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Properties

Label 32.0.53696696980...0000.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{16}\cdot 17^{30}$
Root discriminant $63.69$
Ramified primes $2, 5, 17$
Class number $54400$ (GRH)
Class group $[40, 1360]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2550409, 0, -510024, 0, 103392, 0, -7500, 0, 67392, 0, 186394, 0, 362466, 0, 483196, 0, 459050, 0, 316785, 0, 160295, 0, 59435, 0, 15959, 0, 3020, 0, 382, 0, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 29*x^30 + 382*x^28 + 3020*x^26 + 15959*x^24 + 59435*x^22 + 160295*x^20 + 316785*x^18 + 459050*x^16 + 483196*x^14 + 362466*x^12 + 186394*x^10 + 67392*x^8 - 7500*x^6 + 103392*x^4 - 510024*x^2 + 2550409)
 
gp: K = bnfinit(x^32 + 29*x^30 + 382*x^28 + 3020*x^26 + 15959*x^24 + 59435*x^22 + 160295*x^20 + 316785*x^18 + 459050*x^16 + 483196*x^14 + 362466*x^12 + 186394*x^10 + 67392*x^8 - 7500*x^6 + 103392*x^4 - 510024*x^2 + 2550409, 1)
 

Normalized defining polynomial

\( x^{32} + 29 x^{30} + 382 x^{28} + 3020 x^{26} + 15959 x^{24} + 59435 x^{22} + 160295 x^{20} + 316785 x^{18} + 459050 x^{16} + 483196 x^{14} + 362466 x^{12} + 186394 x^{10} + 67392 x^{8} - 7500 x^{6} + 103392 x^{4} - 510024 x^{2} + 2550409 \)

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:  \(5369669698069964755143230114229268544880640000000000000000=2^{32}\cdot 5^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.69$
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$$\chi_{340}(1,·)$, $\chi_{340}(131,·)$, $\chi_{340}(129,·)$, $\chi_{340}(11,·)$, $\chi_{340}(269,·)$, $\chi_{340}(19,·)$, $\chi_{340}(21,·)$, $\chi_{340}(281,·)$, $\chi_{340}(29,·)$, $\chi_{340}(31,·)$, $\chi_{340}(161,·)$, $\chi_{340}(91,·)$, $\chi_{340}(179,·)$, $\chi_{340}(309,·)$, $\chi_{340}(311,·)$, $\chi_{340}(59,·)$, $\chi_{340}(319,·)$, $\chi_{340}(321,·)$, $\chi_{340}(71,·)$, $\chi_{340}(329,·)$, $\chi_{340}(209,·)$, $\chi_{340}(211,·)$, $\chi_{340}(121,·)$, $\chi_{340}(219,·)$, $\chi_{340}(101,·)$, $\chi_{340}(81,·)$, $\chi_{340}(259,·)$, $\chi_{340}(231,·)$, $\chi_{340}(109,·)$, $\chi_{340}(239,·)$, $\chi_{340}(339,·)$, $\chi_{340}(249,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1597} a^{17} + \frac{17}{1597} a^{15} + \frac{119}{1597} a^{13} + \frac{442}{1597} a^{11} - \frac{662}{1597} a^{9} - \frac{475}{1597} a^{7} + \frac{714}{1597} a^{5} + \frac{204}{1597} a^{3} + \frac{17}{1597} a$, $\frac{1}{5702887} a^{18} - \frac{2178291}{5702887} a^{16} - \frac{635487}{5702887} a^{14} + \frac{1571890}{5702887} a^{12} - \frac{2576623}{5702887} a^{10} - \frac{554634}{5702887} a^{8} + \frac{1819697}{5702887} a^{6} - \frac{1941748}{5702887} a^{4} - \frac{2542407}{5702887} a^{2} + \frac{843}{3571}$, $\frac{1}{5702887} a^{19} + \frac{19}{5702887} a^{17} + \frac{2178461}{5702887} a^{15} - \frac{1542022}{5702887} a^{13} + \frac{2151381}{5702887} a^{11} + \frac{234557}{5702887} a^{9} - \frac{655006}{5702887} a^{7} + \frac{2186328}{5702887} a^{5} + \frac{2710534}{5702887} a^{3} - \frac{1542668}{5702887} a$, $\frac{1}{5702887} a^{20} - \frac{2057106}{5702887} a^{16} - \frac{873543}{5702887} a^{14} + \frac{799906}{5702887} a^{12} - \frac{2135589}{5702887} a^{10} - \frac{1522734}{5702887} a^{8} + \frac{1829407}{5702887} a^{6} - \frac{316463}{5702887} a^{4} + \frac{1139969}{5702887} a^{2} - \frac{1733}{3571}$, $\frac{1}{5702887} a^{21} - \frac{210}{5702887} a^{17} - \frac{123633}{5702887} a^{15} + \frac{346389}{5702887} a^{13} + \frac{253410}{5702887} a^{11} - \frac{197893}{5702887} a^{9} - \frac{2516}{5702887} a^{7} + \frac{2665322}{5702887} a^{5} - \frac{1266885}{5702887} a^{3} - \frac{2017691}{5702887} a$, $\frac{1}{5702887} a^{22} - \frac{1333783}{5702887} a^{16} - \frac{1939480}{5702887} a^{14} - \frac{417136}{5702887} a^{12} + \frac{485542}{5702887} a^{10} - \frac{2417916}{5702887} a^{8} + \frac{2708263}{5702887} a^{6} + \frac{1573899}{5702887} a^{4} + \frac{148217}{5702887} a^{2} - \frac{1520}{3571}$, $\frac{1}{5702887} a^{23} + \frac{1771}{5702887} a^{17} - \frac{2046610}{5702887} a^{15} - \frac{1167046}{5702887} a^{13} - \frac{2299838}{5702887} a^{11} - \frac{2607179}{5702887} a^{9} + \frac{1340570}{5702887} a^{7} + \frac{2777326}{5702887} a^{5} - \frac{1137343}{5702887} a^{3} - \frac{2534570}{5702887} a$, $\frac{1}{5702887} a^{24} + \frac{555139}{5702887} a^{16} + \frac{811692}{5702887} a^{14} + \frac{2594715}{5702887} a^{12} - \frac{1717446}{5702887} a^{10} + \frac{2700820}{5702887} a^{8} + \frac{2225094}{5702887} a^{6} - \frac{1142496}{5702887} a^{4} + \frac{490384}{5702887} a^{2} - \frac{275}{3571}$, $\frac{1}{5702887} a^{25} + \frac{1634}{5702887} a^{17} + \frac{2807881}{5702887} a^{15} - \frac{540623}{5702887} a^{13} - \frac{1142515}{5702887} a^{11} - \frac{1566525}{5702887} a^{9} + \frac{2807167}{5702887} a^{7} - \frac{2845863}{5702887} a^{5} + \frac{1633104}{5702887} a^{3} + \frac{1557014}{5702887} a$, $\frac{1}{5702887} a^{26} - \frac{2169000}{5702887} a^{16} - \frac{80299}{5702887} a^{14} + \frac{2391262}{5702887} a^{12} - \frac{95149}{5702887} a^{10} + \frac{2320090}{5702887} a^{8} + \frac{676253}{5702887} a^{6} - \frac{2058723}{5702887} a^{4} - \frac{1554571}{5702887} a^{2} + \frac{944}{3571}$, $\frac{1}{5702887} a^{27} - \frac{1403}{5702887} a^{17} + \frac{2551528}{5702887} a^{15} - \frac{1997497}{5702887} a^{13} - \frac{102291}{5702887} a^{11} - \frac{1204487}{5702887} a^{9} - \frac{2412662}{5702887} a^{7} + \frac{123158}{5702887} a^{5} + \frac{1512918}{5702887} a^{3} - \frac{1563492}{5702887} a$, $\frac{1}{5702887} a^{28} - \frac{2546200}{5702887} a^{16} + \frac{1767501}{5702887} a^{14} - \frac{1757890}{5702887} a^{12} - \frac{576198}{5702887} a^{10} + \frac{731355}{5702887} a^{8} - \frac{1735327}{5702887} a^{6} - \frac{2482427}{5702887} a^{4} + \frac{1446749}{5702887} a^{2} + \frac{728}{3571}$, $\frac{1}{5702887} a^{29} - \frac{77}{5702887} a^{17} - \frac{571504}{5702887} a^{15} - \frac{1022264}{5702887} a^{13} + \frac{1341429}{5702887} a^{11} - \frac{2450406}{5702887} a^{9} - \frac{2131708}{5702887} a^{7} + \frac{1931329}{5702887} a^{5} + \frac{1893124}{5702887} a^{3} - \frac{1176389}{5702887} a$, $\frac{1}{5702887} a^{30} + \frac{2786699}{5702887} a^{16} + \frac{1371220}{5702887} a^{14} + \frac{2616332}{5702887} a^{12} - \frac{1249332}{5702887} a^{10} + \frac{784570}{5702887} a^{8} - \frac{524177}{5702887} a^{6} + \frac{653590}{5702887} a^{4} + \frac{2659317}{5702887} a^{2} + \frac{633}{3571}$, $\frac{1}{5702887} a^{31} + \frac{1319}{5702887} a^{17} - \frac{357144}{5702887} a^{15} + \frac{1923558}{5702887} a^{13} - \frac{563700}{5702887} a^{11} + \frac{2673629}{5702887} a^{9} - \frac{538461}{5702887} a^{7} + \frac{2199833}{5702887} a^{5} - \frac{972390}{5702887} a^{3} - \frac{717463}{5702887} a$

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

Class group and class number

$C_{40}\times C_{1360}$, which has order $54400$ (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:  \( 73876734347.55115 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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 $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-5}, \sqrt{17})\), 4.4.4913.1, 4.0.1965200.1, 8.0.3862011040000.2, \(\Q(\zeta_{17})^+\), 8.0.65654187680000.2, 16.0.4310472359920663782400000000.1, 16.0.1118134004496021794140625.1, \(\Q(\zeta_{68})^+\)

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 $16^{2}$ R $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
17Data not computed