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Properties

Label 12.12.429...912.1
Degree $12$
Signature $[12, 0]$
Discriminant $4.297\times 10^{21}$
Root discriminant \(63.50\)
Ramified primes $2,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_4:C_4$ (as 12T30)

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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 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8)
 
Copy content gp:K = bnfinit(y^12 - 44*y^10 + 574*y^8 - 2128*y^6 + 1148*y^4 - 176*y^2 + 8, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8)
 

\( x^{12} - 44x^{10} + 574x^{8} - 2128x^{6} + 1148x^{4} - 176x^{2} + 8 \) 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:   \(4297083967217611046912\) \(\medspace = 2^{33}\cdot 29^{8}\) 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:  \(63.50\)
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:  $2^{27/8}29^{2/3}\approx 97.92830419169753$
Ramified primes:   \(2\), \(29\) 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:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{192}a^{8}-\frac{5}{48}a^{6}-\frac{7}{24}a^{2}-\frac{1}{48}$, $\frac{1}{192}a^{9}-\frac{5}{48}a^{7}-\frac{7}{24}a^{3}-\frac{1}{48}a$, $\frac{1}{384}a^{10}+\frac{1}{12}a^{6}+\frac{5}{48}a^{4}-\frac{17}{96}a^{2}+\frac{7}{24}$, $\frac{1}{384}a^{11}+\frac{1}{12}a^{7}+\frac{5}{48}a^{5}-\frac{17}{96}a^{3}+\frac{7}{24}a$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $\frac{1}{8}a^{10}-\frac{527}{96}a^{8}+\frac{1711}{24}a^{6}-260a^{4}+\frac{1451}{12}a^{2}-\frac{217}{24}$, $\frac{313}{384}a^{11}-\frac{1}{4}a^{10}-\frac{3437}{96}a^{9}+\frac{527}{48}a^{8}+\frac{3721}{8}a^{7}-\frac{1711}{12}a^{6}-\frac{81547}{48}a^{5}+520a^{4}+\frac{77327}{96}a^{3}-\frac{1451}{6}a^{2}-\frac{155}{2}a+\frac{277}{12}$, $\frac{313}{384}a^{11}+\frac{1}{4}a^{10}-\frac{3437}{96}a^{9}-\frac{527}{48}a^{8}+\frac{3721}{8}a^{7}+\frac{1711}{12}a^{6}-\frac{81547}{48}a^{5}-520a^{4}+\frac{77327}{96}a^{3}+\frac{1451}{6}a^{2}-\frac{155}{2}a-\frac{277}{12}$, $\frac{1}{192}a^{10}-\frac{11}{48}a^{8}+3a^{6}-\frac{271}{24}a^{4}+\frac{287}{48}a^{2}-\frac{1}{2}$, $\frac{215}{384}a^{11}+\frac{1}{4}a^{10}-\frac{295}{12}a^{9}-\frac{527}{48}a^{8}+\frac{3829}{12}a^{7}+\frac{1711}{12}a^{6}-\frac{55745}{48}a^{5}-520a^{4}+\frac{16947}{32}a^{3}+\frac{1451}{6}a^{2}-\frac{803}{24}a-\frac{205}{12}$, $\frac{241}{384}a^{11}+\frac{109}{192}a^{10}-\frac{5287}{192}a^{9}-\frac{1595}{64}a^{8}+\frac{5709}{16}a^{7}+\frac{15521}{48}a^{6}-\frac{61963}{48}a^{5}-\frac{28213}{24}a^{4}+\frac{53411}{96}a^{3}+\frac{25561}{48}a^{2}-\frac{457}{16}a-\frac{1979}{48}$, $\frac{95}{384}a^{11}+\frac{11}{96}a^{10}-\frac{2087}{192}a^{9}-\frac{161}{32}a^{8}+\frac{2261}{16}a^{7}+\frac{1567}{24}a^{6}-\frac{24833}{48}a^{5}-\frac{2849}{12}a^{4}+\frac{23989}{96}a^{3}+\frac{2591}{24}a^{2}-\frac{389}{16}a-\frac{241}{24}$, $\frac{95}{384}a^{11}-\frac{11}{96}a^{10}-\frac{2087}{192}a^{9}+\frac{161}{32}a^{8}+\frac{2261}{16}a^{7}-\frac{1567}{24}a^{6}-\frac{24833}{48}a^{5}+\frac{2849}{12}a^{4}+\frac{23989}{96}a^{3}-\frac{2591}{24}a^{2}-\frac{389}{16}a+\frac{241}{24}$, $\frac{73}{384}a^{11}-\frac{5}{64}a^{10}-\frac{401}{48}a^{9}+\frac{659}{192}a^{8}+\frac{326}{3}a^{7}-\frac{2143}{48}a^{6}-\frac{19135}{48}a^{5}+\frac{1313}{8}a^{4}+\frac{6237}{32}a^{3}-\frac{3931}{48}a^{2}-\frac{331}{24}a+\frac{289}{48}$, $\frac{193}{96}a^{11}-\frac{57}{64}a^{10}-\frac{2117}{24}a^{9}+\frac{937}{24}a^{8}+1143a^{7}-\frac{3029}{6}a^{6}-\frac{49627}{12}a^{5}+\frac{14503}{8}a^{4}+\frac{42911}{24}a^{3}-\frac{34981}{48}a^{2}-112a+\frac{553}{12}$, $\frac{413}{384}a^{11}-\frac{47}{192}a^{10}-\frac{1509}{32}a^{9}+\frac{343}{32}a^{8}+\frac{14641}{24}a^{7}-\frac{3317}{24}a^{6}-\frac{105323}{48}a^{5}+\frac{11777}{24}a^{4}+\frac{86075}{96}a^{3}-\frac{8141}{48}a^{2}-\frac{821}{12}a+\frac{233}{24}$ 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:  \( 64178206.6169 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 64178206.6169 \cdot 1}{2\cdot\sqrt{4297083967217611046912}}\cr\approx \mathstrut & 2.00507494118 \end{aligned}\] (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 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8) 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 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8, 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 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8); 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 - 44*x^10 + 574*x^8 - 2128*x^6 + 1148*x^4 - 176*x^2 + 8); 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

$A_4:C_4$ (as 12T30):

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 solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.6728.1 x3, 6.6.362127872.1

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]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 12.12.68753343475481776750592.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ R ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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 2.1.4.11a1.13$x^{4} + 4 x^{2} + 8 x + 2$$4$$1$$11$$D_{4}$$$[2, 3, 4]$$
2.1.4.11a1.13$x^{4} + 4 x^{2} + 8 x + 2$$4$$1$$11$$D_{4}$$$[2, 3, 4]$$
2.1.4.11a1.9$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$$[3, 4]$$
\(29\) Copy content Toggle raw display 29.4.3.8a1.2$x^{12} + 6 x^{10} + 45 x^{9} + 18 x^{8} + 180 x^{7} + 707 x^{6} + 360 x^{5} + 1386 x^{4} + 3735 x^{3} + 1374 x^{2} + 180 x + 37$$3$$4$$8$$C_3 : C_4$$$[\ ]_{3}^{4}$$

Spectrum of ring of integers

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