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Properties

Label 12.0.48631121859501.1
Degree $12$
Signature $[0, 6]$
Discriminant $4.863\times 10^{13}$
Root discriminant \(13.82\)
Ramified primes $3,23$
Class number $1$
Class group trivial
Galois group $(C_6\times C_2):C_2$ (as 12T15)

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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 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173)
 
Copy content gp:K = bnfinit(y^12 - y^11 - 3*y^10 + 8*y^9 - y^8 - 42*y^7 + 19*y^6 + 93*y^5 - 105*y^4 - 181*y^3 + 229*y^2 + 435*y + 173, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173)
 

\( x^{12} - x^{11} - 3 x^{10} + 8 x^{9} - x^{8} - 42 x^{7} + 19 x^{6} + 93 x^{5} - 105 x^{4} - 181 x^{3} + \cdots + 173 \) 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:   \(48631121859501\) \(\medspace = 3^{3}\cdot 23^{9}\) 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:  \(13.82\)
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:  $3^{1/2}23^{3/4}\approx 18.19099708945548$
Ramified primes:   \(3\), \(23\) 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:  \(\Q(\sqrt{69}) \)
$\Aut(K/\Q)$:   $C_6$
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.
Maximal CM subfield:  \(\Q(\sqrt{-23}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{30}a^{9}+\frac{7}{30}a^{8}-\frac{1}{3}a^{7}-\frac{1}{15}a^{6}-\frac{1}{3}a^{5}-\frac{1}{5}a^{4}-\frac{1}{30}a^{3}+\frac{1}{10}a^{2}+\frac{1}{5}a+\frac{7}{30}$, $\frac{1}{150}a^{10}+\frac{1}{150}a^{9}-\frac{26}{75}a^{8}+\frac{14}{75}a^{7}+\frac{16}{75}a^{6}-\frac{1}{25}a^{5}-\frac{1}{6}a^{4}-\frac{17}{50}a^{3}+\frac{3}{25}a^{2}+\frac{31}{150}a+\frac{8}{25}$, $\frac{1}{3927750}a^{11}+\frac{1129}{785550}a^{10}+\frac{9482}{654625}a^{9}-\frac{20047}{130925}a^{8}-\frac{90781}{654625}a^{7}-\frac{850949}{1963875}a^{6}+\frac{193361}{3927750}a^{5}-\frac{460051}{3927750}a^{4}+\frac{126429}{654625}a^{3}+\frac{1399273}{3927750}a^{2}+\frac{206006}{1963875}a+\frac{99902}{654625}$ 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:  $5$
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:   $\frac{49709}{1963875}a^{11}-\frac{2326}{26185}a^{10}+\frac{121507}{1309250}a^{9}+\frac{111323}{785550}a^{8}-\frac{1003889}{1963875}a^{7}-\frac{86464}{654625}a^{6}+\frac{1178868}{654625}a^{5}-\frac{3016684}{1963875}a^{4}-\frac{10218233}{3927750}a^{3}+\frac{13934959}{3927750}a^{2}+\frac{5396128}{1963875}a-\frac{5320039}{3927750}$, $\frac{116852}{1963875}a^{11}-\frac{46492}{392775}a^{10}-\frac{200407}{3927750}a^{9}+\frac{391297}{785550}a^{8}-\frac{1078222}{1963875}a^{7}-\frac{3620221}{1963875}a^{6}+\frac{1824899}{654625}a^{5}+\frac{5162548}{1963875}a^{4}-\frac{31913279}{3927750}a^{3}-\frac{10731683}{3927750}a^{2}+\frac{29488724}{1963875}a+\frac{44328923}{3927750}$, $\frac{13313}{157110}a^{11}-\frac{43829}{261850}a^{10}-\frac{24479}{261850}a^{9}+\frac{588589}{785550}a^{8}-\frac{316838}{392775}a^{7}-\frac{360274}{130925}a^{6}+\frac{1091229}{261850}a^{5}+\frac{630503}{157110}a^{4}-\frac{9887573}{785550}a^{3}-\frac{1371418}{392775}a^{2}+\frac{9141964}{392775}a+\frac{12495419}{785550}$, $\frac{223202}{1963875}a^{11}-\frac{9281}{52370}a^{10}-\frac{505081}{1963875}a^{9}+\frac{862159}{785550}a^{8}-\frac{1505992}{1963875}a^{7}-\frac{8652751}{1963875}a^{6}+\frac{9321637}{1963875}a^{5}+\frac{32425021}{3927750}a^{4}-\frac{11264054}{654625}a^{3}-\frac{44242373}{3927750}a^{2}+\frac{135139043}{3927750}a+\frac{39061961}{1309250}$, $\frac{49481}{3927750}a^{11}-\frac{98533}{785550}a^{10}+\frac{1206767}{3927750}a^{9}-\frac{100703}{785550}a^{8}-\frac{1762348}{1963875}a^{7}+\frac{973157}{654625}a^{6}+\frac{7971101}{3927750}a^{5}-\frac{26038781}{3927750}a^{4}+\frac{6946129}{3927750}a^{3}+\frac{27372904}{1963875}a^{2}-\frac{15703469}{1963875}a-\frac{72331633}{3927750}$ 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:  \( 67.2184535757 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 67.2184535757 \cdot 1}{2\cdot\sqrt{48631121859501}}\cr\approx \mathstrut & 0.296538159150 \end{aligned}\]

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 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173) 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 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173, 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 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173); 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 - x^11 - 3*x^10 + 8*x^9 - x^8 - 42*x^7 + 19*x^6 + 93*x^5 - 105*x^4 - 181*x^3 + 229*x^2 + 435*x + 173); 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

$C_3:D_4$ (as 12T15):

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 24
The 9 conjugacy class representatives for $(C_6\times C_2):C_2$
Character table for $(C_6\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 4.0.36501.1, 6.0.12167.1

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

Galois closure: deg 24
Degree 12 sibling: 12.2.1313040290206527.1
Minimal sibling: 12.2.1313040290206527.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ R ${\href{/padicField/5.2.0.1}{2} }^{6}$ ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ R ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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 3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.3.2.3a1.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(23\) Copy content Toggle raw display 23.1.4.3a1.1$x^{4} + 23$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
23.1.4.3a1.1$x^{4} + 23$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
23.1.4.3a1.1$x^{4} + 23$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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