(169,·)$, $\chi_{340}(43,·)$, $\chi_{340}(179,·)$, $\chi_{340}(59,·)$, $\chi_{340}(69,·)$, $\chi_{340}(331,·)$, $\chi_{340}(81,·)$, $\chi_{340}(83,·)$, $\chi_{340}(87,·)$, $\chi_{340}(89,·)$, $\chi_{340}(219,·)$, $\chi_{340}(223,·)$, $\chi_{340}(101,·)$, $\chi_{340}(237,·)$, $\chi_{340}(111,·)$, $\chi_{340}(247,·)$, $\chi_{340}(217,·)$, $\chi_{340}(21,·)$, $\chi_{340}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17} a^{8}$, $\frac{1}{17} a^{9}$, $\frac{1}{17} a^{10}$, $\frac{1}{17} a^{11}$, $\frac{1}{17} a^{12}$, $\frac{1}{17} a^{13}$, $\frac{1}{17} a^{14}$, $\frac{1}{17} a^{15}$, $\frac{1}{289} a^{16}$, $\frac{1}{289} a^{17}$, $\frac{1}{289} a^{18}$, $\frac{1}{289} a^{19}$, $\frac{1}{289} a^{20}$, $\frac{1}{289} a^{21}$, $\frac{1}{289} a^{22}$, $\frac{1}{289} a^{23}$, $\frac{1}{255476} a^{24} + \frac{1}{3757} a^{22} - \frac{5}{3757} a^{20} + \frac{11}{15028} a^{18} + \frac{3}{221} a^{14} + \frac{1}{884} a^{12} - \frac{3}{221} a^{10} - \frac{21}{52} a^{6} - \frac{6}{13} a^{4} + \frac{4}{13} a^{2} + \frac{25}{52}$, $\frac{1}{255476} a^{25} + \frac{1}{3757} a^{23} - \frac{5}{3757} a^{21} + \frac{11}{15028} a^{19} + \frac{3}{221} a^{15} + \frac{1}{884} a^{13} - \frac{3}{221} a^{11} - \frac{21}{52} a^{7} - \frac{6}{13} a^{5} + \frac{4}{13} a^{3} + \frac{25}{52} a$, $\frac{1}{36247802269815916} a^{26} - \frac{61991704677}{36247802269815916} a^{24} - \frac{49831212731}{533055915732587} a^{22} + \frac{3636489284411}{2132223662930348} a^{20} - \frac{3688650149179}{2132223662930348} a^{18} - \frac{620342199946}{533055915732587} a^{16} + \frac{2504304436085}{125424921348844} a^{14} - \frac{674890205777}{125424921348844} a^{12} - \frac{635819889691}{31356230337211} a^{10} - \frac{3209236976705}{125424921348844} a^{8} - \frac{1069851699583}{7377936549932} a^{6} - \frac{663176898482}{1844484137483} a^{4} - \frac{983561091411}{7377936549932} a^{2} + \frac{3511760935003}{7377936549932}$, $\frac{1}{36247802269815916} a^{27} - \frac{61991704677}{36247802269815916} a^{25} - \frac{49831212731}{533055915732587} a^{23} + \frac{3636489284411}{2132223662930348} a^{21} - \frac{3688650149179}{2132223662930348} a^{19} - \frac{620342199946}{533055915732587} a^{17} + \frac{2504304436085}{125424921348844} a^{15} - \frac{674890205777}{125424921348844} a^{13} - \frac{635819889691}{31356230337211} a^{11} - \frac{3209236976705}{125424921348844} a^{9} - \frac{1069851699583}{7377936549932} a^{7} - \frac{663176898482}{1844484137483} a^{5} - \frac{983561091411}{7377936549932} a^{3} + \frac{3511760935003}{7377936549932} a$, $\frac{1}{36247802269815916} a^{28} + \frac{65120407865}{36247802269815916} a^{24} + \frac{1874032384883}{2132223662930348} a^{22} - \frac{651821493686}{533055915732587} a^{20} + \frac{495001280931}{2132223662930348} a^{18} - \frac{2349029184811}{2132223662930348} a^{16} - \frac{718231177373}{31356230337211} a^{14} - \frac{529975297435}{125424921348844} a^{12} + \frac{1658444996947}{125424921348844} a^{10} + \frac{24813687473}{2412017718247} a^{8} + \frac{2724415687387}{7377936549932} a^{6} + \frac{327096262257}{7377936549932} a^{4} - \frac{122595807743}{1844484137483} a^{2} - \frac{3299769935687}{7377936549932}$, $\frac{1}{36247802269815916} a^{29} + \frac{65120407865}{36247802269815916} a^{25} + \frac{1874032384883}{2132223662930348} a^{23} - \frac{651821493686}{533055915732587} a^{21} + \frac{495001280931}{2132223662930348} a^{19} - \frac{2349029184811}{2132223662930348} a^{17} - \frac{718231177373}{31356230337211} a^{15} - \frac{529975297435}{125424921348844} a^{13} + \frac{1658444996947}{125424921348844} a^{11} + \frac{24813687473}{2412017718247} a^{9} + \frac{2724415687387}{7377936549932} a^{7} + \frac{327096262257}{7377936549932} a^{5} - \frac{122595807743}{1844484137483} a^{3} - \frac{3299769935687}{7377936549932} a$, $\frac{1}{36247802269815916} a^{30} + \frac{693410468100}{533055915732587} a^{20} - \frac{563057580707}{31356230337211} a^{10} - \frac{568026535487}{7377936549932}$, $\frac{1}{36247802269815916} a^{31} + \frac{693410468100}{533055915732587} a^{21} - \frac{563057580707}{31356230337211} a^{11} - \frac{568026535487}{7377936549932} a$

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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2045940413}{697073120573383} a^{30} + \frac{34780281210}{697073120573383} a^{28} - \frac{26597225369}{41004301210199} a^{26} + \frac{323258585254}{41004301210199} a^{24} - \frac{3858643618918}{41004301210199} a^{22} + \frac{10229702065}{23881363547} a^{20} - \frac{63241590341631}{41004301210199} a^{18} + \frac{11944200131094}{2412017718247} a^{16} - \frac{34087413220993}{2412017718247} a^{14} + \frac{3875011142222}{141883395191} a^{12} - \frac{6796614051986}{141883395191} a^{10} + \frac{178949706537445}{2412017718247} a^{8} - \frac{12206080503958}{141883395191} a^{6} + \frac{2991164883806}{141883395191} a^{4} - \frac{730400727441}{141883395191} a^{2} + \frac{173904935105}{141883395191} \) (order $10$)
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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.0.614125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.614125.1, 4.4.122825.1, 4.4.4913.1, 4.0.36125.1, \(\Q(\zeta_{5})\), 8.0.377149515625.1, 8.8.15085980625.1, 8.0.1305015625.1, 8.8.1641354692000000.2, 8.8.1641354692000000.1, 8.0.105046700288.1, 8.0.65654187680000.2, 16.0.142241757136172119140625.1, 16.16.2694045224950414864000000000000.1, 16.0.4310472359920663782400000000.4

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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}$
5Data not computed
17Data not computed