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Properties

Label 28.0.902...416.1
Degree $28$
Signature $[0, 14]$
Discriminant $9.021\times 10^{51}$
Root discriminant $71.70$
Ramified primes $2, 29$
Class number $19264$ (GRH)
Class group $[4, 4, 1204]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1)
 
gp: K = bnfinit(x^28 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 1437, 0, 0, 0, 345566, 0, 0, 0, 395946, 0, 0, 0, 111451, 0, 0, 0, 9834, 0, 0, 0, 197, 0, 0, 0, 1]);
 

\( x^{28} + 197 x^{24} + 9834 x^{20} + 111451 x^{16} + 395946 x^{12} + 345566 x^{8} + 1437 x^{4} + 1 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(9020522762586308771279473372699929575905800975548416\)\(\medspace = 2^{56}\cdot 29^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $71.70$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(107,·)$, $\chi_{232}(197,·)$, $\chi_{232}(7,·)$, $\chi_{232}(139,·)$, $\chi_{232}(141,·)$, $\chi_{232}(81,·)$, $\chi_{232}(83,·)$, $\chi_{232}(23,·)$, $\chi_{232}(25,·)$, $\chi_{232}(219,·)$, $\chi_{232}(223,·)$, $\chi_{232}(161,·)$, $\chi_{232}(227,·)$, $\chi_{232}(165,·)$, $\chi_{232}(65,·)$, $\chi_{232}(103,·)$, $\chi_{232}(169,·)$, $\chi_{232}(199,·)$, $\chi_{232}(45,·)$, $\chi_{232}(175,·)$, $\chi_{232}(49,·)$, $\chi_{232}(123,·)$, $\chi_{232}(53,·)$, $\chi_{232}(111,·)$, $\chi_{232}(59,·)$, $\chi_{232}(117,·)$, $\chi_{232}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16} - \frac{1}{17}$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} - \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} - \frac{1}{17} a^{7}$, $\frac{1}{61692893418446977} a^{24} - \frac{555760637084211}{61692893418446977} a^{20} - \frac{1309021780647847}{61692893418446977} a^{16} + \frac{1112648792681163}{3628993730496881} a^{12} - \frac{26589209768599822}{61692893418446977} a^{8} + \frac{14483265078261197}{61692893418446977} a^{4} - \frac{21699345601541164}{61692893418446977}$, $\frac{1}{61692893418446977} a^{25} - \frac{555760637084211}{61692893418446977} a^{21} - \frac{1309021780647847}{61692893418446977} a^{17} + \frac{1112648792681163}{3628993730496881} a^{13} - \frac{26589209768599822}{61692893418446977} a^{9} + \frac{14483265078261197}{61692893418446977} a^{5} - \frac{21699345601541164}{61692893418446977} a$, $\frac{1}{61692893418446977} a^{26} - \frac{555760637084211}{61692893418446977} a^{22} - \frac{1309021780647847}{61692893418446977} a^{18} + \frac{1112648792681163}{3628993730496881} a^{14} - \frac{26589209768599822}{61692893418446977} a^{10} + \frac{14483265078261197}{61692893418446977} a^{6} - \frac{21699345601541164}{61692893418446977} a^{2}$, $\frac{1}{61692893418446977} a^{27} - \frac{555760637084211}{61692893418446977} a^{23} - \frac{1309021780647847}{61692893418446977} a^{19} + \frac{1112648792681163}{3628993730496881} a^{15} - \frac{26589209768599822}{61692893418446977} a^{11} + \frac{14483265078261197}{61692893418446977} a^{7} - \frac{21699345601541164}{61692893418446977} a^{3}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{1204}$, which has order $19264$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{120147492678582}{61692893418446977} a^{25} + \frac{23669843847044992}{61692893418446977} a^{21} + \frac{1181686024924130293}{61692893418446977} a^{17} + \frac{788139880767185850}{3628993730496881} a^{13} + \frac{47662821025571406002}{61692893418446977} a^{9} + \frac{41841580176291853913}{61692893418446977} a^{5} + \frac{394774270161398293}{61692893418446977} a \) (order $8$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 297452739458.56445 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 297452739458.56445 \cdot 19264}{8\sqrt{9020522762586308771279473372699929575905800975548416}}\approx 1.12713713712242$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 7.7.594823321.1, 14.14.742003380228915810271232.1, 14.0.742003380228915810271232.1, 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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$