Properties

Label 32.0.78541128710...5408.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 29^{16}$
Root discriminant $337.27$
Ramified primes $2, 29$
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![500492947360694697575042, 0, 2209072319385135216882944, 0, 1618716785756349081336640, 0, 468869689667356285628544, 0, 71312077184628696151632, 0, 6557432384793443324288, 0, 395707126668569855776, 0, 16494044688724018240, 0, 490555639449119508, 0, 10613779893422464, 0, 168549136238560, 0, 1962506736320, 0, 16550375400, 0, 98336448, 0, 390224, 0, 928, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 928*x^30 + 390224*x^28 + 98336448*x^26 + 16550375400*x^24 + 1962506736320*x^22 + 168549136238560*x^20 + 10613779893422464*x^18 + 490555639449119508*x^16 + 16494044688724018240*x^14 + 395707126668569855776*x^12 + 6557432384793443324288*x^10 + 71312077184628696151632*x^8 + 468869689667356285628544*x^6 + 1618716785756349081336640*x^4 + 2209072319385135216882944*x^2 + 500492947360694697575042)
 
gp: K = bnfinit(x^32 + 928*x^30 + 390224*x^28 + 98336448*x^26 + 16550375400*x^24 + 1962506736320*x^22 + 168549136238560*x^20 + 10613779893422464*x^18 + 490555639449119508*x^16 + 16494044688724018240*x^14 + 395707126668569855776*x^12 + 6557432384793443324288*x^10 + 71312077184628696151632*x^8 + 468869689667356285628544*x^6 + 1618716785756349081336640*x^4 + 2209072319385135216882944*x^2 + 500492947360694697575042, 1)
 

Normalized defining polynomial

\( x^{32} + 928 x^{30} + 390224 x^{28} + 98336448 x^{26} + 16550375400 x^{24} + 1962506736320 x^{22} + 168549136238560 x^{20} + 10613779893422464 x^{18} + 490555639449119508 x^{16} + 16494044688724018240 x^{14} + 395707126668569855776 x^{12} + 6557432384793443324288 x^{10} + 71312077184628696151632 x^{8} + 468869689667356285628544 x^{6} + 1618716785756349081336640 x^{4} + 2209072319385135216882944 x^{2} + 500492947360694697575042 \)

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:  \(785411287106652838033363851140085365286337539393649349536153620008513161719185408=2^{191}\cdot 29^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $337.27$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3712=2^{7}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{3712}(1,·)$, $\chi_{3712}(2435,·)$, $\chi_{3712}(1161,·)$, $\chi_{3712}(3595,·)$, $\chi_{3712}(2321,·)$, $\chi_{3712}(1043,·)$, $\chi_{3712}(3481,·)$, $\chi_{3712}(2203,·)$, $\chi_{3712}(929,·)$, $\chi_{3712}(3363,·)$, $\chi_{3712}(2089,·)$, $\chi_{3712}(811,·)$, $\chi_{3712}(3249,·)$, $\chi_{3712}(1971,·)$, $\chi_{3712}(697,·)$, $\chi_{3712}(3131,·)$, $\chi_{3712}(1857,·)$, $\chi_{3712}(579,·)$, $\chi_{3712}(3017,·)$, $\chi_{3712}(1739,·)$, $\chi_{3712}(465,·)$, $\chi_{3712}(2899,·)$, $\chi_{3712}(1625,·)$, $\chi_{3712}(347,·)$, $\chi_{3712}(2785,·)$, $\chi_{3712}(1507,·)$, $\chi_{3712}(233,·)$, $\chi_{3712}(2667,·)$, $\chi_{3712}(1393,·)$, $\chi_{3712}(115,·)$, $\chi_{3712}(2553,·)$, $\chi_{3712}(1275,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{29} a^{2}$, $\frac{1}{29} a^{3}$, $\frac{1}{841} a^{4}$, $\frac{1}{841} a^{5}$, $\frac{1}{24389} a^{6}$, $\frac{1}{24389} a^{7}$, $\frac{1}{707281} a^{8}$, $\frac{1}{707281} a^{9}$, $\frac{1}{20511149} a^{10}$, $\frac{1}{20511149} a^{11}$, $\frac{1}{594823321} a^{12}$, $\frac{1}{594823321} a^{13}$, $\frac{1}{17249876309} a^{14}$, $\frac{1}{17249876309} a^{15}$, $\frac{1}{500246412961} a^{16}$, $\frac{1}{500246412961} a^{17}$, $\frac{1}{14507145975869} a^{18}$, $\frac{1}{14507145975869} a^{19}$, $\frac{1}{420707233300201} a^{20}$, $\frac{1}{420707233300201} a^{21}$, $\frac{1}{12200509765705829} a^{22}$, $\frac{1}{12200509765705829} a^{23}$, $\frac{1}{353814783205469041} a^{24}$, $\frac{1}{353814783205469041} a^{25}$, $\frac{1}{10260628712958602189} a^{26}$, $\frac{1}{10260628712958602189} a^{27}$, $\frac{1}{297558232675799463481} a^{28}$, $\frac{1}{297558232675799463481} a^{29}$, $\frac{1}{8629188747598184440949} a^{30}$, $\frac{1}{8629188747598184440949} a^{31}$

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

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
29Data not computed