Properties

Label 32.0.62080071703...4288.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 3^{16}\cdot 11^{16}$
Root discriminant $359.77$
Ramified primes $2, 3, 11$
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![3955970402925117755868162, 0, 15344370047709547659124992, 0, 9880844348903875386557760, 0, 2515124016084622825669248, 0, 336166467517804024859472, 0, 27164967072145779786624, 0, 1440566435644094382624, 0, 52768001305644482880, 0, 1379163670488435348, 0, 26222957489976192, 0, 365950363695840, 0, 3744474402240, 0, 27750551400, 0, 144897984, 0, 505296, 0, 1056, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162)
 
gp: K = bnfinit(x^32 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162, 1)
 

Normalized defining polynomial

\( x^{32} + 1056 x^{30} + 505296 x^{28} + 144897984 x^{26} + 27750551400 x^{24} + 3744474402240 x^{22} + 365950363695840 x^{20} + 26222957489976192 x^{18} + 1379163670488435348 x^{16} + 52768001305644482880 x^{14} + 1440566435644094382624 x^{12} + 27164967072145779786624 x^{10} + 336166467517804024859472 x^{8} + 2515124016084622825669248 x^{6} + 9880844348903875386557760 x^{4} + 15344370047709547659124992 x^{2} + 3955970402925117755868162 \)

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:  \(6208007170334900849388551577113100621683540552916899074163477799595412799816204288=2^{191}\cdot 3^{16}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $359.77$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4224=2^{7}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4224}(1,·)$, $\chi_{4224}(131,·)$, $\chi_{4224}(265,·)$, $\chi_{4224}(395,·)$, $\chi_{4224}(529,·)$, $\chi_{4224}(659,·)$, $\chi_{4224}(793,·)$, $\chi_{4224}(923,·)$, $\chi_{4224}(1057,·)$, $\chi_{4224}(1187,·)$, $\chi_{4224}(1321,·)$, $\chi_{4224}(1451,·)$, $\chi_{4224}(1585,·)$, $\chi_{4224}(1715,·)$, $\chi_{4224}(1849,·)$, $\chi_{4224}(1979,·)$, $\chi_{4224}(2113,·)$, $\chi_{4224}(2243,·)$, $\chi_{4224}(2377,·)$, $\chi_{4224}(2507,·)$, $\chi_{4224}(2641,·)$, $\chi_{4224}(2771,·)$, $\chi_{4224}(2905,·)$, $\chi_{4224}(3035,·)$, $\chi_{4224}(3169,·)$, $\chi_{4224}(3299,·)$, $\chi_{4224}(3433,·)$, $\chi_{4224}(3563,·)$, $\chi_{4224}(3697,·)$, $\chi_{4224}(3827,·)$, $\chi_{4224}(3961,·)$, $\chi_{4224}(4091,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{33} a^{2}$, $\frac{1}{33} a^{3}$, $\frac{1}{1089} a^{4}$, $\frac{1}{1089} a^{5}$, $\frac{1}{35937} a^{6}$, $\frac{1}{35937} a^{7}$, $\frac{1}{1185921} a^{8}$, $\frac{1}{1185921} a^{9}$, $\frac{1}{39135393} a^{10}$, $\frac{1}{39135393} a^{11}$, $\frac{1}{1291467969} a^{12}$, $\frac{1}{1291467969} a^{13}$, $\frac{1}{42618442977} a^{14}$, $\frac{1}{42618442977} a^{15}$, $\frac{1}{1406408618241} a^{16}$, $\frac{1}{1406408618241} a^{17}$, $\frac{1}{46411484401953} a^{18}$, $\frac{1}{46411484401953} a^{19}$, $\frac{1}{1531578985264449} a^{20}$, $\frac{1}{1531578985264449} a^{21}$, $\frac{1}{50542106513726817} a^{22}$, $\frac{1}{50542106513726817} a^{23}$, $\frac{1}{1667889514952984961} a^{24}$, $\frac{1}{1667889514952984961} a^{25}$, $\frac{1}{55040353993448503713} a^{26}$, $\frac{1}{55040353993448503713} a^{27}$, $\frac{1}{1816331681783800622529} a^{28}$, $\frac{1}{1816331681783800622529} a^{29}$, $\frac{1}{59938945498865420543457} a^{30}$, $\frac{1}{59938945498865420543457} a^{31}$

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

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
11Data not computed