Properties

Label 12.4.256000000000000.1
Degree $12$
Signature $(4, 4)$
Discriminant $2.560\times 10^{14}$
Root discriminant \(15.87\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $S_5$ (as 12T74)

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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 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5)
 
Copy content gp:K = bnfinit(y^12 - 2*y^11 - 4*y^10 + 10*y^9 - 5*y^8 - 20*y^7 + 40*y^6 - 25*y^4 + 30*y^3 - 10*y + 5, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5)
 

\( x^{12} - 2x^{11} - 4x^{10} + 10x^{9} - 5x^{8} - 20x^{7} + 40x^{6} - 25x^{4} + 30x^{3} - 10x + 5 \) 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:  $(4, 4)$
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:   \(256000000000000\) \(\medspace = 2^{20}\cdot 5^{12}\) 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:  \(15.87\)
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:  $2^{13/6}5^{23/20}\approx 28.57900880593445$
Ramified primes:   \(2\), \(5\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{50468}a^{11}-\frac{137}{12617}a^{10}+\frac{9013}{50468}a^{9}-\frac{229}{25234}a^{8}+\frac{235}{1147}a^{7}+\frac{1689}{12617}a^{6}+\frac{10331}{25234}a^{5}+\frac{5845}{12617}a^{4}+\frac{2899}{50468}a^{3}-\frac{4579}{12617}a^{2}+\frac{20489}{50468}a-\frac{4171}{25234}$ 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 $1$
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:  Trivial group, which has order $1$
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:  $7$
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:   $\frac{67}{407}a^{11}-\frac{751}{1628}a^{10}-\frac{117}{407}a^{9}+\frac{3019}{1628}a^{8}-\frac{173}{74}a^{7}-\frac{1085}{814}a^{6}+\frac{6399}{814}a^{5}-\frac{2525}{407}a^{4}+\frac{595}{814}a^{3}+\frac{7465}{1628}a^{2}-\frac{2945}{814}a-\frac{819}{1628}$, $\frac{67}{407}a^{11}-\frac{751}{1628}a^{10}-\frac{117}{407}a^{9}+\frac{3019}{1628}a^{8}-\frac{173}{74}a^{7}-\frac{1085}{814}a^{6}+\frac{6399}{814}a^{5}-\frac{2525}{407}a^{4}+\frac{595}{814}a^{3}+\frac{7465}{1628}a^{2}-\frac{2131}{814}a+\frac{809}{1628}$, $\frac{8225}{50468}a^{11}-\frac{3912}{12617}a^{10}-\frac{30801}{50468}a^{9}+\frac{34259}{25234}a^{8}-\frac{967}{1147}a^{7}-\frac{61669}{25234}a^{6}+\frac{135767}{25234}a^{5}-\frac{3907}{25234}a^{4}-\frac{27089}{50468}a^{3}+\frac{62025}{25234}a^{2}+\frac{34607}{50468}a-\frac{426}{12617}$, $\frac{20325}{50468}a^{11}-\frac{30187}{25234}a^{10}-\frac{34849}{50468}a^{9}+\frac{63709}{12617}a^{8}-\frac{6615}{1147}a^{7}-\frac{52400}{12617}a^{6}+\frac{510141}{25234}a^{5}-\frac{191302}{12617}a^{4}-\frac{74917}{50468}a^{3}+\frac{254591}{25234}a^{2}-\frac{351053}{50468}a+\frac{11641}{12617}$, $\frac{3225}{12617}a^{11}-\frac{41531}{50468}a^{10}-\frac{2643}{12617}a^{9}+\frac{160577}{50468}a^{8}-\frac{10365}{2294}a^{7}-\frac{14076}{12617}a^{6}+\frac{324571}{25234}a^{5}-\frac{338043}{25234}a^{4}+\frac{88475}{25234}a^{3}+\frac{254499}{50468}a^{2}-\frac{160429}{25234}a+\frac{99529}{50468}$, $\frac{2831}{50468}a^{11}+\frac{503}{50468}a^{10}-\frac{21005}{50468}a^{9}+\frac{2953}{50468}a^{8}+\frac{1197}{2294}a^{7}-\frac{38419}{25234}a^{6}+\frac{6736}{12617}a^{5}+\frac{44159}{12617}a^{4}-\frac{44449}{50468}a^{3}+\frac{66359}{50468}a^{2}+\frac{92329}{50468}a+\frac{15439}{50468}$, $\frac{987}{2294}a^{11}-\frac{893}{1147}a^{10}-\frac{1871}{1147}a^{9}+\frac{7901}{2294}a^{8}-\frac{2507}{1147}a^{7}-\frac{15397}{2294}a^{6}+\frac{15925}{1147}a^{5}-\frac{433}{2294}a^{4}-\frac{3893}{2294}a^{3}+\frac{18327}{2294}a^{2}+\frac{1090}{1147}a-\frac{194}{1147}$ 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:  \( 506.047684117 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 4 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 506.047684117 \cdot 1}{2\cdot\sqrt{256000000000000}}\cr\approx \mathstrut & 0.394349159438 \end{aligned}\]

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 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5) 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 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5, 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 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5); 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 - 2*x^11 - 4*x^10 + 10*x^9 - 5*x^8 - 20*x^7 + 40*x^6 - 25*x^4 + 30*x^3 - 10*x + 5); 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

$S_5$ (as 12T74):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.2.3200000.1

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 5 sibling: 5.1.800000.1
Degree 6 sibling: 6.2.3200000.1
Degree 10 siblings: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.1.800000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.6.20a1.45$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 231 x^{6} + 228 x^{5} + 180 x^{4} + 110 x^{3} + 55 x^{2} + 20 x + 9$$6$$2$$20$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.10.11a2.2$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$$[\frac{5}{4}]_{4}$$

Spectrum of ring of integers

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