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Properties

Label 32.0.20615283744...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $242.56$
Ramified primes $2, 3, 5$
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![13136816711425781250, 0, 112100835937500000000, 0, 158809517578125000000, 0, 88933329843750000000, 0, 26150633894531250000, 0, 4649001581250000000, 0, 542383517812500000, 0, 43708561875000000, 0, 2513242307812500, 0, 105129090000000, 0, 3227647500000, 0, 72657000000, 0, 1184625000, 0, 13608000, 0, 104400, 0, 480, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 480*x^30 + 104400*x^28 + 13608000*x^26 + 1184625000*x^24 + 72657000000*x^22 + 3227647500000*x^20 + 105129090000000*x^18 + 2513242307812500*x^16 + 43708561875000000*x^14 + 542383517812500000*x^12 + 4649001581250000000*x^10 + 26150633894531250000*x^8 + 88933329843750000000*x^6 + 158809517578125000000*x^4 + 112100835937500000000*x^2 + 13136816711425781250)
 
gp: K = bnfinit(x^32 + 480*x^30 + 104400*x^28 + 13608000*x^26 + 1184625000*x^24 + 72657000000*x^22 + 3227647500000*x^20 + 105129090000000*x^18 + 2513242307812500*x^16 + 43708561875000000*x^14 + 542383517812500000*x^12 + 4649001581250000000*x^10 + 26150633894531250000*x^8 + 88933329843750000000*x^6 + 158809517578125000000*x^4 + 112100835937500000000*x^2 + 13136816711425781250, 1)
 

Normalized defining polynomial

\( x^{32} + 480 x^{30} + 104400 x^{28} + 13608000 x^{26} + 1184625000 x^{24} + 72657000000 x^{22} + 3227647500000 x^{20} + 105129090000000 x^{18} + 2513242307812500 x^{16} + 43708561875000000 x^{14} + 542383517812500000 x^{12} + 4649001581250000000 x^{10} + 26150633894531250000 x^{8} + 88933329843750000000 x^{6} + 158809517578125000000 x^{4} + 112100835937500000000 x^{2} + 13136816711425781250 \)

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:  \(20615283744186880032112588280912120153429260426382010514145280000000000000000=2^{191}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.56$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1920=2^{7}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{1920}(1,·)$, $\chi_{1920}(389,·)$, $\chi_{1920}(1801,·)$, $\chi_{1920}(269,·)$, $\chi_{1920}(1681,·)$, $\chi_{1920}(149,·)$, $\chi_{1920}(1561,·)$, $\chi_{1920}(29,·)$, $\chi_{1920}(1441,·)$, $\chi_{1920}(1829,·)$, $\chi_{1920}(1321,·)$, $\chi_{1920}(1709,·)$, $\chi_{1920}(1201,·)$, $\chi_{1920}(1589,·)$, $\chi_{1920}(1081,·)$, $\chi_{1920}(1469,·)$, $\chi_{1920}(961,·)$, $\chi_{1920}(1349,·)$, $\chi_{1920}(841,·)$, $\chi_{1920}(1229,·)$, $\chi_{1920}(721,·)$, $\chi_{1920}(1109,·)$, $\chi_{1920}(601,·)$, $\chi_{1920}(989,·)$, $\chi_{1920}(481,·)$, $\chi_{1920}(869,·)$, $\chi_{1920}(361,·)$, $\chi_{1920}(749,·)$, $\chi_{1920}(241,·)$, $\chi_{1920}(629,·)$, $\chi_{1920}(121,·)$, $\chi_{1920}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{15} a^{2}$, $\frac{1}{15} a^{3}$, $\frac{1}{225} a^{4}$, $\frac{1}{225} a^{5}$, $\frac{1}{3375} a^{6}$, $\frac{1}{3375} a^{7}$, $\frac{1}{50625} a^{8}$, $\frac{1}{50625} a^{9}$, $\frac{1}{759375} a^{10}$, $\frac{1}{759375} a^{11}$, $\frac{1}{11390625} a^{12}$, $\frac{1}{11390625} a^{13}$, $\frac{1}{170859375} a^{14}$, $\frac{1}{170859375} a^{15}$, $\frac{1}{2562890625} a^{16}$, $\frac{1}{2562890625} a^{17}$, $\frac{1}{38443359375} a^{18}$, $\frac{1}{38443359375} a^{19}$, $\frac{1}{576650390625} a^{20}$, $\frac{1}{576650390625} a^{21}$, $\frac{1}{8649755859375} a^{22}$, $\frac{1}{8649755859375} a^{23}$, $\frac{1}{129746337890625} a^{24}$, $\frac{1}{129746337890625} a^{25}$, $\frac{1}{1946195068359375} a^{26}$, $\frac{1}{1946195068359375} a^{27}$, $\frac{1}{29192926025390625} a^{28}$, $\frac{1}{29192926025390625} a^{29}$, $\frac{1}{437893890380859375} a^{30}$, $\frac{1}{437893890380859375} a^{31}$

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

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
3Data not computed
5Data not computed