Properties

Label 32.0.44899126040...3808.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $287.00$
Ramified primes $2, 3, 7$
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![2861137380483970656642, 0, 17439313557235630669056, 0, 17646924432917007224640, 0, 7058769773166802889856, 0, 1482581746575000266832, 0, 188264348771428605312, 0, 15688695730952383776, 0, 903064641760733760, 0, 37090154929458708, 0, 1108202574923136, 0, 24302688046560, 0, 390766783680, 0, 4550855400, 0, 37340352, 0, 204624, 0, 672, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 672*x^30 + 204624*x^28 + 37340352*x^26 + 4550855400*x^24 + 390766783680*x^22 + 24302688046560*x^20 + 1108202574923136*x^18 + 37090154929458708*x^16 + 903064641760733760*x^14 + 15688695730952383776*x^12 + 188264348771428605312*x^10 + 1482581746575000266832*x^8 + 7058769773166802889856*x^6 + 17646924432917007224640*x^4 + 17439313557235630669056*x^2 + 2861137380483970656642)
 
gp: K = bnfinit(x^32 + 672*x^30 + 204624*x^28 + 37340352*x^26 + 4550855400*x^24 + 390766783680*x^22 + 24302688046560*x^20 + 1108202574923136*x^18 + 37090154929458708*x^16 + 903064641760733760*x^14 + 15688695730952383776*x^12 + 188264348771428605312*x^10 + 1482581746575000266832*x^8 + 7058769773166802889856*x^6 + 17646924432917007224640*x^4 + 17439313557235630669056*x^2 + 2861137380483970656642, 1)
 

Normalized defining polynomial

\( x^{32} + 672 x^{30} + 204624 x^{28} + 37340352 x^{26} + 4550855400 x^{24} + 390766783680 x^{22} + 24302688046560 x^{20} + 1108202574923136 x^{18} + 37090154929458708 x^{16} + 903064641760733760 x^{14} + 15688695730952383776 x^{12} + 188264348771428605312 x^{10} + 1482581746575000266832 x^{8} + 7058769773166802889856 x^{6} + 17646924432917007224640 x^{4} + 17439313557235630669056 x^{2} + 2861137380483970656642 \)

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:  \(4489912604053908534055314729400632754872833954383027744299245049859706934263808=2^{191}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $287.00$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2688=2^{7}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{2688}(1,·)$, $\chi_{2688}(2435,·)$, $\chi_{2688}(2185,·)$, $\chi_{2688}(1931,·)$, $\chi_{2688}(1681,·)$, $\chi_{2688}(1427,·)$, $\chi_{2688}(1177,·)$, $\chi_{2688}(923,·)$, $\chi_{2688}(673,·)$, $\chi_{2688}(419,·)$, $\chi_{2688}(169,·)$, $\chi_{2688}(2603,·)$, $\chi_{2688}(2353,·)$, $\chi_{2688}(2099,·)$, $\chi_{2688}(1849,·)$, $\chi_{2688}(1595,·)$, $\chi_{2688}(1345,·)$, $\chi_{2688}(1091,·)$, $\chi_{2688}(841,·)$, $\chi_{2688}(587,·)$, $\chi_{2688}(337,·)$, $\chi_{2688}(83,·)$, $\chi_{2688}(2521,·)$, $\chi_{2688}(2267,·)$, $\chi_{2688}(2017,·)$, $\chi_{2688}(1763,·)$, $\chi_{2688}(1513,·)$, $\chi_{2688}(1259,·)$, $\chi_{2688}(1009,·)$, $\chi_{2688}(755,·)$, $\chi_{2688}(505,·)$, $\chi_{2688}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{21} a^{2}$, $\frac{1}{21} a^{3}$, $\frac{1}{441} a^{4}$, $\frac{1}{441} a^{5}$, $\frac{1}{9261} a^{6}$, $\frac{1}{9261} a^{7}$, $\frac{1}{194481} a^{8}$, $\frac{1}{194481} a^{9}$, $\frac{1}{4084101} a^{10}$, $\frac{1}{4084101} a^{11}$, $\frac{1}{85766121} a^{12}$, $\frac{1}{85766121} a^{13}$, $\frac{1}{1801088541} a^{14}$, $\frac{1}{1801088541} a^{15}$, $\frac{1}{37822859361} a^{16}$, $\frac{1}{37822859361} a^{17}$, $\frac{1}{794280046581} a^{18}$, $\frac{1}{794280046581} a^{19}$, $\frac{1}{16679880978201} a^{20}$, $\frac{1}{16679880978201} a^{21}$, $\frac{1}{350277500542221} a^{22}$, $\frac{1}{350277500542221} a^{23}$, $\frac{1}{7355827511386641} a^{24}$, $\frac{1}{7355827511386641} a^{25}$, $\frac{1}{154472377739119461} a^{26}$, $\frac{1}{154472377739119461} a^{27}$, $\frac{1}{3243919932521508681} a^{28}$, $\frac{1}{3243919932521508681} a^{29}$, $\frac{1}{68122318582951682301} a^{30}$, $\frac{1}{68122318582951682301} a^{31}$

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:  $15$
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:  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 R $32$ R $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
3Data not computed
7Data not computed