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Properties

Label 28.0.55585098816...2896.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 7^{14}\cdot 29^{27}$
Root discriminant $136.07$
Ramified primes $2, 7, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19668469112621, 0, 98342345563105, 0, 146108627693756, 0, 101381496767096, 0, 39828445158502, 0, 9827798155994, 0, 1619966729010, 0, 185139054744, 0, 14974482369, 0, 863190713, 0, 35232274, 0, 994700, 0, 18473, 0, 203, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 203*x^26 + 18473*x^24 + 994700*x^22 + 35232274*x^20 + 863190713*x^18 + 14974482369*x^16 + 185139054744*x^14 + 1619966729010*x^12 + 9827798155994*x^10 + 39828445158502*x^8 + 101381496767096*x^6 + 146108627693756*x^4 + 98342345563105*x^2 + 19668469112621)
 
gp: K = bnfinit(x^28 + 203*x^26 + 18473*x^24 + 994700*x^22 + 35232274*x^20 + 863190713*x^18 + 14974482369*x^16 + 185139054744*x^14 + 1619966729010*x^12 + 9827798155994*x^10 + 39828445158502*x^8 + 101381496767096*x^6 + 146108627693756*x^4 + 98342345563105*x^2 + 19668469112621, 1)
 

Normalized defining polynomial

\( x^{28} + 203 x^{26} + 18473 x^{24} + 994700 x^{22} + 35232274 x^{20} + 863190713 x^{18} + 14974482369 x^{16} + 185139054744 x^{14} + 1619966729010 x^{12} + 9827798155994 x^{10} + 39828445158502 x^{8} + 101381496767096 x^{6} + 146108627693756 x^{4} + 98342345563105 x^{2} + 19668469112621 \)

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:  \(555850988161784919111048015916462199222154398969836020432896=2^{28}\cdot 7^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.07$
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:  $2, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(812=2^{2}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{812}(1,·)$, $\chi_{812}(195,·)$, $\chi_{812}(645,·)$, $\chi_{812}(391,·)$, $\chi_{812}(363,·)$, $\chi_{812}(393,·)$, $\chi_{812}(729,·)$, $\chi_{812}(727,·)$, $\chi_{812}(141,·)$, $\chi_{812}(589,·)$, $\chi_{812}(251,·)$, $\chi_{812}(279,·)$, $\chi_{812}(281,·)$, $\chi_{812}(27,·)$, $\chi_{812}(477,·)$, $\chi_{812}(197,·)$, $\chi_{812}(673,·)$, $\chi_{812}(475,·)$, $\chi_{812}(503,·)$, $\chi_{812}(169,·)$, $\chi_{812}(225,·)$, $\chi_{812}(559,·)$, $\chi_{812}(307,·)$, $\chi_{812}(55,·)$, $\chi_{812}(57,·)$, $\chi_{812}(699,·)$, $\chi_{812}(701,·)$, $\chi_{812}(447,·)$$\rbrace$
This is a CM field.

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

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

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

\(\Q(\sqrt{29}) \), 4.0.19120976.1, 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 $28$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ R $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed