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Properties

Label 32.32.1043032430...7248.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 7^{16}$
Root discriminant $165.70$
Ramified primes $2, 7$
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![66465861139202, 0, -1215375746545408, 0, 3689533516298560, 0, -4427440219558272, 0, 2789739117935952, 0, -1062757759213696, 0, 265689439803424, 0, -45880437014720, 0, 5653125275028, 0, -506722713728, 0, 33337020640, 0, -1608093760, 0, 56183400, 0, -1382976, 0, 22736, 0, -224, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 224*x^30 + 22736*x^28 - 1382976*x^26 + 56183400*x^24 - 1608093760*x^22 + 33337020640*x^20 - 506722713728*x^18 + 5653125275028*x^16 - 45880437014720*x^14 + 265689439803424*x^12 - 1062757759213696*x^10 + 2789739117935952*x^8 - 4427440219558272*x^6 + 3689533516298560*x^4 - 1215375746545408*x^2 + 66465861139202)
 
gp: K = bnfinit(x^32 - 224*x^30 + 22736*x^28 - 1382976*x^26 + 56183400*x^24 - 1608093760*x^22 + 33337020640*x^20 - 506722713728*x^18 + 5653125275028*x^16 - 45880437014720*x^14 + 265689439803424*x^12 - 1062757759213696*x^10 + 2789739117935952*x^8 - 4427440219558272*x^6 + 3689533516298560*x^4 - 1215375746545408*x^2 + 66465861139202, 1)
 

Normalized defining polynomial

\( x^{32} - 224 x^{30} + 22736 x^{28} - 1382976 x^{26} + 56183400 x^{24} - 1608093760 x^{22} + 33337020640 x^{20} - 506722713728 x^{18} + 5653125275028 x^{16} - 45880437014720 x^{14} + 265689439803424 x^{12} - 1062757759213696 x^{10} + 2789739117935952 x^{8} - 4427440219558272 x^{6} + 3689533516298560 x^{4} - 1215375746545408 x^{2} + 66465861139202 \)

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:  \(104303243075213755167445035578915122359095224799654955003407693930037248=2^{191}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.70$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(896=2^{7}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{896}(1,·)$, $\chi_{896}(643,·)$, $\chi_{896}(393,·)$, $\chi_{896}(139,·)$, $\chi_{896}(785,·)$, $\chi_{896}(531,·)$, $\chi_{896}(281,·)$, $\chi_{896}(27,·)$, $\chi_{896}(673,·)$, $\chi_{896}(419,·)$, $\chi_{896}(169,·)$, $\chi_{896}(811,·)$, $\chi_{896}(561,·)$, $\chi_{896}(307,·)$, $\chi_{896}(57,·)$, $\chi_{896}(699,·)$, $\chi_{896}(449,·)$, $\chi_{896}(195,·)$, $\chi_{896}(841,·)$, $\chi_{896}(587,·)$, $\chi_{896}(337,·)$, $\chi_{896}(83,·)$, $\chi_{896}(729,·)$, $\chi_{896}(475,·)$, $\chi_{896}(225,·)$, $\chi_{896}(867,·)$, $\chi_{896}(617,·)$, $\chi_{896}(363,·)$, $\chi_{896}(113,·)$, $\chi_{896}(755,·)$, $\chi_{896}(505,·)$, $\chi_{896}(251,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{16807} a^{11}$, $\frac{1}{117649} a^{12}$, $\frac{1}{117649} a^{13}$, $\frac{1}{823543} a^{14}$, $\frac{1}{823543} a^{15}$, $\frac{1}{5764801} a^{16}$, $\frac{1}{5764801} a^{17}$, $\frac{1}{40353607} a^{18}$, $\frac{1}{40353607} a^{19}$, $\frac{1}{282475249} a^{20}$, $\frac{1}{282475249} a^{21}$, $\frac{1}{1977326743} a^{22}$, $\frac{1}{1977326743} a^{23}$, $\frac{1}{13841287201} a^{24}$, $\frac{1}{13841287201} a^{25}$, $\frac{1}{96889010407} a^{26}$, $\frac{1}{96889010407} a^{27}$, $\frac{1}{678223072849} a^{28}$, $\frac{1}{678223072849} a^{29}$, $\frac{1}{4747561509943} a^{30}$, $\frac{1}{4747561509943} a^{31}$

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:  $31$
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:  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 $32$ $32$ 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
7Data not computed