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Properties

Label 12.0.6775409390765625.3
Degree $12$
Signature $[0, 6]$
Discriminant $6.775\times 10^{15}$
Root discriminant \(20.86\)
Ramified primes $3,5,29$
Class number $4$
Class group [2, 2]
Galois group $D_6$ (as 12T3)

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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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189)
 
gp: K = bnfinit(y^12 - 6*y^11 + 29*y^10 - 90*y^9 + 225*y^8 - 426*y^7 + 667*y^6 - 822*y^5 + 1020*y^4 - 1020*y^3 + 1379*y^2 - 957*y + 1189, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189)
 

\( x^{12} - 6 x^{11} + 29 x^{10} - 90 x^{9} + 225 x^{8} - 426 x^{7} + 667 x^{6} - 822 x^{5} + 1020 x^{4} + \cdots + 1189 \) 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:   \(6775409390765625\) \(\medspace = 3^{6}\cdot 5^{6}\cdot 29^{6}\) 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:  \(20.86\)
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:  $3^{1/2}5^{1/2}29^{1/2}\approx 20.85665361461421$
Ramified primes:   \(3\), \(5\), \(29\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{249527}a^{10}-\frac{5}{249527}a^{9}-\frac{100812}{249527}a^{8}-\frac{95776}{249527}a^{7}-\frac{97172}{249527}a^{6}-\frac{121870}{249527}a^{5}-\frac{111647}{249527}a^{4}+\frac{65155}{249527}a^{3}-\frac{56119}{249527}a^{2}+\frac{19191}{249527}a+\frac{62748}{249527}$, $\frac{1}{498305419}a^{11}+\frac{993}{498305419}a^{10}+\frac{52045341}{498305419}a^{9}-\frac{85734532}{498305419}a^{8}-\frac{227681403}{498305419}a^{7}-\frac{36713992}{498305419}a^{6}-\frac{123983650}{498305419}a^{5}+\frac{193563443}{498305419}a^{4}+\frac{194722611}{498305419}a^{3}-\frac{120116010}{498305419}a^{2}-\frac{206856096}{498305419}a+\frac{243030525}{498305419}$ 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

$C_{2}\times C_{2}$, which has order $4$

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:   $\frac{1936}{872689}a^{11}-\frac{10648}{872689}a^{10}+\frac{46826}{872689}a^{9}-\frac{130857}{872689}a^{8}+\frac{267326}{872689}a^{7}-\frac{399511}{872689}a^{6}+\frac{413575}{872689}a^{5}-\frac{287253}{872689}a^{4}+\frac{496099}{872689}a^{3}-\frac{585385}{872689}a^{2}+\frac{48749}{45931}a+\frac{67175}{872689}$, $\frac{175100}{498305419}a^{11}-\frac{1160753}{498305419}a^{10}+\frac{8270855}{498305419}a^{9}-\frac{26929579}{498305419}a^{8}+\frac{77711925}{498305419}a^{7}-\frac{134485292}{498305419}a^{6}+\frac{198607131}{498305419}a^{5}-\frac{185869455}{498305419}a^{4}+\frac{186442882}{498305419}a^{3}-\frac{140811288}{498305419}a^{2}+\frac{301014028}{498305419}a-\frac{89315146}{498305419}$, $\frac{678620}{498305419}a^{11}-\frac{4075894}{498305419}a^{10}+\frac{19411375}{498305419}a^{9}-\frac{56547335}{498305419}a^{8}+\frac{131627998}{498305419}a^{7}-\frac{222541228}{498305419}a^{6}+\frac{281798416}{498305419}a^{5}-\frac{287249240}{498305419}a^{4}+\frac{12045030}{21665453}a^{3}-\frac{318461904}{498305419}a^{2}+\frac{738776827}{498305419}a-\frac{156874968}{498305419}$, $\frac{179274}{498305419}a^{11}-\frac{1183710}{498305419}a^{10}+\frac{5200200}{498305419}a^{9}-\frac{12939454}{498305419}a^{8}+\frac{30699681}{498305419}a^{7}-\frac{35390387}{498305419}a^{6}+\frac{81006922}{498305419}a^{5}-\frac{106847785}{498305419}a^{4}+\frac{249901617}{498305419}a^{3}-\frac{523987503}{498305419}a^{2}+\frac{192748342}{498305419}a-\frac{634984096}{498305419}$, $\frac{1361414}{498305419}a^{11}-\frac{8174745}{498305419}a^{10}+\frac{35752095}{498305419}a^{9}-\frac{99104545}{498305419}a^{8}+\frac{216243752}{498305419}a^{7}-\frac{15042937}{21665453}a^{6}+\frac{445996623}{498305419}a^{5}-\frac{21542470}{21665453}a^{4}+\frac{32501585}{26226601}a^{3}-\frac{521794604}{498305419}a^{2}+\frac{870982549}{498305419}a-\frac{361113467}{498305419}$ 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:  \( 217.33692993 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 217.33692993 \cdot 4}{2\cdot\sqrt{6775409390765625}}\cr\approx \mathstrut & 0.32491889151 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189)
 
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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189, 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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189);
 
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 + 29*x^10 - 90*x^9 + 225*x^8 - 426*x^7 + 667*x^6 - 822*x^5 + 1020*x^4 - 1020*x^3 + 1379*x^2 - 957*x + 1189);
 
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

$D_6$ (as 12T3):

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 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{-87}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-435}) \), 3.1.87.1 x3, \(\Q(\sqrt{5}, \sqrt{-87})\), 6.0.658503.1, 6.2.946125.1 x3, 6.0.82312875.1 x3

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: 6.2.946125.1, 6.0.82312875.1
Minimal sibling: 6.2.946125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ R ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.1.0.1}{1} }^{12}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(29\) Copy content Toggle raw display 29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$