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Properties

Label 12.0.449...576.4
Degree $12$
Signature $[0, 6]$
Discriminant $4.492\times 10^{18}$
Root discriminant \(35.84\)
Ramified primes $2,17$
Class number $2$
Class group [2]
Galois group $D_6\wr C_2$ (as 12T125)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752)
 
gp: K = bnfinit(y^12 - 2*y^11 - y^10 + 10*y^9 + 43*y^8 - 152*y^7 + 294*y^6 - 348*y^5 + 712*y^4 - 768*y^3 + 868*y^2 - 288*y + 752, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752)
 

\( x^{12} - 2 x^{11} - x^{10} + 10 x^{9} + 43 x^{8} - 152 x^{7} + 294 x^{6} - 348 x^{5} + 712 x^{4} + \cdots + 752 \) 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:   \(4492085992834072576\) \(\medspace = 2^{17}\cdot 17^{11}\) 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:  \(35.84\)
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:  $2^{13/6}17^{11/12}\approx 60.275825724785875$
Ramified primes:   \(2\), \(17\) 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:  \(\Q(\sqrt{34}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{8}-\frac{5}{12}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}a$, $\frac{1}{2728523407440}a^{11}+\frac{4872600053}{227376950620}a^{10}-\frac{55311684713}{2728523407440}a^{9}+\frac{30058404233}{227376950620}a^{8}+\frac{625688622251}{2728523407440}a^{7}+\frac{68741112433}{1364261703720}a^{6}+\frac{73955180967}{454753901240}a^{5}-\frac{27078619682}{170532712965}a^{4}-\frac{78089574574}{170532712965}a^{3}+\frac{9008521295}{34106542593}a^{2}+\frac{107882913099}{227376950620}a-\frac{2447410519}{7256711190}$ 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

$C_{2}$, which has order $2$

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:   $\frac{200477156}{170532712965}a^{11}+\frac{2143349707}{341065425930}a^{10}+\frac{426515454}{56844237655}a^{9}-\frac{1166413484}{170532712965}a^{8}+\frac{3556454702}{56844237655}a^{7}+\frac{17589480322}{56844237655}a^{6}+\frac{61459586387}{170532712965}a^{5}+\frac{22553326437}{113688475310}a^{4}+\frac{122032246196}{170532712965}a^{3}+\frac{45370730605}{34106542593}a^{2}+\frac{234751474088}{170532712965}a+\frac{1375655353}{3628355595}$, $\frac{454159456}{170532712965}a^{11}-\frac{393524538}{56844237655}a^{10}-\frac{860083161}{56844237655}a^{9}+\frac{5243659639}{113688475310}a^{8}+\frac{7441464892}{56844237655}a^{7}-\frac{201815799143}{341065425930}a^{6}+\frac{53088767642}{170532712965}a^{5}+\frac{127712637721}{341065425930}a^{4}-\frac{116114774054}{170532712965}a^{3}+\frac{7696571918}{34106542593}a^{2}-\frac{83197021544}{56844237655}a+\frac{14895659153}{3628355595}$, $\frac{999343072}{170532712965}a^{11}-\frac{2217076123}{170532712965}a^{10}-\frac{1351762511}{170532712965}a^{9}+\frac{15258814792}{170532712965}a^{8}+\frac{34555052252}{170532712965}a^{7}-\frac{176671711853}{170532712965}a^{6}+\frac{335756692594}{170532712965}a^{5}-\frac{50006444753}{56844237655}a^{4}-\frac{3513394168}{170532712965}a^{3}+\frac{35365801610}{34106542593}a^{2}+\frac{31009751696}{170532712965}a+\frac{13158230941}{3628355595}$, $\frac{173838572}{56844237655}a^{11}+\frac{11123486}{170532712965}a^{10}-\frac{499819328}{170532712965}a^{9}+\frac{8672898527}{341065425930}a^{8}+\frac{28185903386}{170532712965}a^{7}-\frac{58497001523}{341065425930}a^{6}+\frac{92701441412}{170532712965}a^{5}+\frac{20492776271}{341065425930}a^{4}+\frac{240852108646}{170532712965}a^{3}-\frac{1348927560}{11368847531}a^{2}+\frac{76065695048}{170532712965}a-\frac{2268054577}{3628355595}$, $\frac{2497403801}{2728523407440}a^{11}+\frac{198786797}{341065425930}a^{10}+\frac{6078241389}{909507802480}a^{9}+\frac{3372073646}{170532712965}a^{8}+\frac{11245901057}{909507802480}a^{7}+\frac{26382618763}{1364261703720}a^{6}+\frac{274263905827}{454753901240}a^{5}+\frac{77371122541}{341065425930}a^{4}+\frac{194682520927}{341065425930}a^{3}+\frac{22249844152}{34106542593}a^{2}+\frac{1270997597017}{682130851860}a+\frac{15447252881}{7256711190}$ 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:  \( 30270.5942168 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 30270.5942168 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 0.878771828087 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752)
 
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 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752, 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 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752);
 
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 - 2*x^11 - x^10 + 10*x^9 + 43*x^8 - 152*x^7 + 294*x^6 - 348*x^5 + 712*x^4 - 768*x^3 + 868*x^2 - 288*x + 752);
 
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

$D_6\wr C_2$ (as 12T125):

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 288
The 27 conjugacy class representatives for $D_6\wr C_2$
Character table for $D_6\wr C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.39304.1, 6.4.45435424.1

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 12 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.2.561510749104259072.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ R ${\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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
\(2\) Copy content Toggle raw display 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.11.5$x^{6} + 4 x^{3} + 10$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
\(17\) Copy content Toggle raw display 17.12.11.2$x^{12} + 34$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$