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Properties

Label 12.12.551...041.1
Degree $12$
Signature $[12, 0]$
Discriminant $5.514\times 10^{18}$
Root discriminant \(36.46\)
Ramified primes $3,71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4$ (as 12T4)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648)
 
Copy content gp:K = bnfinit(y^12 - 3*y^11 - 39*y^10 + 92*y^9 + 543*y^8 - 882*y^7 - 3153*y^6 + 2880*y^5 + 7236*y^4 - 1899*y^3 - 4212*y^2 + 324*y + 648, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648)
 

\( x^{12} - 3 x^{11} - 39 x^{10} + 92 x^{9} + 543 x^{8} - 882 x^{7} - 3153 x^{6} + 2880 x^{5} + 7236 x^{4} + \cdots + 648 \) 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:   \(5514297181968073041\) \(\medspace = 3^{16}\cdot 71^{6}\) 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:  \(36.46\)
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:  $3^{4/3}71^{1/2}\approx 36.45783266912841$
Ramified primes:   \(3\), \(71\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $A_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{7}-\frac{1}{12}a^{6}-\frac{1}{4}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{72}a^{8}-\frac{1}{24}a^{7}+\frac{1}{24}a^{6}+\frac{7}{36}a^{5}+\frac{1}{6}a^{4}+\frac{7}{24}a^{3}+\frac{1}{6}a^{2}-\frac{1}{2}a$, $\frac{1}{432}a^{9}+\frac{1}{72}a^{7}-\frac{13}{432}a^{6}-\frac{1}{8}a^{5}+\frac{1}{48}a^{4}+\frac{49}{144}a^{3}-\frac{1}{12}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{1728}a^{10}+\frac{1}{1728}a^{9}-\frac{1}{288}a^{8}+\frac{65}{1728}a^{7}-\frac{139}{1728}a^{6}-\frac{107}{576}a^{5}-\frac{47}{144}a^{4}-\frac{143}{576}a^{3}-\frac{5}{48}a^{2}-\frac{7}{16}a+\frac{1}{8}$, $\frac{1}{5609088}a^{11}+\frac{79}{467424}a^{10}+\frac{1871}{1869696}a^{9}+\frac{27599}{5609088}a^{8}+\frac{6223}{233712}a^{7}+\frac{15919}{311616}a^{6}+\frac{338611}{1869696}a^{5}+\frac{30919}{69248}a^{4}+\frac{59471}{207744}a^{3}+\frac{3065}{12984}a^{2}+\frac{4191}{17312}a-\frac{223}{8656}$ 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:  No
Index:  Not computed
Inessential primes:  $2$, $3$

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:   $\frac{1295}{1869696}a^{11}-\frac{457}{233712}a^{10}-\frac{15883}{623232}a^{9}+\frac{99625}{1869696}a^{8}+\frac{149069}{467424}a^{7}-\frac{121907}{311616}a^{6}-\frac{905447}{623232}a^{5}+\frac{277363}{623232}a^{4}+\frac{112877}{69248}a^{3}+\frac{13853}{6492}a^{2}+\frac{32559}{17312}a-\frac{11575}{8656}$, $\frac{125}{103872}a^{11}-\frac{1975}{934848}a^{10}-\frac{2731}{51936}a^{9}+\frac{5869}{103872}a^{8}+\frac{774637}{934848}a^{7}-\frac{13063}{34624}a^{6}-\frac{106691}{19476}a^{5}-\frac{2719}{311616}a^{4}+\frac{359995}{25968}a^{3}+\frac{113807}{25968}a^{2}-\frac{28583}{4328}a-\frac{688}{541}$, $\frac{15757}{5609088}a^{11}-\frac{8191}{934848}a^{10}-\frac{22351}{207744}a^{9}+\frac{1516967}{5609088}a^{8}+\frac{1361023}{934848}a^{7}-\frac{307915}{116856}a^{6}-\frac{14943455}{1869696}a^{5}+\frac{615283}{69248}a^{4}+\frac{9928591}{623232}a^{3}-\frac{165817}{25968}a^{2}-\frac{86679}{17312}a+\frac{16755}{8656}$, $\frac{10049}{5609088}a^{11}-\frac{317}{467424}a^{10}-\frac{159289}{1869696}a^{9}-\frac{73793}{5609088}a^{8}+\frac{336157}{233712}a^{7}+\frac{763705}{934848}a^{6}-\frac{18582061}{1869696}a^{5}-\frac{5104201}{623232}a^{4}+\frac{15228469}{623232}a^{3}+\frac{319163}{12984}a^{2}-\frac{69657}{17312}a-\frac{55287}{8656}$, $\frac{4037}{467424}a^{11}-\frac{29671}{934848}a^{10}-\frac{297217}{934848}a^{9}+\frac{59219}{58428}a^{8}+\frac{3842329}{934848}a^{7}-\frac{9809819}{934848}a^{6}-\frac{748253}{34624}a^{5}+\frac{6221557}{155808}a^{4}+\frac{13385747}{311616}a^{3}-\frac{1177429}{25968}a^{2}-\frac{154105}{8656}a+\frac{56643}{4328}$, $\frac{29881}{1869696}a^{11}-\frac{2633}{51936}a^{10}-\frac{1143523}{1869696}a^{9}+\frac{2928359}{1869696}a^{8}+\frac{322801}{38952}a^{7}-\frac{14236357}{934848}a^{6}-\frac{28627453}{623232}a^{5}+\frac{3543709}{69248}a^{4}+\frac{59126567}{623232}a^{3}-\frac{171581}{4328}a^{2}-\frac{543555}{17312}a+\frac{121827}{8656}$, $\frac{5521}{1869696}a^{11}-\frac{1403}{116856}a^{10}-\frac{63901}{623232}a^{9}+\frac{707975}{1869696}a^{8}+\frac{566885}{467424}a^{7}-\frac{131609}{34624}a^{6}-\frac{3412241}{623232}a^{5}+\frac{8182765}{623232}a^{4}+\frac{560475}{69248}a^{3}-\frac{5825}{541}a^{2}-\frac{33719}{17312}a+\frac{17711}{8656}$, $\frac{6329}{1869696}a^{11}-\frac{6391}{934848}a^{10}-\frac{260309}{1869696}a^{9}+\frac{331411}{1869696}a^{8}+\frac{1899763}{934848}a^{7}-\frac{501961}{467424}a^{6}-\frac{2499317}{207744}a^{5}-\frac{688187}{623232}a^{4}+\frac{15919333}{623232}a^{3}+\frac{374483}{25968}a^{2}-\frac{122357}{17312}a-\frac{37023}{8656}$, $\frac{49}{77904}a^{11}-\frac{763}{233712}a^{10}-\frac{4783}{233712}a^{9}+\frac{8507}{77904}a^{8}+\frac{51277}{233712}a^{7}-\frac{69665}{58428}a^{6}-\frac{16493}{19476}a^{5}+\frac{174631}{38952}a^{4}+\frac{66341}{77904}a^{3}-\frac{5261}{2164}a^{2}+\frac{175}{2164}a+\frac{119}{1082}$, $\frac{2959}{2804544}a^{11}-\frac{869}{155808}a^{10}-\frac{26273}{934848}a^{9}+\frac{469997}{2804544}a^{8}+\frac{24853}{155808}a^{7}-\frac{363023}{233712}a^{6}+\frac{776431}{934848}a^{5}+\frac{1443113}{311616}a^{4}-\frac{1841527}{311616}a^{3}-\frac{12075}{4328}a^{2}+\frac{22007}{8656}a+\frac{1245}{4328}$, $\frac{63589}{5609088}a^{11}-\frac{38801}{934848}a^{10}-\frac{779659}{1869696}a^{9}+\frac{7430807}{5609088}a^{8}+\frac{5022773}{934848}a^{7}-\frac{711763}{51936}a^{6}-\frac{52325891}{1869696}a^{5}+\frac{32484523}{623232}a^{4}+\frac{11280905}{207744}a^{3}-\frac{1539895}{25968}a^{2}-\frac{353243}{17312}a+\frac{155071}{8656}$ 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:  \( 1188211.01781 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1188211.01781 \cdot 1}{2\cdot\sqrt{5514297181968073041}}\cr\approx \mathstrut & 1.03628316651 \end{aligned}\] (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 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648) 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 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648, 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 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648); 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 - 3*x^11 - 39*x^10 + 92*x^9 + 543*x^8 - 882*x^7 - 3153*x^6 + 2880*x^5 + 7236*x^4 - 1899*x^3 - 4212*x^2 + 324*x + 648); 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

$A_4$ (as 12T4):

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 solvable group of order 12
The 4 conjugacy class representatives for $A_4$
Character table for $A_4$

Intermediate fields

\(\Q(\zeta_{9})^+\), 4.4.408321.1 x4, 6.6.33074001.1 x3

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]
 

Sibling fields

Degree 4 sibling: 4.4.408321.1
Degree 6 sibling: 6.6.33074001.1
Minimal sibling: 4.4.408321.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{4}$ R ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(3\) Copy content Toggle raw display 3.1.3.4a2.1$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$$[2]$$
3.1.3.4a2.1$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$$[2]$$
3.1.3.4a2.1$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$$[2]$$
3.1.3.4a2.1$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$$[2]$$
\(71\) Copy content Toggle raw display 71.2.2.2a1.2$x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
71.2.2.2a1.2$x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
71.2.2.2a1.2$x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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