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Properties

Label 32.0.53696696980...0000.3
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 $544$ (GRH)
Class group $[2, 2, 136]$ (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![12752041, 0, -12751752, 0, 12744816, 0, -12678924, 0, 12349464, 0, -11350102, 0, 9351378, 0, -6572932, 0, 3794486, 0, -1751511, 0, 633251, 0, -175781, 0, 36551, 0, -5492, 0, 562, 0, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 35*x^30 + 562*x^28 - 5492*x^26 + 36551*x^24 - 175781*x^22 + 633251*x^20 - 1751511*x^18 + 3794486*x^16 - 6572932*x^14 + 9351378*x^12 - 11350102*x^10 + 12349464*x^8 - 12678924*x^6 + 12744816*x^4 - 12751752*x^2 + 12752041)
 
gp: K = bnfinit(x^32 - 35*x^30 + 562*x^28 - 5492*x^26 + 36551*x^24 - 175781*x^22 + 633251*x^20 - 1751511*x^18 + 3794486*x^16 - 6572932*x^14 + 9351378*x^12 - 11350102*x^10 + 12349464*x^8 - 12678924*x^6 + 12744816*x^4 - 12751752*x^2 + 12752041, 1)
 

Normalized defining polynomial

\( x^{32} - 35 x^{30} + 562 x^{28} - 5492 x^{26} + 36551 x^{24} - 175781 x^{22} + 633251 x^{20} - 1751511 x^{18} + 3794486 x^{16} - 6572932 x^{14} + 9351378 x^{12} - 11350102 x^{10} + 12349464 x^{8} - 12678924 x^{6} + 12744816 x^{4} - 12751752 x^{2} + 12752041 \)

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$$\chi_{340}(1,·)$, $\chi_{340}(171,·)$, $\chi_{340}(129,·)$, $\chi_{340}(139,·)$, $\chi_{340}(269,·)$, $\chi_{340}(271,·)$, $\chi_{340}(21,·)$, $\chi_{340}(279,·)$, $\chi_{340}(281,·)$, $\chi_{340}(29,·)$, $\chi_{340}(159,·)$, $\chi_{340}(161,·)$, $\chi_{340}(291,·)$, $\chi_{340}(39,·)$, $\chi_{340}(299,·)$, $\chi_{340}(309,·)$, $\chi_{340}(151,·)$, $\chi_{340}(191,·)$, $\chi_{340}(321,·)$, $\chi_{340}(199,·)$, $\chi_{340}(329,·)$, $\chi_{340}(331,·)$, $\chi_{340}(79,·)$, $\chi_{340}(81,·)$, $\chi_{340}(249,·)$, $\chi_{340}(99,·)$, $\chi_{340}(101,·)$, $\chi_{340}(209,·)$, $\chi_{340}(109,·)$, $\chi_{340}(111,·)$, $\chi_{340}(121,·)$, $\chi_{340}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3571} a^{17} - \frac{17}{3571} a^{15} + \frac{119}{3571} a^{13} - \frac{442}{3571} a^{11} + \frac{935}{3571} a^{9} - \frac{1122}{3571} a^{7} + \frac{714}{3571} a^{5} - \frac{204}{3571} a^{3} + \frac{17}{3571} a$, $\frac{1}{5702887} a^{18} - \frac{2178327}{5702887} a^{16} + \frac{635757}{5702887} a^{14} + \frac{1570798}{5702887} a^{12} + \frac{2579197}{5702887} a^{10} - \frac{558198}{5702887} a^{8} - \frac{1816925}{5702887} a^{6} - \frac{1942828}{5702887} a^{4} + \frac{2542569}{5702887} a^{2} + \frac{377}{1597}$, $\frac{1}{5702887} a^{19} - \frac{19}{5702887} a^{17} - \frac{2178157}{5702887} a^{15} - \frac{1543352}{5702887} a^{13} - \frac{2147923}{5702887} a^{11} + \frac{229123}{5702887} a^{9} + \frac{660022}{5702887} a^{7} + \frac{2183820}{5702887} a^{5} - \frac{2709964}{5702887} a^{3} - \frac{1542706}{5702887} a$, $\frac{1}{5702887} a^{20} + \frac{2056726}{5702887} a^{16} - \frac{869743}{5702887} a^{14} - \frac{817196}{5702887} a^{12} - \frac{2092117}{5702887} a^{10} + \frac{1460034}{5702887} a^{8} + \frac{1879567}{5702887} a^{6} + \frac{296513}{5702887} a^{4} + \frac{1143009}{5702887} a^{2} + \frac{775}{1597}$, $\frac{1}{5702887} a^{21} - \frac{210}{5702887} a^{17} - \frac{119153}{5702887} a^{15} - \frac{368439}{5702887} a^{13} + \frac{314562}{5702887} a^{11} + \frac{97793}{5702887} a^{9} + \frac{92524}{5702887} a^{7} - \frac{2713832}{5702887} a^{5} - \frac{1255685}{5702887} a^{3} + \frac{2016935}{5702887} a$, $\frac{1}{5702887} a^{22} - \frac{1336863}{5702887} a^{16} + \frac{1974130}{5702887} a^{14} - \frac{585304}{5702887} a^{12} - \frac{45102}{5702887} a^{10} + \frac{2631571}{5702887} a^{8} - \frac{2174653}{5702887} a^{6} + \frac{1358299}{5702887} a^{4} - \frac{114953}{5702887} a^{2} - \frac{680}{1597}$, $\frac{1}{5702887} a^{23} - \frac{174}{5702887} a^{17} + \frac{2061965}{5702887} a^{15} - \frac{1200149}{5702887} a^{13} + \frac{2238608}{5702887} a^{11} - \frac{2199354}{5702887} a^{9} - \frac{2080430}{5702887} a^{7} - \frac{2330771}{5702887} a^{5} + \frac{939067}{5702887} a^{3} - \frac{2516115}{5702887} a$, $\frac{1}{5702887} a^{24} - \frac{576391}{5702887} a^{16} + \frac{1066716}{5702887} a^{14} + \frac{1818884}{5702887} a^{12} + \frac{1755738}{5702887} a^{10} - \frac{2257803}{5702887} a^{8} + \frac{885951}{5702887} a^{6} - \frac{642672}{5702887} a^{4} + \frac{768592}{5702887} a^{2} + \frac{121}{1597}$, $\frac{1}{5702887} a^{25} + \frac{126}{5702887} a^{17} + \frac{2671701}{5702887} a^{15} + \frac{1989763}{5702887} a^{13} - \frac{2137748}{5702887} a^{11} + \frac{714214}{5702887} a^{9} - \frac{1539892}{5702887} a^{7} + \frac{382602}{5702887} a^{5} - \frac{2783136}{5702887} a^{3} - \frac{1172894}{5702887} a$, $\frac{1}{5702887} a^{26} - \frac{2300560}{5702887} a^{16} + \frac{1724799}{5702887} a^{14} - \frac{457251}{5702887} a^{12} + \frac{799951}{5702887} a^{10} + \frac{358412}{5702887} a^{8} + \frac{1199672}{5702887} a^{6} + \frac{2491938}{5702887} a^{4} - \frac{2174916}{5702887} a^{2} + \frac{408}{1597}$, $\frac{1}{5702887} a^{27} + \frac{717}{5702887} a^{17} + \frac{2523299}{5702887} a^{15} - \frac{343864}{5702887} a^{13} - \frac{1250597}{5702887} a^{11} + \frac{2064008}{5702887} a^{9} + \frac{2574689}{5702887} a^{7} - \frac{2530627}{5702887} a^{5} + \frac{1704197}{5702887} a^{3} + \frac{658468}{5702887} a$, $\frac{1}{5702887} a^{28} + \frac{1792720}{5702887} a^{16} + \frac{49327}{5702887} a^{14} + \frac{1658863}{5702887} a^{12} + \frac{515147}{5702887} a^{10} - \frac{2102322}{5702887} a^{8} - \frac{53638}{5702887} a^{6} - \frac{2495442}{5702887} a^{4} + \frac{2560335}{5702887} a^{2} - \frac{416}{1597}$, $\frac{1}{5702887} a^{29} - \frac{711}{5702887} a^{17} + \frac{2023219}{5702887} a^{15} - \frac{752607}{5702887} a^{13} + \frac{510356}{5702887} a^{11} - \frac{2311529}{5702887} a^{9} - \frac{943167}{5702887} a^{7} + \frac{144399}{5702887} a^{5} - \frac{2267396}{5702887} a^{3} + \frac{2243459}{5702887} a$, $\frac{1}{5702887} a^{30} - \frac{1284901}{5702887} a^{16} + \frac{742547}{5702887} a^{14} - \frac{418118}{5702887} a^{12} + \frac{870811}{5702887} a^{10} + \frac{1380145}{5702887} a^{8} - \frac{2836814}{5702887} a^{6} + \frac{2183437}{5702887} a^{4} + \frac{2194839}{5702887} a^{2} - \frac{249}{1597}$, $\frac{1}{5702887} a^{31} + \frac{684}{5702887} a^{17} + \frac{1699150}{5702887} a^{15} - \frac{1411452}{5702887} a^{13} - \frac{2771946}{5702887} a^{11} + \frac{92963}{5702887} a^{9} - \frac{2432773}{5702887} a^{7} + \frac{1926320}{5702887} a^{5} + \frac{2268301}{5702887} a^{3} - \frac{1845782}{5702887} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{136}$, which has order $544$ (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:  \( \frac{1}{3571} a^{17} - \frac{17}{3571} a^{15} + \frac{119}{3571} a^{13} - \frac{442}{3571} a^{11} + \frac{935}{3571} a^{9} - \frac{1122}{3571} a^{7} + \frac{714}{3571} a^{5} - \frac{204}{3571} a^{3} + \frac{17}{3571} a \) (order $4$)
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:  \( 10897700797646.188 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{17})\), 4.4.4913.1, 4.0.78608.1, 8.0.6179217664.1, \(\Q(\zeta_{17})^+\), 8.0.105046700288.1, 16.0.11034809241396899282944.1, 16.16.73278030118651284300800000000.1, 16.0.1118134004496021794140625.1

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 $16^{2}$ 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