Properties

Label 32.32.3138550867...6448.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}$
Root discriminant $62.63$
Ramified prime $2$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{32}$ (as 32T33)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -256, 0, 5440, 0, -45696, 0, 201552, 0, -537472, 0, 940576, 0, -1136960, 0, 980628, 0, -615296, 0, 283360, 0, -95680, 0, 23400, 0, -4032, 0, 464, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 980628*x^16 - 1136960*x^14 + 940576*x^12 - 537472*x^10 + 201552*x^8 - 45696*x^6 + 5440*x^4 - 256*x^2 + 2)
 
gp: K = bnfinit(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 980628*x^16 - 1136960*x^14 + 940576*x^12 - 537472*x^10 + 201552*x^8 - 45696*x^6 + 5440*x^4 - 256*x^2 + 2, 1)
 

Normalized defining polynomial

\( x^{32} - 32 x^{30} + 464 x^{28} - 4032 x^{26} + 23400 x^{24} - 95680 x^{22} + 283360 x^{20} - 615296 x^{18} + 980628 x^{16} - 1136960 x^{14} + 940576 x^{12} - 537472 x^{10} + 201552 x^{8} - 45696 x^{6} + 5440 x^{4} - 256 x^{2} + 2 \)

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:  \(3138550867693340381917894711603833208051177722232017256448=2^{191}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.63$
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:  $2$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(128=2^{7}\)
Dirichlet character group:    $\lbrace$$\chi_{128}(1,·)$, $\chi_{128}(5,·)$, $\chi_{128}(9,·)$, $\chi_{128}(13,·)$, $\chi_{128}(17,·)$, $\chi_{128}(21,·)$, $\chi_{128}(25,·)$, $\chi_{128}(29,·)$, $\chi_{128}(33,·)$, $\chi_{128}(37,·)$, $\chi_{128}(41,·)$, $\chi_{128}(45,·)$, $\chi_{128}(49,·)$, $\chi_{128}(53,·)$, $\chi_{128}(57,·)$, $\chi_{128}(61,·)$, $\chi_{128}(65,·)$, $\chi_{128}(69,·)$, $\chi_{128}(73,·)$, $\chi_{128}(77,·)$, $\chi_{128}(81,·)$, $\chi_{128}(85,·)$, $\chi_{128}(89,·)$, $\chi_{128}(93,·)$, $\chi_{128}(97,·)$, $\chi_{128}(101,·)$, $\chi_{128}(105,·)$, $\chi_{128}(109,·)$, $\chi_{128}(113,·)$, $\chi_{128}(117,·)$, $\chi_{128}(121,·)$, $\chi_{128}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $31$
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:  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:  \( 7254970870890416000 \) (assuming GRH)
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 $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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