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Properties

Label 14.0.759...827.1
Degree $14$
Signature $[0, 7]$
Discriminant $-7.597\times 10^{31}$
Root discriminant \(189.32\)
Ramified primes $3,239$
Class number $7624$ (GRH)
Class group [2, 2, 1906] (GRH)
Galois group $C_{14}$ (as 14T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489)
 
Copy content gp:K = bnfinit(y^14 - y^13 + 103*y^12 - 288*y^11 + 8749*y^10 - 17168*y^9 + 207881*y^8 + 164988*y^7 + 2712410*y^6 + 2203236*y^5 + 18493219*y^4 + 27913574*y^3 + 74729515*y^2 + 46299739*y + 26863489, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489)
 

\( x^{14} - x^{13} + 103 x^{12} - 288 x^{11} + 8749 x^{10} - 17168 x^{9} + 207881 x^{8} + 164988 x^{7} + \cdots + 26863489 \) 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:  $14$
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:  $[0, 7]$
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:   \(-75966758316706681853567921242827\) \(\medspace = -\,3^{7}\cdot 239^{12}\) 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:  \(189.32\)
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:  $3^{1/2}239^{6/7}\approx 189.31603477905156$
Ramified primes:   \(3\), \(239\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{14}$
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:  \(717=3\cdot 239\)
Dirichlet character group:    $\lbrace$$\chi_{717}(1,·)$, $\chi_{717}(98,·)$, $\chi_{717}(100,·)$, $\chi_{717}(263,·)$, $\chi_{717}(488,·)$, $\chi_{717}(10,·)$, $\chi_{717}(679,·)$, $\chi_{717}(44,·)$, $\chi_{717}(578,·)$, $\chi_{717}(337,·)$, $\chi_{717}(502,·)$, $\chi_{717}(440,·)$, $\chi_{717}(283,·)$, $\chi_{717}(479,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{64}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23}a^{10}+\frac{10}{23}a^{9}+\frac{4}{23}a^{8}+\frac{10}{23}a^{7}+\frac{6}{23}a^{6}+\frac{6}{23}a^{5}+\frac{9}{23}a^{3}+\frac{11}{23}a^{2}+\frac{4}{23}a+\frac{4}{23}$, $\frac{1}{23}a^{11}-\frac{4}{23}a^{9}-\frac{7}{23}a^{8}-\frac{2}{23}a^{7}-\frac{8}{23}a^{6}+\frac{9}{23}a^{5}+\frac{9}{23}a^{4}-\frac{10}{23}a^{3}+\frac{9}{23}a^{2}+\frac{10}{23}a+\frac{6}{23}$, $\frac{1}{10323343}a^{12}-\frac{110647}{10323343}a^{11}+\frac{110749}{10323343}a^{10}+\frac{4562766}{10323343}a^{9}-\frac{4924956}{10323343}a^{8}+\frac{4955134}{10323343}a^{7}+\frac{88467}{10323343}a^{6}-\frac{4445720}{10323343}a^{5}-\frac{4061917}{10323343}a^{4}-\frac{3777775}{10323343}a^{3}+\frac{4718280}{10323343}a^{2}+\frac{2463237}{10323343}a-\frac{2030878}{10323343}$, $\frac{1}{24\cdots 53}a^{13}+\frac{10\cdots 77}{24\cdots 53}a^{12}+\frac{39\cdots 41}{24\cdots 53}a^{11}+\frac{13\cdots 02}{24\cdots 53}a^{10}-\frac{34\cdots 09}{24\cdots 53}a^{9}+\frac{28\cdots 60}{24\cdots 53}a^{8}+\frac{32\cdots 99}{24\cdots 53}a^{7}+\frac{51\cdots 65}{24\cdots 53}a^{6}-\frac{85\cdots 11}{24\cdots 53}a^{5}+\frac{42\cdots 97}{10\cdots 11}a^{4}-\frac{81\cdots 97}{24\cdots 53}a^{3}+\frac{25\cdots 81}{24\cdots 53}a^{2}+\frac{82\cdots 55}{24\cdots 53}a-\frac{43\cdots 16}{47\cdots 91}$ 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:  $C_{2}\times C_{2}\times C_{1906}$, which has order $7624$ (assuming GRH)
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:  $C_{2}\times C_{2}\times C_{1906}$, which has order $7624$ (assuming GRH)
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:   $7624$ (assuming GRH)

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:  $6$
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:   \( -\frac{51681703564915017913449529854}{2394852806549510911428985803596202971} a^{13} + \frac{46843315289024321442656022402}{2394852806549510911428985803596202971} a^{12} - \frac{5358650797744434753172185991476}{2394852806549510911428985803596202971} a^{11} + \frac{14573682317120861823764732798125}{2394852806549510911428985803596202971} a^{10} - \frac{454792850258874498675525989963197}{2394852806549510911428985803596202971} a^{9} + \frac{871051472790390299538871409824021}{2394852806549510911428985803596202971} a^{8} - \frac{11030316836184649876375740931857632}{2394852806549510911428985803596202971} a^{7} - \frac{7656329132663771064524569747070397}{2394852806549510911428985803596202971} a^{6} - \frac{149692132002401055709378168984894288}{2394852806549510911428985803596202971} a^{5} - \frac{109313984993274354294755745336581708}{2394852806549510911428985803596202971} a^{4} - \frac{1016230578137625732262077417341134849}{2394852806549510911428985803596202971} a^{3} - \frac{1112072949085062945830542153982085523}{2394852806549510911428985803596202971} a^{2} - \frac{4211182356954355500384202450813021931}{2394852806549510911428985803596202971} a - \frac{42375743076623400257224069730742}{462059194780920492268760525486437} \)  (order $6$) 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:   $\frac{15\cdots 58}{24\cdots 53}a^{13}-\frac{14\cdots 44}{24\cdots 53}a^{12}+\frac{16\cdots 13}{24\cdots 53}a^{11}-\frac{44\cdots 34}{24\cdots 53}a^{10}+\frac{13\cdots 44}{24\cdots 53}a^{9}-\frac{26\cdots 96}{24\cdots 53}a^{8}+\frac{32\cdots 53}{24\cdots 53}a^{7}+\frac{25\cdots 21}{24\cdots 53}a^{6}+\frac{43\cdots 78}{24\cdots 53}a^{5}+\frac{34\cdots 55}{24\cdots 53}a^{4}+\frac{29\cdots 68}{24\cdots 53}a^{3}+\frac{43\cdots 68}{24\cdots 53}a^{2}+\frac{11\cdots 39}{24\cdots 53}a+\frac{14\cdots 89}{47\cdots 91}$, $\frac{77\cdots 76}{24\cdots 53}a^{13}-\frac{15\cdots 11}{24\cdots 53}a^{12}+\frac{79\cdots 13}{24\cdots 53}a^{11}-\frac{30\cdots 27}{24\cdots 53}a^{10}+\frac{69\cdots 46}{24\cdots 53}a^{9}-\frac{20\cdots 93}{24\cdots 53}a^{8}+\frac{17\cdots 79}{24\cdots 53}a^{7}-\frac{20\cdots 04}{24\cdots 53}a^{6}+\frac{18\cdots 69}{24\cdots 53}a^{5}-\frac{24\cdots 64}{24\cdots 53}a^{4}+\frac{12\cdots 67}{24\cdots 53}a^{3}+\frac{10\cdots 69}{24\cdots 53}a^{2}+\frac{39\cdots 46}{24\cdots 53}a+\frac{30\cdots 96}{47\cdots 91}$, $\frac{39\cdots 37}{10\cdots 11}a^{13}-\frac{29\cdots 45}{24\cdots 53}a^{12}+\frac{96\cdots 11}{24\cdots 53}a^{11}-\frac{46\cdots 68}{24\cdots 53}a^{10}+\frac{86\cdots 50}{24\cdots 53}a^{9}-\frac{32\cdots 67}{24\cdots 53}a^{8}+\frac{22\cdots 90}{24\cdots 53}a^{7}-\frac{18\cdots 60}{24\cdots 53}a^{6}+\frac{18\cdots 83}{24\cdots 53}a^{5}-\frac{10\cdots 89}{24\cdots 53}a^{4}+\frac{10\cdots 93}{24\cdots 53}a^{3}+\frac{67\cdots 99}{24\cdots 53}a^{2}+\frac{41\cdots 87}{24\cdots 53}a-\frac{16\cdots 68}{47\cdots 91}$, $\frac{17\cdots 57}{24\cdots 53}a^{13}+\frac{62\cdots 21}{24\cdots 53}a^{12}+\frac{17\cdots 43}{24\cdots 53}a^{11}+\frac{21\cdots 62}{24\cdots 53}a^{10}+\frac{12\cdots 07}{24\cdots 53}a^{9}+\frac{30\cdots 05}{24\cdots 53}a^{8}+\frac{21\cdots 54}{24\cdots 53}a^{7}+\frac{11\cdots 71}{24\cdots 53}a^{6}+\frac{57\cdots 07}{24\cdots 53}a^{5}+\frac{92\cdots 73}{24\cdots 53}a^{4}-\frac{11\cdots 65}{10\cdots 11}a^{3}-\frac{68\cdots 38}{24\cdots 53}a^{2}-\frac{38\cdots 62}{10\cdots 11}a-\frac{11\cdots 86}{47\cdots 91}$, $\frac{39\cdots 92}{24\cdots 53}a^{13}-\frac{20\cdots 42}{24\cdots 53}a^{12}+\frac{30\cdots 63}{24\cdots 53}a^{11}-\frac{22\cdots 57}{24\cdots 53}a^{10}+\frac{26\cdots 63}{24\cdots 53}a^{9}-\frac{12\cdots 52}{24\cdots 53}a^{8}-\frac{12\cdots 67}{24\cdots 53}a^{7}+\frac{34\cdots 84}{24\cdots 53}a^{6}-\frac{27\cdots 27}{24\cdots 53}a^{5}+\frac{23\cdots 23}{24\cdots 53}a^{4}-\frac{12\cdots 57}{24\cdots 53}a^{3}-\frac{79\cdots 69}{24\cdots 53}a^{2}-\frac{85\cdots 52}{24\cdots 53}a-\frac{10\cdots 90}{47\cdots 91}$, $\frac{10\cdots 13}{24\cdots 53}a^{13}-\frac{11\cdots 11}{24\cdots 53}a^{12}+\frac{10\cdots 43}{24\cdots 53}a^{11}-\frac{30\cdots 42}{24\cdots 53}a^{10}+\frac{89\cdots 63}{24\cdots 53}a^{9}-\frac{18\cdots 28}{24\cdots 53}a^{8}+\frac{21\cdots 36}{24\cdots 53}a^{7}+\frac{16\cdots 89}{24\cdots 53}a^{6}+\frac{26\cdots 54}{24\cdots 53}a^{5}+\frac{22\cdots 07}{24\cdots 53}a^{4}+\frac{18\cdots 45}{24\cdots 53}a^{3}+\frac{13\cdots 55}{10\cdots 11}a^{2}+\frac{75\cdots 16}{24\cdots 53}a+\frac{89\cdots 76}{47\cdots 91}$ Copy content Toggle raw display (assuming GRH)
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:  \( 3022802.0673343516 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 3022802.0673343516 \cdot 7624}{6\cdot\sqrt{75966758316706681853567921242827}}\cr\approx \mathstrut & 0.170368236937020 \end{aligned}\] (assuming GRH)

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^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489) 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^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489, 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^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489); 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^14 - x^13 + 103*x^12 - 288*x^11 + 8749*x^10 - 17168*x^9 + 207881*x^8 + 164988*x^7 + 2712410*x^6 + 2203236*x^5 + 18493219*x^4 + 27913574*x^3 + 74729515*x^2 + 46299739*x + 26863489); 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

$C_{14}$ (as 14T1):

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)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.7.186374892382561.1

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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.7.0.1}{7} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.7.0.1}{7} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.14.0.1}{14} }$

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
\(3\) Copy content Toggle raw display 3.7.2.7a1.2$x^{14} + 4 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{2} + 4$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(239\) Copy content Toggle raw display Deg $14$$7$$2$$12$

Spectrum of ring of integers

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