Properties

Label 12.0.1624959306694656.34
Degree $12$
Signature $[0, 6]$
Discriminant $1.625\times 10^{15}$
Root discriminant \(18.52\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $S_3\times D_6$ (as 12T37)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112)
 
gp: K = bnfinit(y^12 - 12*y^10 - 12*y^9 + 54*y^8 + 108*y^7 - 40*y^6 - 324*y^5 - 327*y^4 + 180*y^3 + 612*y^2 + 432*y + 112, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112)
 

\( x^{12} - 12 x^{10} - 12 x^{9} + 54 x^{8} + 108 x^{7} - 40 x^{6} - 324 x^{5} - 327 x^{4} + 180 x^{3} + 612 x^{2} + 432 x + 112 \) 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:   \(1624959306694656\) \(\medspace = 2^{22}\cdot 3^{18}\) 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:  \(18.52\)
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:  $2^{2}3^{11/6}\approx 29.976594395601754$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{40}a^{9}+\frac{1}{40}a^{7}+\frac{1}{10}a^{6}+\frac{7}{40}a^{5}-\frac{1}{10}a^{4}-\frac{1}{8}a^{3}+\frac{2}{5}a^{2}+\frac{1}{10}a+\frac{1}{5}$, $\frac{1}{40}a^{10}+\frac{1}{40}a^{8}+\frac{1}{10}a^{7}-\frac{3}{40}a^{6}-\frac{1}{10}a^{5}-\frac{1}{8}a^{4}-\frac{1}{10}a^{3}+\frac{7}{20}a^{2}-\frac{3}{10}a$, $\frac{1}{560}a^{11}+\frac{1}{560}a^{10}+\frac{3}{560}a^{9}+\frac{1}{112}a^{8}-\frac{67}{560}a^{7}-\frac{29}{560}a^{6}-\frac{11}{112}a^{5}-\frac{57}{560}a^{4}-\frac{9}{56}a^{3}+\frac{13}{70}a^{2}-\frac{33}{70}a-\frac{2}{5}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:   \( \frac{1}{20} a^{9} - \frac{9}{20} a^{7} - \frac{11}{20} a^{6} + \frac{27}{20} a^{5} + \frac{33}{10} a^{4} + \frac{5}{4} a^{3} - \frac{99}{20} a^{2} - \frac{39}{5} a - \frac{13}{5} \)  (order $6$) 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:   $\frac{1}{40}a^{9}-\frac{9}{40}a^{7}-\frac{3}{20}a^{6}+\frac{27}{40}a^{5}+\frac{9}{10}a^{4}+\frac{1}{8}a^{3}-\frac{27}{20}a^{2}-\frac{12}{5}a-\frac{4}{5}$, $\frac{13}{280}a^{11}+\frac{13}{280}a^{10}-\frac{27}{70}a^{9}-\frac{25}{28}a^{8}+\frac{137}{280}a^{7}+\frac{213}{56}a^{6}+\frac{167}{35}a^{5}-\frac{199}{140}a^{4}-\frac{299}{28}a^{3}-\frac{345}{28}a^{2}-\frac{48}{7}a-\frac{8}{5}$, $\frac{1}{80}a^{11}-\frac{1}{80}a^{10}-\frac{3}{16}a^{9}+\frac{3}{80}a^{8}+\frac{87}{80}a^{7}+\frac{13}{16}a^{6}-\frac{213}{80}a^{5}-\frac{75}{16}a^{4}-\frac{13}{20}a^{3}+\frac{147}{20}a^{2}+\frac{81}{10}a+\frac{12}{5}$, $\frac{1}{112}a^{11}-\frac{23}{560}a^{10}-\frac{27}{560}a^{9}+\frac{137}{560}a^{8}+\frac{211}{560}a^{7}-\frac{369}{560}a^{6}-\frac{737}{560}a^{5}-\frac{117}{560}a^{4}+\frac{353}{140}a^{3}+\frac{319}{140}a^{2}-\frac{109}{70}a-\frac{8}{5}$, $\frac{9}{560}a^{11}-\frac{1}{112}a^{10}-\frac{127}{560}a^{9}+\frac{31}{560}a^{8}+\frac{587}{560}a^{7}+\frac{57}{112}a^{6}-\frac{1237}{560}a^{5}-\frac{1227}{560}a^{4}-\frac{34}{35}a^{3}+\frac{619}{140}a^{2}+\frac{207}{70}a+\frac{6}{5}$ 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:  \( 2457.79965637 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2457.79965637 \cdot 1}{6\cdot\sqrt{1624959306694656}}\cr\approx \mathstrut & 0.625249271158 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112)
 
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 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112, 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 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112);
 
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 - 12*x^10 - 12*x^9 + 54*x^8 + 108*x^7 - 40*x^6 - 324*x^5 - 327*x^4 + 180*x^3 + 612*x^2 + 432*x + 112);
 
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

$S_3\times D_6$ (as 12T37):

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 72
The 18 conjugacy class representatives for $S_3\times D_6$
Character table for $S_3\times D_6$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.2.13436928.6

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.12.18.87$x^{12} - 6 x^{11} + 45 x^{10} + 30 x^{9} + 99 x^{8} + 36 x^{7} + 51 x^{6} + 198 x^{5} + 108 x^{4} + 198 x^{3} + 333$$6$$2$$18$$C_6\times S_3$$[2, 2]_{2}^{2}$