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Properties

Label 12.8.289369624500000000.1
Degree $12$
Signature $[8, 2]$
Discriminant $2.894\times 10^{17}$
Root discriminant \(28.52\)
Ramified primes $2,3,5,11$
Class number $1$
Class group trivial
Galois group $C_3:S_3^3:C_4$ (as 12T245)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 1)
 
Copy content gp:K = bnfinit(y^12 - 3*y^11 + 9*y^10 + 19*y^9 - 75*y^8 + 192*y^7 - 187*y^6 - 204*y^5 + 279*y^4 + 11*y^3 - 42*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 1)
 

\( x^{12} - 3 x^{11} + 9 x^{10} + 19 x^{9} - 75 x^{8} + 192 x^{7} - 187 x^{6} - 204 x^{5} + 279 x^{4} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $12$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[8, 2]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(289369624500000000\) \(\medspace = 2^{8}\cdot 3^{14}\cdot 5^{9}\cdot 11^{2}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(28.52\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{7/4}3^{25/18}5^{3/4}11^{1/2}\approx 171.55158998906003$
Ramified primes:   \(2\), \(3\), \(5\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{10416035878}a^{11}-\frac{836254367}{10416035878}a^{10}+\frac{371880705}{5208017939}a^{9}-\frac{302462314}{5208017939}a^{8}-\frac{4718389115}{10416035878}a^{7}-\frac{646456219}{10416035878}a^{6}+\frac{2538267793}{5208017939}a^{5}-\frac{2007222878}{5208017939}a^{4}+\frac{2920445751}{10416035878}a^{3}+\frac{1884850193}{10416035878}a^{2}-\frac{2894076927}{10416035878}a-\frac{1217649595}{10416035878}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{2028417}{7536929}a^{11}-\frac{5602752}{7536929}a^{10}+\frac{33919013}{15073858}a^{9}+\frac{85103007}{15073858}a^{8}-\frac{283852317}{15073858}a^{7}+\frac{714279219}{15073858}a^{6}-\frac{295453545}{7536929}a^{5}-\frac{482898261}{7536929}a^{4}+\frac{459607689}{7536929}a^{3}+\frac{107552034}{7536929}a^{2}-\frac{125691891}{15073858}a+\frac{2154043}{15073858}$, $\frac{792117377}{10416035878}a^{11}+\frac{1762294011}{10416035878}a^{10}-\frac{2561246638}{5208017939}a^{9}-\frac{21025950757}{10416035878}a^{8}+\frac{49123304621}{10416035878}a^{7}-\frac{51284680758}{5208017939}a^{6}+\frac{9506883082}{5208017939}a^{5}+\frac{152713849019}{5208017939}a^{4}-\frac{120043971721}{10416035878}a^{3}-\frac{215758632777}{10416035878}a^{2}+\frac{57680204059}{10416035878}a+\frac{13826964387}{5208017939}$, $\frac{3671548135}{10416035878}a^{11}-\frac{5690370395}{5208017939}a^{10}+\frac{17197751466}{5208017939}a^{9}+\frac{32800117662}{5208017939}a^{8}-\frac{279750547271}{10416035878}a^{7}+\frac{368146892726}{5208017939}a^{6}-\frac{388883956093}{5208017939}a^{5}-\frac{312524189659}{5208017939}a^{4}+\frac{1033044806835}{10416035878}a^{3}-\frac{44790602813}{5208017939}a^{2}-\frac{76003059807}{10416035878}a-\frac{701881797}{5208017939}$, $a$, $\frac{2079245101}{10416035878}a^{11}-\frac{3268225400}{5208017939}a^{10}+\frac{19807421123}{10416035878}a^{9}+\frac{36022417739}{10416035878}a^{8}-\frac{79616304814}{5208017939}a^{7}+\frac{423267554033}{10416035878}a^{6}-\frac{232037207489}{5208017939}a^{5}-\frac{158885962595}{5208017939}a^{4}+\frac{558634761593}{10416035878}a^{3}-\frac{22682915904}{5208017939}a^{2}-\frac{23766069369}{5208017939}a+\frac{2362364771}{10416035878}$, $\frac{2056227829}{5208017939}a^{11}+\frac{4978702320}{5208017939}a^{10}-\frac{15017903588}{5208017939}a^{9}-\frac{99328903635}{10416035878}a^{8}+\frac{131346361377}{5208017939}a^{7}-\frac{617054080121}{10416035878}a^{6}+\frac{160527472583}{5208017939}a^{5}+\frac{637282888857}{5208017939}a^{4}-\frac{345675468367}{5208017939}a^{3}-\frac{295852227237}{5208017939}a^{2}+\frac{54501333957}{5208017939}a+\frac{37031422741}{10416035878}$, $\frac{5160727824}{5208017939}a^{11}-\frac{32922766363}{10416035878}a^{10}+\frac{99088960539}{10416035878}a^{9}+\frac{88680198131}{5208017939}a^{8}-\frac{808071428331}{10416035878}a^{7}+\frac{2133593965591}{10416035878}a^{6}-\frac{1167163424268}{5208017939}a^{5}-\frac{833968007381}{5208017939}a^{4}+\frac{1597822728545}{5208017939}a^{3}-\frac{490842268225}{10416035878}a^{2}-\frac{311083801535}{10416035878}a+\frac{47899229231}{10416035878}$, $\frac{2402767364}{5208017939}a^{11}+\frac{13764067125}{10416035878}a^{10}-\frac{41984609137}{10416035878}a^{9}-\frac{47474126113}{5208017939}a^{8}+\frac{341580928807}{10416035878}a^{7}-\frac{885717188597}{10416035878}a^{6}+\frac{414757205395}{5208017939}a^{5}+\frac{483189105542}{5208017939}a^{4}-\frac{526489890428}{5208017939}a^{3}-\frac{94221635131}{10416035878}a^{2}+\frac{1589500123}{10416035878}a+\frac{10310592559}{10416035878}$, $\frac{6834604987}{5208017939}a^{11}-\frac{39823750317}{10416035878}a^{10}+\frac{60389684869}{5208017939}a^{9}+\frac{133367134837}{5208017939}a^{8}-\frac{495795251003}{5208017939}a^{7}+\frac{2562399507803}{10416035878}a^{6}-\frac{1209664683537}{5208017939}a^{5}-\frac{1388870531617}{5208017939}a^{4}+\frac{1682500877972}{5208017939}a^{3}+\frac{204291364283}{10416035878}a^{2}-\frac{157956784266}{5208017939}a+\frac{26603040661}{10416035878}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 23766.397630604413 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 23766.397630604413 \cdot 1}{2\cdot\sqrt{289369624500000000}}\cr\approx \mathstrut & 0.223257776396234 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 9*x^10 + 19*x^9 - 75*x^8 + 192*x^7 - 187*x^6 - 204*x^5 + 279*x^4 + 11*x^3 - 42*x^2 + 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

$C_3:S_3^3:C_4$ (as 12T245):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 2592
The 30 conjugacy class representatives for $C_3:S_3^3:C_4$
Character table for $C_3:S_3^3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: 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 R ${\href{/padicField/7.12.0.1}{12} }$ R ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{5}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.2.8a5.2$x^{8} + 2 x^{7} + 4 x^{5} + 6 x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 5$$2$$4$$8$$C_2^3: C_4$$$[2, 2, 2]^{4}$$
\(3\) Copy content Toggle raw display 3.2.6.14a1.2$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2630 x^{6} + 3303 x^{5} + 3240 x^{4} + 2462 x^{3} + 1410 x^{2} + 567 x + 124$$6$$2$$14$$(C_3\times C_3):C_4$$$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.3.4.9a1.1$x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$$4$$3$$9$$C_{12}$$$[\ ]_{4}^{3}$$
\(11\) Copy content Toggle raw display 11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
11.2.2.2a1.1$x^{4} + 14 x^{3} + 53 x^{2} + 39 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)