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Properties

Label 12.0.295...000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2.952\times 10^{18}$
Root discriminant \(34.61\)
Ramified primes $2,5,7$
Class number $146$
Class group [146]
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 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 28*y^10 - 76*y^9 + 359*y^8 - 772*y^7 + 2662*y^6 - 4216*y^5 + 11540*y^4 - 13284*y^3 + 37260*y^2 - 26468*y + 52681, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681)
 

\( x^{12} - 4 x^{11} + 28 x^{10} - 76 x^{9} + 359 x^{8} - 772 x^{7} + 2662 x^{6} - 4216 x^{5} + \cdots + 52681 \) 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:  $[0, 6]$
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$$\chi_{280}(1,·)$, $\chi_{280}(197,·)$, $\chi_{280}(9,·)$, $\chi_{280}(93,·)$, $\chi_{280}(81,·)$, $\chi_{280}(37,·)$, $\chi_{280}(53,·)$, $\chi_{280}(169,·)$, $\chi_{280}(121,·)$, $\chi_{280}(249,·)$, $\chi_{280}(253,·)$, $\chi_{280}(277,·)$$\rbrace$
This is a CM field.
Reflex fields:  4.0.8000.2$^{2}$, 12.0.2951578112000000000.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{3008402}a^{10}+\frac{1479}{51869}a^{9}+\frac{600847}{3008402}a^{8}+\frac{473855}{3008402}a^{7}+\frac{530297}{3008402}a^{6}+\frac{519167}{3008402}a^{5}-\frac{598564}{1504201}a^{4}+\frac{201921}{1504201}a^{3}-\frac{674466}{1504201}a^{2}-\frac{430749}{3008402}a-\frac{238993}{3008402}$, $\frac{1}{411536520647842}a^{11}-\frac{5418327}{205768260323921}a^{10}+\frac{24069810190907}{411536520647842}a^{9}+\frac{88115747819655}{411536520647842}a^{8}-\frac{12911393564899}{205768260323921}a^{7}-\frac{21185525877925}{411536520647842}a^{6}+\frac{33157227985965}{411536520647842}a^{5}-\frac{19595352091404}{205768260323921}a^{4}-\frac{1420563531549}{5018738056681}a^{3}-\frac{3202154870861}{10037476113362}a^{2}-\frac{68262809540052}{205768260323921}a+\frac{59840150427593}{205768260323921}$ 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:  $C_{146}$, which has order $146$
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:  $C_{146}$, which has order $146$
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:   $146$

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:  $5$
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:   $\frac{14}{1504201}a^{11}+\frac{165}{1504201}a^{10}-\frac{375}{1504201}a^{9}+\frac{6225}{3008402}a^{8}-\frac{160}{51869}a^{7}+\frac{29911}{1504201}a^{6}-\frac{42098}{1504201}a^{5}+\frac{300915}{3008402}a^{4}-\frac{116140}{1504201}a^{3}+\frac{1145081}{3008402}a^{2}-\frac{321094}{1504201}a+\frac{4000071}{3008402}$, $\frac{824540520}{205768260323921}a^{11}-\frac{1305871736}{205768260323921}a^{10}+\frac{59760879800}{205768260323921}a^{9}-\frac{140414664715}{205768260323921}a^{8}+\frac{1035927455440}{205768260323921}a^{7}-\frac{1724766859760}{205768260323921}a^{6}+\frac{9864225104176}{205768260323921}a^{5}-\frac{12741280683680}{205768260323921}a^{4}+\frac{1315681787320}{5018738056681}a^{3}-\frac{977245359380}{5018738056681}a^{2}+\frac{88634281926320}{205768260323921}a+\frac{112461252296087}{205768260323921}$, $\frac{154300500}{205768260323921}a^{11}+\frac{28204902550}{205768260323921}a^{10}-\frac{52902379640}{205768260323921}a^{9}+\frac{628281454295}{205768260323921}a^{8}-\frac{906978865760}{205768260323921}a^{7}+\frac{7155202858210}{205768260323921}a^{6}-\frac{7012383177404}{205768260323921}a^{5}+\frac{47691626973925}{205768260323921}a^{4}-\frac{645190655120}{5018738056681}a^{3}+\frac{4023167893360}{5018738056681}a^{2}-\frac{55117568611400}{205768260323921}a+\frac{394773466855174}{205768260323921}$, $\frac{539958798}{205768260323921}a^{11}+\frac{15209159527}{411536520647842}a^{10}-\frac{79614638602}{205768260323921}a^{9}+\frac{529791529758}{205768260323921}a^{8}-\frac{2327744913182}{205768260323921}a^{7}+\frac{7970855182214}{205768260323921}a^{6}-\frac{25139958849811}{205768260323921}a^{5}+\frac{64789277101415}{205768260323921}a^{4}-\frac{3469337850348}{5018738056681}a^{3}+\frac{10560012128153}{10037476113362}a^{2}-\frac{238363837378709}{205768260323921}a+\frac{457812868320745}{411536520647842}$, $\frac{2739680614}{205768260323921}a^{11}+\frac{21265422229}{205768260323921}a^{10}+\frac{8462484425}{205768260323921}a^{9}+\frac{570724033795}{411536520647842}a^{8}+\frac{401195310000}{205768260323921}a^{7}+\frac{2366929951071}{205768260323921}a^{6}+\frac{4105398841518}{205768260323921}a^{5}+\frac{15681323017355}{411536520647842}a^{4}+\frac{928182883980}{5018738056681}a^{3}+\frac{1866050281201}{10037476113362}a^{2}+\frac{44709996687546}{205768260323921}a+\frac{772115101088365}{411536520647842}$ Copy content Toggle raw display
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:  \( 104.882003477 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 146}{2\cdot\sqrt{2951578112000000000}}\cr\approx \mathstrut & 0.274205337652 \end{aligned}\]

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^12 - 4*x^11 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681) 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 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681, 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 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681); 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 + 28*x^10 - 76*x^9 + 359*x^8 - 772*x^7 + 2662*x^6 - 4216*x^5 + 11540*x^4 - 13284*x^3 + 37260*x^2 - 26468*x + 52681); 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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.8000.2, 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 ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.1.0.1}{1} }^{12}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.1.0.1}{1} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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 2.6.2.18a1.18$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 7 x^{6} + 2 x^{5} + 8 x^{4} + 14 x^{3} + x^{2} + 6 x + 7$$2$$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)