Properties

Label 12.0.2025170913606201.2
Degree $12$
Signature $(0, 6)$
Discriminant $2.025\times 10^{15}$
Root discriminant \(18.86\)
Ramified primes $3,19$
Class number $4$
Class group [4]
Galois group $A_4$ (as 12T4)

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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 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327)
 
Copy content gp:K = bnfinit(y^12 - 6*y^10 - 11*y^9 + 21*y^8 + 39*y^7 + 14*y^6 - 27*y^5 + 72*y^4 + 20*y^3 - 129*y^2 - 297*y + 327, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327)
 

\( x^{12} - 6x^{10} - 11x^{9} + 21x^{8} + 39x^{7} + 14x^{6} - 27x^{5} + 72x^{4} + 20x^{3} - 129x^{2} - 297x + 327 \) 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:   \(2025170913606201\) \(\medspace = 3^{16}\cdot 19^{6}\) 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:  \(18.86\)
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:  $3^{4/3}19^{1/2}\approx 18.859860385004858$
Ramified primes:   \(3\), \(19\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{6}a^{9}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2280}a^{10}-\frac{17}{1140}a^{9}-\frac{93}{760}a^{8}-\frac{153}{760}a^{7}-\frac{107}{380}a^{6}-\frac{87}{380}a^{5}+\frac{1}{6}a^{4}+\frac{1111}{2280}a^{3}+\frac{113}{380}a^{2}+\frac{183}{760}a-\frac{139}{760}$, $\frac{1}{1198459200}a^{11}+\frac{156083}{1198459200}a^{10}-\frac{80357117}{1198459200}a^{9}+\frac{41613}{199743200}a^{8}+\frac{11883953}{79897280}a^{7}+\frac{3126927}{99871600}a^{6}+\frac{178133653}{599229600}a^{5}+\frac{24578371}{1198459200}a^{4}-\frac{403667}{239691840}a^{3}-\frac{33021239}{79897280}a^{2}-\frac{13590957}{99871600}a+\frac{137154177}{399486400}$ 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:  $C_{4}$, which has order $4$
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:  $C_{4}$, which has order $4$
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:  $5$
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:   $\frac{40647}{23499200}a^{11}-\frac{74417}{70497600}a^{10}-\frac{800417}{70497600}a^{9}-\frac{135887}{11749600}a^{8}+\frac{32961}{939968}a^{7}+\frac{298027}{5874800}a^{6}+\frac{724531}{11749600}a^{5}-\frac{329089}{70497600}a^{4}-\frac{624431}{14099520}a^{3}-\frac{540963}{4699840}a^{2}-\frac{1501377}{5874800}a-\frac{28493163}{23499200}$, $\frac{40647}{23499200}a^{11}-\frac{74417}{70497600}a^{10}-\frac{800417}{70497600}a^{9}-\frac{135887}{11749600}a^{8}+\frac{32961}{939968}a^{7}+\frac{298027}{5874800}a^{6}+\frac{724531}{11749600}a^{5}-\frac{329089}{70497600}a^{4}-\frac{624431}{14099520}a^{3}-\frac{540963}{4699840}a^{2}-\frac{1501377}{5874800}a-\frac{4993963}{23499200}$, $\frac{1931397}{399486400}a^{11}+\frac{581053}{1198459200}a^{10}-\frac{36195347}{1198459200}a^{9}-\frac{13167117}{199743200}a^{8}+\frac{8079263}{79897280}a^{7}+\frac{26735857}{99871600}a^{6}+\frac{32264041}{199743200}a^{5}-\frac{331720339}{1198459200}a^{4}+\frac{22917283}{239691840}a^{3}-\frac{16383609}{79897280}a^{2}-\frac{104322587}{99871600}a-\frac{1175994993}{399486400}$, $\frac{461963}{239691840}a^{11}+\frac{1375721}{239691840}a^{10}-\frac{366319}{239691840}a^{9}-\frac{2510269}{39948640}a^{8}-\frac{4430481}{79897280}a^{7}+\frac{2097569}{19974320}a^{6}+\frac{2257973}{6307680}a^{5}+\frac{55592833}{239691840}a^{4}+\frac{125958067}{239691840}a^{3}+\frac{34203327}{79897280}a^{2}+\frac{6555603}{19974320}a-\frac{8227823}{4205120}$, $\frac{252041}{23499200}a^{11}+\frac{820529}{70497600}a^{10}-\frac{1330757}{23499200}a^{9}-\frac{2088281}{11749600}a^{8}+\frac{240507}{4699840}a^{7}+\frac{3087901}{5874800}a^{6}+\frac{7973133}{11749600}a^{5}+\frac{23292433}{70497600}a^{4}+\frac{3245861}{4699840}a^{3}+\frac{3313427}{4699840}a^{2}+\frac{1348229}{5874800}a-\frac{43019509}{23499200}$ 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:  \( 538.58951047 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 538.58951047 \cdot 4}{2\cdot\sqrt{2025170913606201}}\cr\approx \mathstrut & 1.472774500 \end{aligned}\]

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 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327) 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 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327, 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 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327); 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^12 - 6*x^10 - 11*x^9 + 21*x^8 + 39*x^7 + 14*x^6 - 27*x^5 + 72*x^4 + 20*x^3 - 129*x^2 - 297*x + 327); 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()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 12
The 4 conjugacy class representatives for $A_4$
Character table for $A_4$

Intermediate fields

\(\Q(\zeta_{9})^+\), 4.0.29241.1 x4, 6.2.2368521.1 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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 4 sibling: 4.0.29241.1
Degree 6 sibling: 6.2.2368521.1
Minimal sibling: 4.0.29241.1

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 ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ R ${\href{/padicField/23.3.0.1}{3} }^{4}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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$$$[2]$$
\(19\) Copy content Toggle raw display 19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_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)