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Properties

Label 12.0.2304866385432576.5
Degree $12$
Signature $[0, 6]$
Discriminant $2.305\times 10^{15}$
Root discriminant \(19.06\)
Ramified primes $2,3,7$
Class number $2$
Class group [2]
Galois group $C_2 \times S_4$ (as 12T24)

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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 - 3*x^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247)
 
Copy content gp:K = bnfinit(y^12 - 3*y^10 - 8*y^9 + 12*y^8 - 30*y^7 + 129*y^6 + 24*y^5 - 546*y^4 + 466*y^3 + 381*y^2 - 672*y + 247, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247)
 

\( x^{12} - 3 x^{10} - 8 x^{9} + 12 x^{8} - 30 x^{7} + 129 x^{6} + 24 x^{5} - 546 x^{4} + 466 x^{3} + \cdots + 247 \) 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:   \(2304866385432576\) \(\medspace = 2^{12}\cdot 3^{14}\cdot 7^{6}\) 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:  \(19.06\)
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:  $2^{7/6}3^{7/6}7^{1/2}\approx 21.398934305093306$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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.
Maximal CM subfield:  6.0.1714608.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{9}$, $\frac{1}{9}a^{10}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{33327}a^{11}-\frac{526}{33327}a^{10}-\frac{1052}{33327}a^{9}-\frac{234}{3703}a^{8}-\frac{348}{3703}a^{7}+\frac{1090}{11109}a^{6}-\frac{3035}{11109}a^{5}+\frac{425}{11109}a^{4}+\frac{717}{3703}a^{3}-\frac{5567}{33327}a^{2}-\frac{15262}{33327}a+\frac{6442}{33327}$ 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:  $C_{2}$, which has order $2$
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:  $5$
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:   \( \frac{466}{529} a^{11} + \frac{2078}{1587} a^{10} - \frac{378}{529} a^{9} - \frac{4333}{529} a^{8} - \frac{2648}{1587} a^{7} - \frac{45652}{1587} a^{6} + \frac{37738}{529} a^{5} + \frac{202856}{1587} a^{4} - \frac{460502}{1587} a^{3} - \frac{14818}{529} a^{2} + \frac{156368}{529} a - \frac{223018}{1587} \)  (order $6$) 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:   $\frac{6277}{33327}a^{11}-\frac{2329}{33327}a^{10}-\frac{34282}{33327}a^{9}-\frac{25805}{11109}a^{8}+\frac{27043}{11109}a^{7}-\frac{9045}{3703}a^{6}+\frac{338213}{11109}a^{5}+\frac{157091}{11109}a^{4}-\frac{1424966}{11109}a^{3}+\frac{1382371}{33327}a^{2}+\frac{4236971}{33327}a-\frac{2762764}{33327}$, $\frac{4658}{1449}a^{11}+\frac{6913}{1449}a^{10}-\frac{3884}{1449}a^{9}-\frac{4832}{161}a^{8}-\frac{1002}{161}a^{7}-\frac{50630}{483}a^{6}+\frac{41933}{161}a^{5}+\frac{75507}{161}a^{4}-\frac{513101}{483}a^{3}-\frac{146614}{1449}a^{2}+\frac{1575505}{1449}a-\frac{759137}{1449}$, $\frac{10471}{33327}a^{11}+\frac{24536}{33327}a^{10}+\frac{1338}{3703}a^{9}-\frac{9937}{3703}a^{8}-\frac{11265}{3703}a^{7}-\frac{47891}{3703}a^{6}+\frac{162593}{11109}a^{5}+\frac{658303}{11109}a^{4}-\frac{568787}{11109}a^{3}-\frac{1858337}{33327}a^{2}+\frac{1783112}{33327}a-\frac{38153}{11109}$, $\frac{57788}{33327}a^{11}+\frac{27635}{11109}a^{10}-\frac{18790}{11109}a^{9}-\frac{182258}{11109}a^{8}-\frac{10340}{3703}a^{7}-\frac{205837}{3703}a^{6}+\frac{1602008}{11109}a^{5}+\frac{2789963}{11109}a^{4}-\frac{6558329}{11109}a^{3}-\frac{1511089}{33327}a^{2}+\frac{6721468}{11109}a-\frac{1097289}{3703}$, $\frac{177}{529}a^{11}-\frac{523}{1587}a^{10}-\frac{3691}{1587}a^{9}-\frac{6857}{1587}a^{8}+\frac{10135}{1587}a^{7}-\frac{2453}{1587}a^{6}+\frac{100286}{1587}a^{5}+\frac{25826}{1587}a^{4}-\frac{430396}{1587}a^{3}+\frac{158146}{1587}a^{2}+\frac{143588}{529}a-\frac{96039}{529}$ 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:  \( 1277.50517362 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1277.50517362 \cdot 2}{6\cdot\sqrt{2304866385432576}}\cr\approx \mathstrut & 0.545755028568 \end{aligned}\]

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^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247) 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^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247, 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^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247); 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^10 - 8*x^9 + 12*x^8 - 30*x^7 + 129*x^6 + 24*x^5 - 546*x^4 + 466*x^3 + 381*x^2 - 672*x + 247); 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

$C_2\times S_4$ (as 12T24):

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 48
The 10 conjugacy class representatives for $C_2 \times S_4$
Character table for $C_2 \times S_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.1714608.1, 6.4.16003008.3, 6.2.48009024.3

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 6 siblings: 6.4.6858432.1, some data not computed
Degree 8 siblings: 8.0.448084224.9, 8.0.82301184.1
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 6.4.6858432.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.2.6.12a1.3$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 23 x^{2} + 8 x + 5$$6$$2$$12$$S_4$$$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.2.6.14a2.1$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$$6$$2$$14$$D_6$$$[\frac{3}{2}]_{2}^{2}$$
\(7\) Copy content Toggle raw display 7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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