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Properties

Label 12.0.3176493481984.1
Degree $12$
Signature $[0, 6]$
Discriminant $3.176\times 10^{12}$
Root discriminant \(11.01\)
Ramified primes $2,59$
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 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^12 - 6*y^11 + 15*y^10 - 20*y^9 + 25*y^8 - 46*y^7 + 59*y^6 - 34*y^5 + 20*y^4 - 30*y^3 + 24*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*x + 1)
 

\( x^{12} - 6 x^{11} + 15 x^{10} - 20 x^{9} + 25 x^{8} - 46 x^{7} + 59 x^{6} - 34 x^{5} + 20 x^{4} + \cdots + 1 \) 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:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3176493481984\) \(\medspace = 2^{18}\cdot 59^{4}\) 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:  \(11.01\)
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:  $2^{3/2}59^{1/2}\approx 21.72556098240043$
Ramified primes:   \(2\), \(59\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{51}a^{11}+\frac{1}{17}a^{10}+\frac{8}{51}a^{9}+\frac{1}{51}a^{8}+\frac{5}{51}a^{6}+\frac{19}{51}a^{5}-\frac{16}{51}a^{4}-\frac{22}{51}a^{3}-\frac{7}{51}a^{2}-\frac{22}{51}a+\frac{5}{17}$ 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:  $5$
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:   $\frac{130}{51}a^{11}-\frac{244}{17}a^{10}+\frac{562}{17}a^{9}-\frac{2012}{51}a^{8}+\frac{152}{3}a^{7}-\frac{5113}{51}a^{6}+\frac{1968}{17}a^{5}-\frac{2488}{51}a^{4}+\frac{1934}{51}a^{3}-\frac{3256}{51}a^{2}+\frac{2036}{51}a-\frac{430}{51}$, $\frac{38}{51}a^{11}-\frac{64}{17}a^{10}+\frac{389}{51}a^{9}-\frac{404}{51}a^{8}+12a^{7}-\frac{1238}{51}a^{6}+\frac{1130}{51}a^{5}-\frac{112}{17}a^{4}+\frac{643}{51}a^{3}-\frac{623}{51}a^{2}+\frac{269}{51}a-\frac{76}{51}$, $\frac{76}{51}a^{11}-\frac{418}{51}a^{10}+\frac{316}{17}a^{9}-\frac{377}{17}a^{8}+\frac{88}{3}a^{7}-\frac{2884}{51}a^{6}+\frac{1082}{17}a^{5}-\frac{1454}{51}a^{4}+\frac{1184}{51}a^{3}-\frac{1637}{51}a^{2}+\frac{1201}{51}a-\frac{113}{17}$, $a-1$, $\frac{98}{51}a^{11}-\frac{556}{51}a^{10}+\frac{1277}{51}a^{9}-\frac{1483}{51}a^{8}+36a^{7}-\frac{3760}{51}a^{6}+\frac{1448}{17}a^{5}-\frac{1517}{51}a^{4}+\frac{358}{17}a^{3}-\frac{2437}{51}a^{2}+\frac{460}{17}a-\frac{196}{51}$ 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:  \( 17.0537538946 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 17.0537538946 \cdot 1}{2\cdot\sqrt{3176493481984}}\cr\approx \mathstrut & 0.294371134447 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^12 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 15*x^10 - 20*x^9 + 25*x^8 - 46*x^7 + 59*x^6 - 34*x^5 + 20*x^4 - 30*x^3 + 24*x^2 - 8*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

$C_2\times S_4$ (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

\(\Q(\sqrt{-2}) \), 3.1.59.1, 6.0.27848.1, 6.0.1782272.1, 6.2.222784.2

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: 6.2.1643032.1, 6.0.27848.1
Degree 8 siblings: 8.0.14258176.2, 8.4.49632710656.3
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Minimal sibling: 6.0.27848.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{4}$ ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ R

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 2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(59\) Copy content Toggle raw display $\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$