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Properties

Label 28.0.116...000.8
Degree $28$
Signature $[0, 14]$
Discriminant $1.170\times 10^{68}$
Root discriminant \(269.78\)
Ramified primes $2,5,7,29$
Class number not computed
Class group not computed
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241)
 
Copy content gp:K = bnfinit(y^28 - 21*y^26 + 1343*y^24 - 59011*y^22 + 2332758*y^20 - 67729146*y^18 + 1575869078*y^16 - 28987252103*y^14 + 430595937122*y^12 - 5100533631706*y^10 + 47981136220056*y^8 - 347485180692227*y^6 + 1868276740193552*y^4 - 6719535410008896*y^2 + 13289863668552241, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241)
 

\( x^{28} - 21 x^{26} + 1343 x^{24} - 59011 x^{22} + 2332758 x^{20} - 67729146 x^{18} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $28$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 14]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(116987765221280837299435954319702611299113180009478553600000000000000\) \(\medspace = 2^{28}\cdot 5^{14}\cdot 7^{14}\cdot 29^{26}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(269.78\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2\cdot 5^{1/2}7^{1/2}29^{13/14}\approx 269.77705901914624$
Ramified primes:   \(2\), \(5\), \(7\), \(29\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2\times C_{14}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4060=2^{2}\cdot 5\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4060}(1,·)$, $\chi_{4060}(2379,·)$, $\chi_{4060}(2519,·)$, $\chi_{4060}(3009,·)$, $\chi_{4060}(3081,·)$, $\chi_{4060}(209,·)$, $\chi_{4060}(139,·)$, $\chi_{4060}(141,·)$, $\chi_{4060}(2449,·)$, $\chi_{4060}(2659,·)$, $\chi_{4060}(1749,·)$, $\chi_{4060}(1049,·)$, $\chi_{4060}(281,·)$, $\chi_{4060}(71,·)$, $\chi_{4060}(1821,·)$, $\chi_{4060}(3359,·)$, $\chi_{4060}(3431,·)$, $\chi_{4060}(1889,·)$, $\chi_{4060}(1891,·)$, $\chi_{4060}(1959,·)$, $\chi_{4060}(2731,·)$, $\chi_{4060}(2029,·)$, $\chi_{4060}(1399,·)$, $\chi_{4060}(3571,·)$, $\chi_{4060}(631,·)$, $\chi_{4060}(3641,·)$, $\chi_{4060}(3711,·)$, $\chi_{4060}(981,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{41}a^{20}-\frac{9}{41}a^{16}+\frac{20}{41}a^{14}-\frac{13}{41}a^{12}-\frac{3}{41}a^{10}+\frac{1}{41}a^{8}-\frac{2}{41}a^{6}+\frac{2}{41}a^{4}+\frac{4}{41}a^{2}-\frac{10}{41}$, $\frac{1}{41}a^{21}-\frac{9}{41}a^{17}+\frac{20}{41}a^{15}-\frac{13}{41}a^{13}-\frac{3}{41}a^{11}+\frac{1}{41}a^{9}-\frac{2}{41}a^{7}+\frac{2}{41}a^{5}+\frac{4}{41}a^{3}-\frac{10}{41}a$, $\frac{1}{120581}a^{22}+\frac{686}{120581}a^{20}+\frac{35251}{120581}a^{18}+\frac{39356}{120581}a^{16}+\frac{24490}{120581}a^{14}+\frac{26216}{120581}a^{12}+\frac{14261}{120581}a^{10}-\frac{4031}{120581}a^{8}+\frac{47051}{120581}a^{6}+\frac{22737}{120581}a^{4}-\frac{1366}{120581}a^{2}-\frac{56388}{120581}$, $\frac{1}{120581}a^{23}+\frac{686}{120581}a^{21}+\frac{35251}{120581}a^{19}+\frac{39356}{120581}a^{17}+\frac{24490}{120581}a^{15}+\frac{26216}{120581}a^{13}+\frac{14261}{120581}a^{11}-\frac{4031}{120581}a^{9}+\frac{47051}{120581}a^{7}+\frac{22737}{120581}a^{5}-\frac{1366}{120581}a^{3}-\frac{56388}{120581}a$, $\frac{1}{106955347}a^{24}-\frac{384}{106955347}a^{22}+\frac{165885}{106955347}a^{20}+\frac{1389030}{106955347}a^{18}+\frac{42737892}{106955347}a^{16}-\frac{37583282}{106955347}a^{14}+\frac{1611996}{6291491}a^{12}+\frac{5607}{2608667}a^{10}-\frac{38787190}{106955347}a^{8}+\frac{14268409}{106955347}a^{6}-\frac{40446636}{106955347}a^{4}-\frac{34566139}{106955347}a^{2}-\frac{45379085}{106955347}$, $\frac{1}{106955347}a^{25}-\frac{384}{106955347}a^{23}+\frac{165885}{106955347}a^{21}+\frac{1389030}{106955347}a^{19}+\frac{42737892}{106955347}a^{17}-\frac{37583282}{106955347}a^{15}+\frac{1611996}{6291491}a^{13}+\frac{5607}{2608667}a^{11}-\frac{38787190}{106955347}a^{9}+\frac{14268409}{106955347}a^{7}-\frac{40446636}{106955347}a^{5}-\frac{34566139}{106955347}a^{3}-\frac{45379085}{106955347}a$, $\frac{1}{10\cdots 97}a^{26}+\frac{40\cdots 49}{10\cdots 97}a^{24}+\frac{77\cdots 02}{10\cdots 97}a^{22}+\frac{65\cdots 06}{10\cdots 97}a^{20}-\frac{52\cdots 92}{10\cdots 97}a^{18}+\frac{63\cdots 19}{64\cdots 41}a^{16}+\frac{48\cdots 90}{10\cdots 97}a^{14}-\frac{95\cdots 98}{10\cdots 97}a^{12}+\frac{17\cdots 94}{10\cdots 97}a^{10}-\frac{94\cdots 08}{10\cdots 97}a^{8}+\frac{24\cdots 45}{10\cdots 97}a^{6}-\frac{41\cdots 95}{10\cdots 97}a^{4}-\frac{30\cdots 19}{10\cdots 97}a^{2}+\frac{40\cdots 31}{10\cdots 97}$, $\frac{1}{12\cdots 87}a^{27}-\frac{26\cdots 85}{12\cdots 87}a^{25}-\frac{10\cdots 00}{12\cdots 87}a^{23}+\frac{66\cdots 97}{12\cdots 87}a^{21}+\frac{28\cdots 19}{12\cdots 87}a^{19}-\frac{51\cdots 44}{12\cdots 87}a^{17}-\frac{88\cdots 81}{12\cdots 87}a^{15}-\frac{59\cdots 49}{12\cdots 87}a^{13}+\frac{10\cdots 38}{12\cdots 87}a^{11}-\frac{58\cdots 34}{12\cdots 87}a^{9}+\frac{59\cdots 49}{12\cdots 87}a^{7}-\frac{17\cdots 74}{12\cdots 87}a^{5}+\frac{31\cdots 83}{12\cdots 87}a^{3}+\frac{25\cdots 46}{12\cdots 87}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   data not computed

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $13$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot R \cdot h}{2\cdot\sqrt{116987765221280837299435954319702611299113180009478553600000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^28 - 21*x^26 + 1343*x^24 - 59011*x^22 + 2332758*x^20 - 67729146*x^18 + 1575869078*x^16 - 28987252103*x^14 + 430595937122*x^12 - 5100533631706*x^10 + 47981136220056*x^8 - 347485180692227*x^6 + 1868276740193552*x^4 - 6719535410008896*x^2 + 13289863668552241); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$

Intermediate fields

\(\Q(\sqrt{-1015}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{-29}, \sqrt{35})\), 7.7.594823321.1, 14.0.660161636887192665823095859375.1, 14.0.168110140833113738264576.1, 14.14.372968560646888435753296640000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.14.0.1}{14} }^{2}$ R R ${\href{/padicField/11.14.0.1}{14} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{14}$ ${\href{/padicField/19.7.0.1}{7} }^{4}$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R ${\href{/padicField/31.7.0.1}{7} }^{4}$ ${\href{/padicField/37.14.0.1}{14} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{14}$ ${\href{/padicField/43.7.0.1}{7} }^{4}$ ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.7.2.14a1.2$x^{14} + 2 x^{8} + 4 x^{7} + x^{2} + 4 x + 9$$2$$7$$14$$C_{14}$$$[2]^{7}$$
2.7.2.14a1.2$x^{14} + 2 x^{8} + 4 x^{7} + x^{2} + 4 x + 9$$2$$7$$14$$C_{14}$$$[2]^{7}$$
\(5\) Copy content Toggle raw display 5.7.2.7a1.1$x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
5.7.2.7a1.1$x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(7\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(29\) Copy content Toggle raw display 29.1.14.13a1.1$x^{14} + 29$$14$$1$$13$$C_{14}$$$[\ ]_{14}$$
29.1.14.13a1.1$x^{14} + 29$$14$$1$$13$$C_{14}$$$[\ ]_{14}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)