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Properties

Label 28.0.160...384.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.607\times 10^{50}$
Root discriminant $62.10$
Ramified primes $2, 3, 29$
Class number $448$ (GRH)
Class group $[4, 4, 28]$ (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 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 1)
 
gp: K = bnfinit(x^28 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -53, 0, 2123, 0, -33970, 0, 406547, 0, -737996, 0, 888182, 0, -623516, 0, 315729, 0, -105124, 0, 25427, 0, -3816, 0, 411, 0, -25, 0, 1]);
 

\( x^{28} - 25 x^{26} + 411 x^{24} - 3816 x^{22} + 25427 x^{20} - 105124 x^{18} + 315729 x^{16} - 623516 x^{14} + 888182 x^{12} - 737996 x^{10} + 406547 x^{8} - 33970 x^{6} + 2123 x^{4} - 53 x^{2} + 1 \)

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:  \(160727205638753900965518547139872645824181262352384\)\(\medspace = 2^{28}\cdot 3^{14}\cdot 29^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $62.10$
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:  $2, 3, 29$
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:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(343,·)$, $\chi_{348}(107,·)$, $\chi_{348}(197,·)$, $\chi_{348}(7,·)$, $\chi_{348}(139,·)$, $\chi_{348}(335,·)$, $\chi_{348}(83,·)$, $\chi_{348}(277,·)$, $\chi_{348}(23,·)$, $\chi_{348}(25,·)$, $\chi_{348}(239,·)$, $\chi_{348}(223,·)$, $\chi_{348}(161,·)$, $\chi_{348}(227,·)$, $\chi_{348}(65,·)$, $\chi_{348}(103,·)$, $\chi_{348}(169,·)$, $\chi_{348}(199,·)$, $\chi_{348}(257,·)$, $\chi_{348}(175,·)$, $\chi_{348}(49,·)$, $\chi_{348}(53,·)$, $\chi_{348}(233,·)$, $\chi_{348}(313,·)$, $\chi_{348}(59,·)$, $\chi_{348}(281,·)$, $\chi_{348}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{17} a^{20} - \frac{7}{17} a^{18} + \frac{8}{17} a^{16} - \frac{3}{17} a^{14} - \frac{2}{17} a^{12} - \frac{5}{17} a^{10} - \frac{4}{17} a^{8} + \frac{2}{17} a^{6} + \frac{6}{17} a^{4} - \frac{2}{17} a^{2} - \frac{1}{17}$, $\frac{1}{17} a^{21} - \frac{7}{17} a^{19} + \frac{8}{17} a^{17} - \frac{3}{17} a^{15} - \frac{2}{17} a^{13} - \frac{5}{17} a^{11} - \frac{4}{17} a^{9} + \frac{2}{17} a^{7} + \frac{6}{17} a^{5} - \frac{2}{17} a^{3} - \frac{1}{17} a$, $\frac{1}{17} a^{22} - \frac{7}{17} a^{18} + \frac{2}{17} a^{16} - \frac{6}{17} a^{14} - \frac{2}{17} a^{12} - \frac{5}{17} a^{10} + \frac{8}{17} a^{8} + \frac{3}{17} a^{6} + \frac{6}{17} a^{4} + \frac{2}{17} a^{2} - \frac{7}{17}$, $\frac{1}{17} a^{23} - \frac{7}{17} a^{19} + \frac{2}{17} a^{17} - \frac{6}{17} a^{15} - \frac{2}{17} a^{13} - \frac{5}{17} a^{11} + \frac{8}{17} a^{9} + \frac{3}{17} a^{7} + \frac{6}{17} a^{5} + \frac{2}{17} a^{3} - \frac{7}{17} a$, $\frac{1}{11849} a^{24} + \frac{52}{11849} a^{22} - \frac{120}{11849} a^{20} - \frac{47}{11849} a^{18} - \frac{2744}{11849} a^{16} + \frac{3374}{11849} a^{14} - \frac{2484}{11849} a^{12} - \frac{1234}{11849} a^{10} - \frac{4178}{11849} a^{8} - \frac{1679}{11849} a^{6} + \frac{5416}{11849} a^{4} - \frac{44}{697} a^{2} + \frac{5189}{11849}$, $\frac{1}{11849} a^{25} + \frac{52}{11849} a^{23} - \frac{120}{11849} a^{21} - \frac{47}{11849} a^{19} - \frac{2744}{11849} a^{17} + \frac{3374}{11849} a^{15} - \frac{2484}{11849} a^{13} - \frac{1234}{11849} a^{11} - \frac{4178}{11849} a^{9} - \frac{1679}{11849} a^{7} + \frac{5416}{11849} a^{5} - \frac{44}{697} a^{3} + \frac{5189}{11849} a$, $\frac{1}{8488749944424829218280470588419} a^{26} - \frac{77506411692536330767487256}{8488749944424829218280470588419} a^{24} - \frac{170856184520780757665841945443}{8488749944424829218280470588419} a^{22} + \frac{169875133373943106298216586181}{8488749944424829218280470588419} a^{20} - \frac{356609968711475728568365004556}{8488749944424829218280470588419} a^{18} + \frac{1887021880545886406710689965557}{8488749944424829218280470588419} a^{16} - \frac{16001948078436466468522667695}{8488749944424829218280470588419} a^{14} + \frac{1560707166805695903226476889338}{8488749944424829218280470588419} a^{12} - \frac{5316256687436986737907093723}{499338232024989954016498269907} a^{10} + \frac{565147394519281087701464619927}{8488749944424829218280470588419} a^{8} - \frac{2889395042677315015115962350215}{8488749944424829218280470588419} a^{6} - \frac{3699134021812782009042680941372}{8488749944424829218280470588419} a^{4} + \frac{2682151932112203912448451823380}{8488749944424829218280470588419} a^{2} - \frac{3647559283339362376732335834482}{8488749944424829218280470588419}$, $\frac{1}{8488749944424829218280470588419} a^{27} - \frac{77506411692536330767487256}{8488749944424829218280470588419} a^{25} - \frac{170856184520780757665841945443}{8488749944424829218280470588419} a^{23} + \frac{169875133373943106298216586181}{8488749944424829218280470588419} a^{21} - \frac{356609968711475728568365004556}{8488749944424829218280470588419} a^{19} + \frac{1887021880545886406710689965557}{8488749944424829218280470588419} a^{17} - \frac{16001948078436466468522667695}{8488749944424829218280470588419} a^{15} + \frac{1560707166805695903226476889338}{8488749944424829218280470588419} a^{13} - \frac{5316256687436986737907093723}{499338232024989954016498269907} a^{11} + \frac{565147394519281087701464619927}{8488749944424829218280470588419} a^{9} - \frac{2889395042677315015115962350215}{8488749944424829218280470588419} a^{7} - \frac{3699134021812782009042680941372}{8488749944424829218280470588419} a^{5} + \frac{2682151932112203912448451823380}{8488749944424829218280470588419} a^{3} - \frac{3647559283339362376732335834482}{8488749944424829218280470588419} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{28}$, which has order $448$ (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:  \( -\frac{288590093062887570060361338540}{8488749944424829218280470588419} a^{27} + \frac{7192302348233041285334129907620}{8488749944424829218280470588419} a^{25} - \frac{118049944760298021357460582714080}{8488749944424829218280470588419} a^{23} + \frac{1092049354853136431580153280231920}{8488749944424829218280470588419} a^{21} - \frac{7252581220191819876020875790151593}{8488749944424829218280470588419} a^{19} + \frac{29769390625439443012468847663524410}{8488749944424829218280470588419} a^{17} - \frac{88772612343268179347793001574123310}{8488749944424829218280470588419} a^{15} + \frac{172918802109188044818442110393529645}{8488749944424829218280470588419} a^{13} - \frac{242518112588426636198495167467734970}{8488749944424829218280470588419} a^{11} + \frac{193410159954734900696486829247578240}{8488749944424829218280470588419} a^{9} - \frac{2469852703732143960323866927526123}{207042681571337298006840746059} a^{7} + \frac{1049219412973485518423970884632950}{8488749944424829218280470588419} a^{5} - \frac{26295418437133815605295921542250}{8488749944424829218280470588419} a^{3} - \frac{50965411896753748653089392725391}{8488749944424829218280470588419} a \) (order $12$)
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:  \( 266164993237.31995 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 266164993237.31995 \cdot 448}{12\sqrt{160727205638753900965518547139872645824181262352384}}\approx 0.117144300415993$ (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 7.7.594823321.1, 14.14.12677823379379991227056128.1, 14.0.773792930870360792667.1, 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 R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ 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.

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
3Data 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}$