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Properties

Label 32.32.1527256259...5488.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 17^{16}$
Root discriminant $258.22$
Ramified primes $2, 17$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![97322383751333736962, 0, -732780301186512843008, 0, 915975376483141053760, 0, -452599597791669697152, 0, 117428677158132789072, 0, -18420184652256123776, 0, 1896195478908718624, 0, -134829672062029120, 0, 6840623067852948, 0, -252479744142208, 0, 6839621551840, 0, -135851917760, 0, 1954391400, 0, -19809216, 0, 134096, 0, -544, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 544*x^30 + 134096*x^28 - 19809216*x^26 + 1954391400*x^24 - 135851917760*x^22 + 6839621551840*x^20 - 252479744142208*x^18 + 6840623067852948*x^16 - 134829672062029120*x^14 + 1896195478908718624*x^12 - 18420184652256123776*x^10 + 117428677158132789072*x^8 - 452599597791669697152*x^6 + 915975376483141053760*x^4 - 732780301186512843008*x^2 + 97322383751333736962)
 
gp: K = bnfinit(x^32 - 544*x^30 + 134096*x^28 - 19809216*x^26 + 1954391400*x^24 - 135851917760*x^22 + 6839621551840*x^20 - 252479744142208*x^18 + 6840623067852948*x^16 - 134829672062029120*x^14 + 1896195478908718624*x^12 - 18420184652256123776*x^10 + 117428677158132789072*x^8 - 452599597791669697152*x^6 + 915975376483141053760*x^4 - 732780301186512843008*x^2 + 97322383751333736962, 1)
 

Normalized defining polynomial

\( x^{32} - 544 x^{30} + 134096 x^{28} - 19809216 x^{26} + 1954391400 x^{24} - 135851917760 x^{22} + 6839621551840 x^{20} - 252479744142208 x^{18} + 6840623067852948 x^{16} - 134829672062029120 x^{14} + 1896195478908718624 x^{12} - 18420184652256123776 x^{10} + 117428677158132789072 x^{8} - 452599597791669697152 x^{6} + 915975376483141053760 x^{4} - 732780301186512843008 x^{2} + 97322383751333736962 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(152725625984366375633872344931707463035520335286488090842213114237373265215488=2^{191}\cdot 17^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $258.22$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2176=2^{7}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2176}(1,·)$, $\chi_{2176}(645,·)$, $\chi_{2176}(137,·)$, $\chi_{2176}(781,·)$, $\chi_{2176}(273,·)$, $\chi_{2176}(917,·)$, $\chi_{2176}(409,·)$, $\chi_{2176}(1053,·)$, $\chi_{2176}(545,·)$, $\chi_{2176}(1189,·)$, $\chi_{2176}(681,·)$, $\chi_{2176}(1325,·)$, $\chi_{2176}(817,·)$, $\chi_{2176}(1461,·)$, $\chi_{2176}(953,·)$, $\chi_{2176}(1597,·)$, $\chi_{2176}(1089,·)$, $\chi_{2176}(1733,·)$, $\chi_{2176}(1225,·)$, $\chi_{2176}(1869,·)$, $\chi_{2176}(1361,·)$, $\chi_{2176}(2005,·)$, $\chi_{2176}(1497,·)$, $\chi_{2176}(2141,·)$, $\chi_{2176}(1633,·)$, $\chi_{2176}(101,·)$, $\chi_{2176}(1769,·)$, $\chi_{2176}(237,·)$, $\chi_{2176}(1905,·)$, $\chi_{2176}(373,·)$, $\chi_{2176}(2041,·)$, $\chi_{2176}(509,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{17} a^{2}$, $\frac{1}{17} a^{3}$, $\frac{1}{289} a^{4}$, $\frac{1}{289} a^{5}$, $\frac{1}{4913} a^{6}$, $\frac{1}{4913} a^{7}$, $\frac{1}{83521} a^{8}$, $\frac{1}{83521} a^{9}$, $\frac{1}{1419857} a^{10}$, $\frac{1}{1419857} a^{11}$, $\frac{1}{24137569} a^{12}$, $\frac{1}{24137569} a^{13}$, $\frac{1}{410338673} a^{14}$, $\frac{1}{410338673} a^{15}$, $\frac{1}{6975757441} a^{16}$, $\frac{1}{6975757441} a^{17}$, $\frac{1}{118587876497} a^{18}$, $\frac{1}{118587876497} a^{19}$, $\frac{1}{2015993900449} a^{20}$, $\frac{1}{2015993900449} a^{21}$, $\frac{1}{34271896307633} a^{22}$, $\frac{1}{34271896307633} a^{23}$, $\frac{1}{582622237229761} a^{24}$, $\frac{1}{582622237229761} a^{25}$, $\frac{1}{9904578032905937} a^{26}$, $\frac{1}{9904578032905937} a^{27}$, $\frac{1}{168377826559400929} a^{28}$, $\frac{1}{168377826559400929} a^{29}$, $\frac{1}{2862423051509815793} a^{30}$, $\frac{1}{2862423051509815793} a^{31}$

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:  $31$
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:  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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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 $32$ $32$ $16^{2}$ $32$ $32$ R $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
$17$17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$