↑

Properties

Label 28.0.39199758183...3376.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 3^{14}\cdot 29^{27}$
Root discriminant $89.08$
Ramified primes $2, 3, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![138706101, 0, 1618237845, 0, 5609891196, 0, 9082680984, 0, 8325790902, 0, 4793637186, 0, 1843706610, 0, 491655096, 0, 92787849, 0, 12480237, 0, 1188594, 0, 78300, 0, 3393, 0, 87, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 87*x^26 + 3393*x^24 + 78300*x^22 + 1188594*x^20 + 12480237*x^18 + 92787849*x^16 + 491655096*x^14 + 1843706610*x^12 + 4793637186*x^10 + 8325790902*x^8 + 9082680984*x^6 + 5609891196*x^4 + 1618237845*x^2 + 138706101)
 
gp: K = bnfinit(x^28 + 87*x^26 + 3393*x^24 + 78300*x^22 + 1188594*x^20 + 12480237*x^18 + 92787849*x^16 + 491655096*x^14 + 1843706610*x^12 + 4793637186*x^10 + 8325790902*x^8 + 9082680984*x^6 + 5609891196*x^4 + 1618237845*x^2 + 138706101, 1)
 

Normalized defining polynomial

\( x^{28} + 87 x^{26} + 3393 x^{24} + 78300 x^{22} + 1188594 x^{20} + 12480237 x^{18} + 92787849 x^{16} + 491655096 x^{14} + 1843706610 x^{12} + 4793637186 x^{10} + 8325790902 x^{8} + 9082680984 x^{6} + 5609891196 x^{4} + 1618237845 x^{2} + 138706101 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3919975818323568890648031846194353959005956807512293376=2^{28}\cdot 3^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.08$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(131,·)$, $\chi_{348}(325,·)$, $\chi_{348}(263,·)$, $\chi_{348}(265,·)$, $\chi_{348}(11,·)$, $\chi_{348}(13,·)$, $\chi_{348}(143,·)$, $\chi_{348}(275,·)$, $\chi_{348}(121,·)$, $\chi_{348}(277,·)$, $\chi_{348}(215,·)$, $\chi_{348}(25,·)$, $\chi_{348}(155,·)$, $\chi_{348}(287,·)$, $\chi_{348}(289,·)$, $\chi_{348}(251,·)$, $\chi_{348}(241,·)$, $\chi_{348}(119,·)$, $\chi_{348}(169,·)$, $\chi_{348}(109,·)$, $\chi_{348}(47,·)$, $\chi_{348}(49,·)$, $\chi_{348}(181,·)$, $\chi_{348}(311,·)$, $\chi_{348}(313,·)$, $\chi_{348}(95,·)$, $\chi_{348}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$

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:  $13$
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

$C_{28}$ (as 28T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.3512016.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
29Data not computed