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Properties

Label 28.0.75862596433...7856.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{56}\cdot 29^{26}$
Root discriminant $91.20$
Ramified primes $2, 29$
Class number $38528$ (GRH)
Class group $[2, 4, 4, 1204]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![841, 0, 0, 0, 51301, 0, 0, 0, 819134, 0, 0, 0, 1428018, 0, 0, 0, 333123, 0, 0, 0, 16298, 0, 0, 0, 261, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 261*x^24 + 16298*x^20 + 333123*x^16 + 1428018*x^12 + 819134*x^8 + 51301*x^4 + 841)
 
gp: K = bnfinit(x^28 + 261*x^24 + 16298*x^20 + 333123*x^16 + 1428018*x^12 + 819134*x^8 + 51301*x^4 + 841, 1)
 

Normalized defining polynomial

\( x^{28} + 261 x^{24} + 16298 x^{20} + 333123 x^{16} + 1428018 x^{12} + 819134 x^{8} + 51301 x^{4} + 841 \)

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:  \(7586259643335085676646037106440640773336778620436217856=2^{56}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.20$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(67,·)$, $\chi_{232}(5,·)$, $\chi_{232}(51,·)$, $\chi_{232}(7,·)$, $\chi_{232}(111,·)$, $\chi_{232}(13,·)$, $\chi_{232}(109,·)$, $\chi_{232}(115,·)$, $\chi_{232}(81,·)$, $\chi_{232}(149,·)$, $\chi_{232}(23,·)$, $\chi_{232}(25,·)$, $\chi_{232}(91,·)$, $\chi_{232}(93,·)$, $\chi_{232}(223,·)$, $\chi_{232}(161,·)$, $\chi_{232}(35,·)$, $\chi_{232}(65,·)$, $\chi_{232}(103,·)$, $\chi_{232}(169,·)$, $\chi_{232}(199,·)$, $\chi_{232}(173,·)$, $\chi_{232}(175,·)$, $\chi_{232}(49,·)$, $\chi_{232}(179,·)$, $\chi_{232}(187,·)$, $\chi_{232}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14}$, $\frac{1}{29} a^{15}$, $\frac{1}{493} a^{16} - \frac{7}{17}$, $\frac{1}{493} a^{17} - \frac{7}{17} a$, $\frac{1}{493} a^{18} - \frac{7}{17} a^{2}$, $\frac{1}{493} a^{19} - \frac{7}{17} a^{3}$, $\frac{1}{343621} a^{20} - \frac{202}{343621} a^{16} - \frac{183}{697} a^{12} - \frac{319}{697} a^{8} + \frac{1302}{11849} a^{4} + \frac{4015}{11849}$, $\frac{1}{343621} a^{21} - \frac{202}{343621} a^{17} - \frac{183}{697} a^{13} - \frac{319}{697} a^{9} + \frac{1302}{11849} a^{5} + \frac{4015}{11849} a$, $\frac{1}{343621} a^{22} - \frac{202}{343621} a^{18} + \frac{269}{20213} a^{14} - \frac{319}{697} a^{10} + \frac{1302}{11849} a^{6} + \frac{4015}{11849} a^{2}$, $\frac{1}{343621} a^{23} - \frac{202}{343621} a^{19} + \frac{269}{20213} a^{15} - \frac{319}{697} a^{11} + \frac{1302}{11849} a^{7} + \frac{4015}{11849} a^{3}$, $\frac{1}{23194036424311} a^{24} - \frac{2766032}{23194036424311} a^{20} + \frac{250687478}{23194036424311} a^{16} + \frac{21754533101}{47046727027} a^{12} - \frac{219037445122}{799794359459} a^{8} + \frac{177921994957}{799794359459} a^{4} + \frac{392035722031}{799794359459}$, $\frac{1}{23194036424311} a^{25} - \frac{2766032}{23194036424311} a^{21} + \frac{250687478}{23194036424311} a^{17} + \frac{21754533101}{47046727027} a^{13} - \frac{219037445122}{799794359459} a^{9} + \frac{177921994957}{799794359459} a^{5} + \frac{392035722031}{799794359459} a$, $\frac{1}{23194036424311} a^{26} - \frac{2766032}{23194036424311} a^{22} + \frac{250687478}{23194036424311} a^{18} + \frac{19274008578}{1364355083783} a^{14} - \frac{219037445122}{799794359459} a^{10} + \frac{177921994957}{799794359459} a^{6} + \frac{392035722031}{799794359459} a^{2}$, $\frac{1}{23194036424311} a^{27} - \frac{2766032}{23194036424311} a^{23} + \frac{250687478}{23194036424311} a^{19} + \frac{19274008578}{1364355083783} a^{15} - \frac{219037445122}{799794359459} a^{11} + \frac{177921994957}{799794359459} a^{7} + \frac{392035722031}{799794359459} a^{3}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{1204}$, which has order $38528$ (assuming GRH)

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:  \( -\frac{41243647}{393119261429} a^{26} - \frac{10765496237}{393119261429} a^{22} - \frac{672425181863}{393119261429} a^{18} - \frac{809058513268}{23124662437} a^{14} - \frac{2041413081834}{13555836601} a^{10} - \frac{1211760837578}{13555836601} a^{6} - \frac{112760781878}{13555836601} a^{2} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 306576699955.91235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{58}) \), \(\Q(\sqrt{-58}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{58})\), 7.7.594823321.1, 14.14.21518098026638558497865728.1, 14.0.21518098026638558497865728.1, 14.0.5796901408038404767744.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$