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Properties

Label 12.2.166168191041536.4
Degree $12$
Signature $[2, 5]$
Discriminant $-1.662\times 10^{14}$
Root discriminant \(15.31\)
Ramified primes $2,19$
Class number $1$
Class group trivial
Galois group $S_4$ (as 12T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2)
 
gp: K = bnfinit(y^12 + 4*y^10 - 12*y^9 - 6*y^8 - 4*y^7 - 8*y^6 - 4*y^5 + 4*y^4 + 4*y^3 + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2)
 

\( x^{12} + 4x^{10} - 12x^{9} - 6x^{8} - 4x^{7} - 8x^{6} - 4x^{5} + 4x^{4} + 4x^{3} + 2 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-166168191041536\) \(\medspace = -\,2^{26}\cdot 19^{5}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(15.31\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{13/6}19^{1/2}\approx 19.570794546160904$
Ramified primes:   \(2\), \(19\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-19}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2132489}a^{11}+\frac{320877}{2132489}a^{10}-\frac{917254}{2132489}a^{9}+\frac{420010}{2132489}a^{8}+\frac{376453}{2132489}a^{7}+\frac{269872}{2132489}a^{6}-\frac{395576}{2132489}a^{5}+\frac{902591}{2132489}a^{4}+\frac{963754}{2132489}a^{3}-\frac{665051}{2132489}a^{2}+\frac{736992}{2132489}a-\frac{718160}{2132489}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{9550}{2132489}a^{11}-\frac{11343}{2132489}a^{10}+\frac{489112}{2132489}a^{9}-\frac{116309}{2132489}a^{8}+\frac{1882185}{2132489}a^{7}-\frac{5166579}{2132489}a^{6}-\frac{3245270}{2132489}a^{5}-\frac{1908977}{2132489}a^{4}-\frac{4236802}{2132489}a^{3}-\frac{2817297}{2132489}a^{2}+\frac{1059900}{2132489}a+\frac{1789113}{2132489}$, $\frac{33908}{2132489}a^{11}+\frac{338438}{2132489}a^{10}+\frac{103433}{2132489}a^{9}+\frac{937538}{2132489}a^{8}-\frac{4575808}{2132489}a^{7}-\frac{1823012}{2132489}a^{6}-\frac{1967687}{2132489}a^{5}-\frac{426500}{2132489}a^{4}+\frac{709196}{2132489}a^{3}+\frac{2654356}{2132489}a^{2}+\frac{1418634}{2132489}a-\frac{477389}{2132489}$, $\frac{11343}{2132489}a^{11}-\frac{450912}{2132489}a^{10}+\frac{1709}{2132489}a^{9}-\frac{1939485}{2132489}a^{8}+\frac{5128379}{2132489}a^{7}+\frac{3168870}{2132489}a^{6}+\frac{1870777}{2132489}a^{5}+\frac{4275002}{2132489}a^{4}+\frac{2855497}{2132489}a^{3}-\frac{1059900}{2132489}a^{2}-\frac{3921602}{2132489}a-\frac{2113389}{2132489}$, $\frac{720539}{2132489}a^{11}-\frac{64677}{2132489}a^{10}+\frac{2903375}{2132489}a^{9}-\frac{9253490}{2132489}a^{8}-\frac{3532633}{2132489}a^{7}-\frac{4105924}{2132489}a^{6}-\frac{1588213}{2132489}a^{5}+\frac{448752}{2132489}a^{4}+\frac{5022913}{2132489}a^{3}+\frac{2818168}{2132489}a^{2}+\frac{3332886}{2132489}a-\frac{3169945}{2132489}$, $\frac{76121}{2132489}a^{11}-\frac{50889}{2132489}a^{10}-\frac{336896}{2132489}a^{9}-\frac{826367}{2132489}a^{8}-\frac{2540858}{2132489}a^{7}+\frac{9189931}{2132489}a^{6}-\frac{896016}{2132489}a^{5}+\frac{3731398}{2132489}a^{4}+\frac{31656}{2132489}a^{3}+\frac{3074178}{2132489}a^{2}-\frac{952580}{2132489}a-\frac{701845}{2132489}$, $\frac{88709}{2132489}a^{11}+\frac{214621}{2132489}a^{10}+\frac{697687}{2132489}a^{9}-\frac{180718}{2132489}a^{8}-\frac{2141052}{2132489}a^{7}-\frac{5643733}{2132489}a^{6}-\frac{5309867}{2132489}a^{5}+\frac{1513025}{2132489}a^{4}+\frac{37087}{2132489}a^{3}-\frac{700974}{2132489}a^{2}+\frac{2108055}{2132489}a+\frac{853435}{2132489}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 313.364356743 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 313.364356743 \cdot 1}{2\cdot\sqrt{166168191041536}}\cr\approx \mathstrut & 0.476107461864 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 4*x^10 - 12*x^9 - 6*x^8 - 4*x^7 - 8*x^6 - 4*x^5 + 4*x^4 + 4*x^3 + 2);
 
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

$S_4$ (as 12T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

3.1.76.1, 4.2.4864.1 x2, 6.2.369664.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 24
Degree 4 sibling: 4.2.4864.1
Degree 6 siblings: 6.2.369664.1, 6.0.7023616.2
Degree 8 sibling: 8.0.8540717056.5
Degree 12 sibling: 12.0.49331181715456.7
Minimal sibling: 4.2.4864.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ R ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.12.26.65$x^{12} + 4 x^{3} + 2$$12$$1$$26$$S_4$$[8/3, 8/3]_{3}^{2}$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$