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Properties

Label 12.12.2755304652953125.1
Degree $12$
Signature $[12, 0]$
Discriminant $2.755\times 10^{15}$
Root discriminant \(19.35\)
Ramified primes $5,7,13,181$
Class number $1$
Class group trivial
Galois group $C_2\wr C_6$ (as 12T134)

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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 - 3*x^11 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1)
 
Copy content gp:K = bnfinit(y^12 - 3*y^11 - 11*y^10 + 35*y^9 + 37*y^8 - 140*y^7 - 19*y^6 + 217*y^5 - 79*y^4 - 88*y^3 + 58*y^2 - 6*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1)
 

\( x^{12} - 3 x^{11} - 11 x^{10} + 35 x^{9} + 37 x^{8} - 140 x^{7} - 19 x^{6} + 217 x^{5} - 79 x^{4} + \cdots - 1 \) 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:   \(2755304652953125\) \(\medspace = 5^{6}\cdot 7^{8}\cdot 13^{2}\cdot 181\) 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:  \(19.35\)
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:  $5^{1/2}7^{2/3}13^{1/2}181^{1/2}\approx 396.91239279873855$
Ramified primes:   \(5\), \(7\), \(13\), \(181\) 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:  \(\Q(\sqrt{181}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{41}a^{11}+\frac{7}{41}a^{10}+\frac{18}{41}a^{9}+\frac{10}{41}a^{8}+\frac{14}{41}a^{7}-\frac{19}{41}a^{5}-\frac{14}{41}a^{4}-\frac{14}{41}a^{3}+\frac{18}{41}a^{2}-\frac{8}{41}a-\frac{4}{41}$ 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:  $C_{2}$, which has order $2$
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:   $2a^{11}-5a^{10}-24a^{9}+56a^{8}+99a^{7}-213a^{6}-145a^{5}+318a^{4}+23a^{3}-139a^{2}+30a+3$, $\frac{58}{41}a^{11}-\frac{127}{41}a^{10}-\frac{760}{41}a^{9}+\frac{1482}{41}a^{8}+\frac{3477}{41}a^{7}-144a^{6}-\frac{5981}{41}a^{5}+\frac{9274}{41}a^{4}+\frac{2345}{41}a^{3}-\frac{4163}{41}a^{2}+\frac{479}{41}a+\frac{55}{41}$, $\frac{27}{41}a^{11}-\frac{57}{41}a^{10}-\frac{375}{41}a^{9}+\frac{721}{41}a^{8}+\frac{1772}{41}a^{7}-76a^{6}-\frac{2973}{41}a^{5}+\frac{5239}{41}a^{4}+\frac{688}{41}a^{3}-\frac{2343}{41}a^{2}+\frac{563}{41}a+\frac{15}{41}$, $\frac{55}{41}a^{11}+\frac{148}{41}a^{10}+\frac{609}{41}a^{9}-\frac{1575}{41}a^{8}-\frac{2287}{41}a^{7}+137a^{6}+\frac{2972}{41}a^{5}-\frac{7799}{41}a^{4}-\frac{255}{41}a^{3}+\frac{3356}{41}a^{2}-\frac{667}{41}a-\frac{108}{41}$, $\frac{51}{41}a^{11}+\frac{135}{41}a^{10}+\frac{599}{41}a^{9}-\frac{1535}{41}a^{8}-\frac{2354}{41}a^{7}+145a^{6}+\frac{2937}{41}a^{5}-\frac{9003}{41}a^{4}+\frac{714}{41}a^{3}+\frac{3879}{41}a^{2}-\frac{1314}{41}a-\frac{42}{41}$, $\frac{76}{41}a^{11}+\frac{206}{41}a^{10}+\frac{887}{41}a^{9}-\frac{2359}{41}a^{8}-\frac{3483}{41}a^{7}+226a^{6}+\frac{4478}{41}a^{5}-\frac{14393}{41}a^{4}+\frac{654}{41}a^{3}+\frac{6668}{41}a^{2}-\frac{1770}{41}a-\frac{188}{41}$, $\frac{140}{41}a^{11}+\frac{332}{41}a^{10}+\frac{1744}{41}a^{9}-\frac{3778}{41}a^{8}-\frac{7536}{41}a^{7}+357a^{6}+\frac{11926}{41}a^{5}-\frac{22312}{41}a^{4}-\frac{3288}{41}a^{3}+\frac{9862}{41}a^{2}-\frac{1668}{41}a-\frac{178}{41}$, $\frac{113}{41}a^{11}+\frac{275}{41}a^{10}+\frac{1369}{41}a^{9}-\frac{3057}{41}a^{8}-\frac{5764}{41}a^{7}+281a^{6}+\frac{8953}{41}a^{5}-\frac{17073}{41}a^{4}-\frac{2600}{41}a^{3}+\frac{7519}{41}a^{2}-\frac{1105}{41}a-\frac{163}{41}$, $\frac{70}{41}a^{11}+\frac{166}{41}a^{10}+\frac{872}{41}a^{9}-\frac{1889}{41}a^{8}-\frac{3768}{41}a^{7}+178a^{6}+\frac{5963}{41}a^{5}-\frac{10992}{41}a^{4}-\frac{1603}{41}a^{3}+\frac{4685}{41}a^{2}-\frac{875}{41}a-\frac{89}{41}$, $\frac{23}{41}a^{11}+\frac{44}{41}a^{10}+\frac{324}{41}a^{9}-\frac{558}{41}a^{8}-\frac{1552}{41}a^{7}+59a^{6}+\frac{2692}{41}a^{5}-\frac{4106}{41}a^{4}-\frac{908}{41}a^{3}+\frac{1882}{41}a^{2}-\frac{267}{41}a-\frac{31}{41}$, $\frac{3}{41}a^{11}-\frac{21}{41}a^{10}+\frac{110}{41}a^{9}+\frac{216}{41}a^{8}-\frac{862}{41}a^{7}-19a^{6}+\frac{2394}{41}a^{5}+\frac{1108}{41}a^{4}-\frac{2377}{41}a^{3}-\frac{587}{41}a^{2}+\frac{680}{41}a+\frac{53}{41}$ 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:  \( 4164.646983587647 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 4164.646983587647 \cdot 1}{2\cdot\sqrt{2755304652953125}}\cr\approx \mathstrut & 0.162488763500577 \end{aligned}\]

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 - 3*x^11 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1) 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 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1, 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 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1); 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 - 11*x^10 + 35*x^9 + 37*x^8 - 140*x^7 - 19*x^6 + 217*x^5 - 79*x^4 - 88*x^3 + 58*x^2 - 6*x - 1); 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

$C_2\wr C_6$ (as 12T134):

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 384
The 28 conjugacy class representatives for $C_2\wr C_6$
Character table for $C_2\wr C_6$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 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]
 

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.6.0.1}{6} }^{2}$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ R ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(5\) Copy content Toggle raw display 5.6.2.6a1.2$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(7\) 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}$$$[\ ]_{3}^{4}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.2.2.2a1.1$x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
\(181\) Copy content Toggle raw display $\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
181.1.2.1a1.2$x^{2} + 362$$2$$1$$1$$C_2$$$[\ ]_{2}$$
181.2.1.0a1.1$x^{2} + 177 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
181.2.1.0a1.1$x^{2} + 177 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
181.2.1.0a1.1$x^{2} + 177 x + 2$$1$$2$$0$$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)