Properties

Label 12.4.1339147769319424.1
Degree $12$
Signature $[4, 4]$
Discriminant $1.339\times 10^{15}$
Root discriminant \(18.22\)
Ramified primes $2,83$
Class number $1$
Class group trivial
Galois group $C_2 \times S_4$ (as 12T23)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4)
 
gp: K = bnfinit(y^12 - 8*y^10 + 10*y^8 - 55*y^6 - 7*y^4 - 12*y^2 + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4)
 

\( x^{12} - 8x^{10} + 10x^{8} - 55x^{6} - 7x^{4} - 12x^{2} + 4 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1339147769319424\) \(\medspace = 2^{12}\cdot 83^{6}\) 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:  \(18.22\)
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:  $2\cdot 83^{1/2}\approx 18.2208671582886$
Ramified primes:   \(2\), \(83\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{34228}a^{10}-\frac{1}{2}a^{9}-\frac{3957}{17114}a^{8}-\frac{345}{17114}a^{6}+\frac{12833}{34228}a^{4}-\frac{1}{2}a^{3}-\frac{5913}{34228}a^{2}-\frac{1}{2}a-\frac{3641}{17114}$, $\frac{1}{68456}a^{11}-\frac{3957}{34228}a^{9}-\frac{1}{2}a^{8}-\frac{345}{34228}a^{7}-\frac{21395}{68456}a^{5}-\frac{5913}{68456}a^{3}-\frac{1}{2}a^{2}-\frac{3641}{34228}a-\frac{1}{2}$ 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:  $7$
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:   $\frac{6507}{34228}a^{11}-\frac{25857}{17114}a^{9}+\frac{31247}{17114}a^{7}-\frac{354269}{34228}a^{5}-\frac{72075}{34228}a^{3}-\frac{40439}{17114}a$, $\frac{7957}{68456}a^{11}-\frac{1025}{17114}a^{10}-\frac{30317}{34228}a^{9}+\frac{8371}{17114}a^{8}+\frac{27303}{34228}a^{7}-\frac{5769}{8557}a^{6}-\frac{400679}{68456}a^{5}+\frac{58183}{17114}a^{4}-\frac{225837}{68456}a^{3}-\frac{3044}{8557}a^{2}-\frac{48777}{34228}a+\frac{10903}{17114}$, $\frac{5057}{68456}a^{11}-\frac{1473}{17114}a^{10}-\frac{21397}{34228}a^{9}+\frac{11245}{17114}a^{8}+\frac{35191}{34228}a^{7}-\frac{5235}{8557}a^{6}-\frac{307859}{68456}a^{5}+\frac{76417}{17114}a^{4}+\frac{81687}{68456}a^{3}+\frac{20804}{8557}a^{2}-\frac{32101}{34228}a+\frac{4465}{17114}$, $\frac{7957}{68456}a^{11}-\frac{651}{8557}a^{10}-\frac{30317}{34228}a^{9}+\frac{9957}{17114}a^{8}+\frac{27303}{34228}a^{7}-\frac{4331}{8557}a^{6}-\frac{400679}{68456}a^{5}+\frac{31577}{8557}a^{4}-\frac{225837}{68456}a^{3}+\frac{23097}{17114}a^{2}-\frac{48777}{34228}a+\frac{8565}{17114}$, $\frac{7395}{68456}a^{11}-\frac{374}{8557}a^{10}-\frac{31303}{34228}a^{9}+\frac{6785}{17114}a^{8}+\frac{50053}{34228}a^{7}-\frac{7207}{8557}a^{6}-\frac{424945}{68456}a^{5}+\frac{26606}{8557}a^{4}+\frac{85205}{68456}a^{3}-\frac{35273}{17114}a^{2}+\frac{12241}{34228}a+\frac{13241}{17114}$, $\frac{7957}{68456}a^{11}+\frac{1473}{17114}a^{10}-\frac{30317}{34228}a^{9}-\frac{11245}{17114}a^{8}+\frac{27303}{34228}a^{7}+\frac{5235}{8557}a^{6}-\frac{400679}{68456}a^{5}-\frac{76417}{17114}a^{4}-\frac{225837}{68456}a^{3}-\frac{20804}{8557}a^{2}-\frac{83005}{34228}a-\frac{21579}{17114}$, $\frac{7395}{68456}a^{11}+\frac{374}{8557}a^{10}-\frac{31303}{34228}a^{9}-\frac{6785}{17114}a^{8}+\frac{50053}{34228}a^{7}+\frac{7207}{8557}a^{6}-\frac{424945}{68456}a^{5}-\frac{26606}{8557}a^{4}+\frac{85205}{68456}a^{3}+\frac{35273}{17114}a^{2}+\frac{12241}{34228}a-\frac{13241}{17114}$ 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:  \( 766.771216361 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 766.771216361 \cdot 1}{2\cdot\sqrt{1339147769319424}}\cr\approx \mathstrut & 0.261252834418 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4)
 
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 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4, 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 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4);
 
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 - 8*x^10 + 10*x^8 - 55*x^6 - 7*x^4 - 12*x^2 + 4);
 
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 12T23):

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

Intermediate fields

\(\Q(\sqrt{83}) \), 3.1.83.1, 6.2.36594368.1, 6.2.2287148.1, 6.2.110224.1

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

Degree 6 siblings: 6.2.2287148.1, 6.0.27556.1
Degree 8 siblings: 8.4.12149330176.1, 8.0.1763584.1
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 6.0.27556.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

# 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 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(83\) Copy content Toggle raw display 83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.1$x^{2} + 166$$2$$1$$1$$C_2$$[\ ]_{2}$
83.4.2.1$x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$