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Properties

Label 12.12.295...000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2.952\times 10^{18}$
Root discriminant \(34.61\)
Ramified primes $2,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 - 32*y^10 + 124*y^9 + 339*y^8 - 1252*y^7 - 1458*y^6 + 4864*y^5 + 2480*y^4 - 6484*y^3 - 1580*y^2 + 2052*y - 239, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239)
 

\( x^{12} - 4 x^{11} - 32 x^{10} + 124 x^{9} + 339 x^{8} - 1252 x^{7} - 1458 x^{6} + 4864 x^{5} + \cdots - 239 \) 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:  $12$
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:  $[12, 0]$
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:   \(2951578112000000000\) \(\medspace = 2^{18}\cdot 5^{9}\cdot 7^{8}\) 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:  \(34.61\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{3/4}7^{2/3}\approx 34.607576700316336$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{12}$
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:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(107,·)$, $\chi_{280}(1,·)$, $\chi_{280}(67,·)$, $\chi_{280}(9,·)$, $\chi_{280}(267,·)$, $\chi_{280}(81,·)$, $\chi_{280}(163,·)$, $\chi_{280}(43,·)$, $\chi_{280}(121,·)$, $\chi_{280}(249,·)$, $\chi_{280}(123,·)$, $\chi_{280}(169,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{62078}a^{10}-\frac{2752}{31039}a^{9}-\frac{3793}{31039}a^{8}+\frac{7015}{62078}a^{7}-\frac{4625}{31039}a^{6}-\frac{19443}{62078}a^{5}-\frac{15108}{31039}a^{4}-\frac{12720}{31039}a^{3}-\frac{9617}{62078}a^{2}+\frac{8987}{62078}a+\frac{23721}{62078}$, $\frac{1}{114548870798}a^{11}+\frac{91610}{57274435399}a^{10}+\frac{339356079}{114548870798}a^{9}+\frac{3991356675}{114548870798}a^{8}+\frac{229671743}{114548870798}a^{7}-\frac{11731433345}{57274435399}a^{6}-\frac{2045884830}{57274435399}a^{5}-\frac{48958056717}{114548870798}a^{4}+\frac{963044744}{57274435399}a^{3}-\frac{51946411089}{114548870798}a^{2}+\frac{27099397157}{114548870798}a+\frac{6993022695}{114548870798}$ 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:  Trivial group, which has order $1$ (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:  $C_{2}$, which has order $2$ (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);
 

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:  $11$
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:   $\frac{14}{31039}a^{11}-\frac{115}{31039}a^{10}-\frac{235}{31039}a^{9}+\frac{7085}{62078}a^{8}-\frac{1400}{31039}a^{7}-\frac{34909}{31039}a^{6}+\frac{34342}{31039}a^{5}+\frac{255655}{62078}a^{4}-\frac{132260}{31039}a^{3}-\frac{281779}{62078}a^{2}+\frac{96746}{31039}a+\frac{17483}{62078}$, $\frac{60436500}{57274435399}a^{11}-\frac{274401726}{57274435399}a^{10}-\frac{1811428760}{57274435399}a^{9}+\frac{8547641730}{57274435399}a^{8}+\frac{16848769680}{57274435399}a^{7}-\frac{86380707330}{57274435399}a^{6}-\frac{55946446740}{57274435399}a^{5}+\frac{326669986695}{57274435399}a^{4}+\frac{73146172440}{57274435399}a^{3}-\frac{355661439720}{57274435399}a^{2}-\frac{133263895880}{57274435399}a+\frac{56962042131}{57274435399}$, $\frac{60436500}{57274435399}a^{11}-\frac{274401726}{57274435399}a^{10}-\frac{1811428760}{57274435399}a^{9}+\frac{8547641730}{57274435399}a^{8}+\frac{16848769680}{57274435399}a^{7}-\frac{86380707330}{57274435399}a^{6}-\frac{55946446740}{57274435399}a^{5}+\frac{326669986695}{57274435399}a^{4}+\frac{73146172440}{57274435399}a^{3}-\frac{355661439720}{57274435399}a^{2}-\frac{133263895880}{57274435399}a-\frac{312393268}{57274435399}$, $\frac{54681866}{57274435399}a^{11}-\frac{73985501}{57274435399}a^{10}-\frac{2360952005}{57274435399}a^{9}+\frac{5043828885}{114548870798}a^{8}+\frac{36670289480}{57274435399}a^{7}-\frac{30353808611}{57274435399}a^{6}-\frac{249370314086}{57274435399}a^{5}+\frac{308191098785}{114548870798}a^{4}+\frac{711250967100}{57274435399}a^{3}-\frac{618221397901}{114548870798}a^{2}-\frac{643783799706}{57274435399}a+\frac{208664479291}{114548870798}$, $\frac{45912114}{57274435399}a^{11}-\frac{223989205}{57274435399}a^{10}-\frac{1416786515}{57274435399}a^{9}+\frac{14095610395}{114548870798}a^{8}+\frac{14654845000}{57274435399}a^{7}-\frac{72804137419}{57274435399}a^{6}-\frac{66685334502}{57274435399}a^{5}+\frac{598341301105}{114548870798}a^{4}+\frac{150001645340}{57274435399}a^{3}-\frac{946798845939}{114548870798}a^{2}-\frac{153480532254}{57274435399}a+\frac{388358833431}{114548870798}$, $\frac{394539783}{57274435399}a^{11}-\frac{2546593745}{114548870798}a^{10}-\frac{27018191227}{114548870798}a^{9}+\frac{77363768053}{114548870798}a^{8}+\frac{159479112919}{57274435399}a^{7}-\frac{378418971301}{57274435399}a^{6}-\frac{1621633116225}{114548870798}a^{5}+\frac{2773762221325}{114548870798}a^{4}+\frac{3509275983731}{114548870798}a^{3}-\frac{1634806489860}{57274435399}a^{2}-\frac{2886389196533}{114548870798}a+\frac{234440121447}{57274435399}$, $\frac{222037205}{114548870798}a^{11}-\frac{1203830591}{114548870798}a^{10}-\frac{3251015890}{57274435399}a^{9}+\frac{19624409068}{57274435399}a^{8}+\frac{58465406667}{114548870798}a^{7}-\frac{426569182153}{114548870798}a^{6}-\frac{183682260803}{114548870798}a^{5}+\frac{1826689205605}{114548870798}a^{4}+\frac{200164766395}{114548870798}a^{3}-\frac{2605500543481}{114548870798}a^{2}-\frac{120608991064}{57274435399}a+\frac{415383660851}{57274435399}$, $\frac{90901659}{114548870798}a^{11}-\frac{370581052}{57274435399}a^{10}-\frac{2380005807}{114548870798}a^{9}+\frac{11422688841}{57274435399}a^{8}+\frac{19117601291}{114548870798}a^{7}-\frac{109477388721}{57274435399}a^{6}-\frac{26423828768}{57274435399}a^{5}+\frac{372765394644}{57274435399}a^{4}-\frac{10901107901}{57274435399}a^{3}-\frac{340869411476}{57274435399}a^{2}+\frac{133449510397}{114548870798}a+\frac{1943316816}{57274435399}$, $\frac{181979877}{114548870798}a^{11}-\frac{703412497}{114548870798}a^{10}-\frac{5550094797}{114548870798}a^{9}+\frac{21043597409}{114548870798}a^{8}+\frac{53638593325}{114548870798}a^{7}-\frac{102094184989}{57274435399}a^{6}-\frac{94725699870}{57274435399}a^{5}+\frac{753593115223}{114548870798}a^{4}+\frac{101401910552}{57274435399}a^{3}-\frac{435430896862}{57274435399}a^{2}-\frac{67432232755}{114548870798}a+\frac{62135160650}{57274435399}$, $\frac{4282009}{57274435399}a^{11}+\frac{52205599}{114548870798}a^{10}-\frac{331289254}{57274435399}a^{9}-\frac{622658736}{57274435399}a^{8}+\frac{14812667523}{114548870798}a^{7}+\frac{5283090127}{114548870798}a^{6}-\frac{129601856935}{114548870798}a^{5}+\frac{38064364941}{114548870798}a^{4}+\frac{215000520176}{57274435399}a^{3}-\frac{211740893341}{114548870798}a^{2}-\frac{357348303611}{114548870798}a+\frac{34550033012}{57274435399}$, $\frac{246403893}{114548870798}a^{11}-\frac{1067354365}{114548870798}a^{10}-\frac{6734369901}{114548870798}a^{9}+\frac{29012576233}{114548870798}a^{8}+\frac{54551932559}{114548870798}a^{7}-\frac{228582713555}{114548870798}a^{6}-\frac{85246945802}{57274435399}a^{5}+\frac{280840132055}{57274435399}a^{4}+\frac{169536513040}{57274435399}a^{3}-\frac{240107325205}{57274435399}a^{2}-\frac{152572756977}{114548870798}a+\frac{101610212191}{114548870798}$ 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:  \( 193697.65851 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 193697.65851 \cdot 1}{2\cdot\sqrt{2951578112000000000}}\cr\approx \mathstrut & 0.23090172775 \end{aligned}\] (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^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239) 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^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239, 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^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239); 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 = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 32*x^10 + 124*x^9 + 339*x^8 - 1252*x^7 - 1458*x^6 + 4864*x^5 + 2480*x^4 - 6484*x^3 - 1580*x^2 + 2052*x - 239); 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

$C_{12}$ (as 12T1):

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); G, transitive_group_identification(G)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.8000.1, 6.6.300125.1

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(b)]
 
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 ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.1.0.1}{1} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.1.0.1}{1} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.6.2.18a1.2$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 10 x^{3} + x^{2} + 2 x + 3$$2$$6$$18$$C_{12}$$$[3]^{6}$$
\(5\) Copy content Toggle raw display 5.3.4.9a1.1$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(7\) Copy content Toggle raw display 7.4.3.8a1.3$x^{12} + 15 x^{10} + 12 x^{9} + 84 x^{8} + 120 x^{7} + 263 x^{6} + 372 x^{5} + 492 x^{4} + 424 x^{3} + 279 x^{2} + 108 x + 34$$3$$4$$8$$C_{12}$$$[\ ]_{3}^{4}$$

Spectrum of ring of integers

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