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Properties

Label 32.0.22830544699...5488.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 31^{16}$
Root discriminant $348.70$
Ramified primes $2, 31$
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![1454846243494370527656962, 0, 6007107069912239598067456, 0, 4117775007601131982546240, 0, 1115784195608048666238336, 0, 158755009398149781429072, 0, 13656344894464497327232, 0, 770922695655253881376, 0, 30060792811796500160, 0, 836368832263692948, 0, 16928401412041856, 0, 251483043048160, 0, 2739237167680, 0, 21610391400, 0, 120117312, 0, 445904, 0, 992, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 992*x^30 + 445904*x^28 + 120117312*x^26 + 21610391400*x^24 + 2739237167680*x^22 + 251483043048160*x^20 + 16928401412041856*x^18 + 836368832263692948*x^16 + 30060792811796500160*x^14 + 770922695655253881376*x^12 + 13656344894464497327232*x^10 + 158755009398149781429072*x^8 + 1115784195608048666238336*x^6 + 4117775007601131982546240*x^4 + 6007107069912239598067456*x^2 + 1454846243494370527656962)
 
gp: K = bnfinit(x^32 + 992*x^30 + 445904*x^28 + 120117312*x^26 + 21610391400*x^24 + 2739237167680*x^22 + 251483043048160*x^20 + 16928401412041856*x^18 + 836368832263692948*x^16 + 30060792811796500160*x^14 + 770922695655253881376*x^12 + 13656344894464497327232*x^10 + 158755009398149781429072*x^8 + 1115784195608048666238336*x^6 + 4117775007601131982546240*x^4 + 6007107069912239598067456*x^2 + 1454846243494370527656962, 1)
 

Normalized defining polynomial

\( x^{32} + 992 x^{30} + 445904 x^{28} + 120117312 x^{26} + 21610391400 x^{24} + 2739237167680 x^{22} + 251483043048160 x^{20} + 16928401412041856 x^{18} + 836368832263692948 x^{16} + 30060792811796500160 x^{14} + 770922695655253881376 x^{12} + 13656344894464497327232 x^{10} + 158755009398149781429072 x^{8} + 1115784195608048666238336 x^{6} + 4117775007601131982546240 x^{4} + 6007107069912239598067456 x^{2} + 1454846243494370527656962 \)

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:  \(2283054469939826689646603434893544205211768622361793362696380065480141214463295488=2^{191}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $348.70$
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:  $2, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3968=2^{7}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3968}(1,·)$, $\chi_{3968}(1797,·)$, $\chi_{3968}(3721,·)$, $\chi_{3968}(1549,·)$, $\chi_{3968}(3473,·)$, $\chi_{3968}(1301,·)$, $\chi_{3968}(3225,·)$, $\chi_{3968}(1053,·)$, $\chi_{3968}(2977,·)$, $\chi_{3968}(805,·)$, $\chi_{3968}(2729,·)$, $\chi_{3968}(557,·)$, $\chi_{3968}(2481,·)$, $\chi_{3968}(309,·)$, $\chi_{3968}(2233,·)$, $\chi_{3968}(61,·)$, $\chi_{3968}(1985,·)$, $\chi_{3968}(3781,·)$, $\chi_{3968}(1737,·)$, $\chi_{3968}(3533,·)$, $\chi_{3968}(1489,·)$, $\chi_{3968}(3285,·)$, $\chi_{3968}(1241,·)$, $\chi_{3968}(3037,·)$, $\chi_{3968}(993,·)$, $\chi_{3968}(2789,·)$, $\chi_{3968}(745,·)$, $\chi_{3968}(2541,·)$, $\chi_{3968}(497,·)$, $\chi_{3968}(2293,·)$, $\chi_{3968}(249,·)$, $\chi_{3968}(2045,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{31} a^{2}$, $\frac{1}{31} a^{3}$, $\frac{1}{961} a^{4}$, $\frac{1}{961} a^{5}$, $\frac{1}{29791} a^{6}$, $\frac{1}{29791} a^{7}$, $\frac{1}{923521} a^{8}$, $\frac{1}{923521} a^{9}$, $\frac{1}{28629151} a^{10}$, $\frac{1}{28629151} a^{11}$, $\frac{1}{887503681} a^{12}$, $\frac{1}{887503681} a^{13}$, $\frac{1}{27512614111} a^{14}$, $\frac{1}{27512614111} a^{15}$, $\frac{1}{852891037441} a^{16}$, $\frac{1}{852891037441} a^{17}$, $\frac{1}{26439622160671} a^{18}$, $\frac{1}{26439622160671} a^{19}$, $\frac{1}{819628286980801} a^{20}$, $\frac{1}{819628286980801} a^{21}$, $\frac{1}{25408476896404831} a^{22}$, $\frac{1}{25408476896404831} a^{23}$, $\frac{1}{787662783788549761} a^{24}$, $\frac{1}{787662783788549761} a^{25}$, $\frac{1}{24417546297445042591} a^{26}$, $\frac{1}{24417546297445042591} a^{27}$, $\frac{1}{756943935220796320321} a^{28}$, $\frac{1}{756943935220796320321} a^{29}$, $\frac{1}{23465261991844685929951} a^{30}$, $\frac{1}{23465261991844685929951} a^{31}$

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

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
$31$31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$