↑

Properties

Label 12.8.1130350095703125.1
Degree $12$
Signature $[8, 2]$
Discriminant $1.130\times 10^{15}$
Root discriminant \(17.97\)
Ramified primes $3,5,11$
Class number $1$
Class group trivial
Galois group $S_3^2:C_4$ (as 12T79)

Related objects

Downloads

Learn more

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 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*x + 1)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 + 9*y^10 + 9*y^9 - 33*y^8 + 30*y^7 - 10*y^6 - 48*y^5 + 78*y^4 + 3*y^3 - 30*y^2 - 3*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*x + 1)
 

\( x^{12} - 6 x^{11} + 9 x^{10} + 9 x^{9} - 33 x^{8} + 30 x^{7} - 10 x^{6} - 48 x^{5} + 78 x^{4} + 3 x^{3} + \cdots + 1 \) 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:   \(1130350095703125\) \(\medspace = 3^{14}\cdot 5^{9}\cdot 11^{2}\) 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:  \(17.97\)
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:  $3^{25/18}5^{3/4}11^{1/2}\approx 51.00259285125494$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{119719}a^{11}+\frac{44774}{119719}a^{10}+\frac{45636}{119719}a^{9}-\frac{23241}{119719}a^{8}-\frac{14746}{119719}a^{7}+\frac{44154}{119719}a^{6}+\frac{56825}{119719}a^{5}-\frac{3893}{119719}a^{4}-\frac{17598}{119719}a^{3}-\frac{47979}{119719}a^{2}-\frac{22476}{119719}a+\frac{2350}{119719}$ 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:   $\frac{1263}{6301}a^{11}-\frac{8214}{6301}a^{10}+\frac{15623}{6301}a^{9}+\frac{2976}{6301}a^{8}-\frac{42549}{6301}a^{7}+\frac{59361}{6301}a^{6}-\frac{42522}{6301}a^{5}-\frac{39885}{6301}a^{4}+\frac{117072}{6301}a^{3}-\frac{51168}{6301}a^{2}-\frac{13785}{6301}a+\frac{12881}{6301}$, $\frac{60073}{119719}a^{11}+\frac{377428}{119719}a^{10}-\frac{644642}{119719}a^{9}-\frac{365542}{119719}a^{8}+\frac{2070800}{119719}a^{7}-\frac{2363458}{119719}a^{6}+\frac{1336250}{119719}a^{5}+\frac{2447362}{119719}a^{4}-\frac{5341471}{119719}a^{3}+\frac{1324451}{119719}a^{2}+\frac{1087337}{119719}a-\frac{22849}{119719}$, $\frac{5822}{119719}a^{11}+\frac{73754}{119719}a^{10}-\frac{275769}{119719}a^{9}+\frac{266070}{119719}a^{8}+\frac{491565}{119719}a^{7}-\frac{1225085}{119719}a^{6}+\frac{1145637}{119719}a^{5}-\frac{321002}{119719}a^{4}-\frac{1939412}{119719}a^{3}+\frac{2303972}{119719}a^{2}+\frac{2405}{119719}a-\frac{273172}{119719}$, $\frac{35686}{119719}a^{11}-\frac{204248}{119719}a^{10}+\frac{268177}{119719}a^{9}+\frac{394063}{119719}a^{8}-\frac{1138222}{119719}a^{7}+\frac{895918}{119719}a^{6}-\frac{63191}{119719}a^{5}-\frac{1967062}{119719}a^{4}+\frac{2557745}{119719}a^{3}+\frac{401701}{119719}a^{2}-\frac{1278145}{119719}a+\frac{58800}{119719}$, $\frac{636}{6301}a^{11}-\frac{4256}{6301}a^{10}+\frac{8391}{6301}a^{9}+\frac{870}{6301}a^{8}-\frac{21471}{6301}a^{7}+\frac{29892}{6301}a^{6}-\frac{20739}{6301}a^{5}-\frac{18558}{6301}a^{4}+\frac{54957}{6301}a^{3}-\frac{24105}{6301}a^{2}-\frac{16670}{6301}a+\frac{1263}{6301}$, $\frac{18175}{119719}a^{11}+\frac{82312}{119719}a^{10}-\frac{21068}{119719}a^{9}-\frac{322614}{119719}a^{8}+\frac{316866}{119719}a^{7}+\frac{97226}{119719}a^{6}-\frac{337719}{119719}a^{5}+\frac{1078817}{119719}a^{4}-\frac{284956}{119719}a^{3}-\frac{1331780}{119719}a^{2}+\frac{259510}{119719}a+\frac{28433}{119719}$, $\frac{64119}{119719}a^{11}+\frac{356671}{119719}a^{10}-\frac{441762}{119719}a^{9}-\frac{671028}{119719}a^{8}+\frac{1753897}{119719}a^{7}-\frac{1432042}{119719}a^{6}+\frac{444747}{119719}a^{5}+\frac{2994127}{119719}a^{4}-\frac{3696702}{119719}a^{3}-\frac{771956}{119719}a^{2}+\frac{919074}{119719}a+\frac{166290}{119719}$, $\frac{14825}{119719}a^{11}-\frac{67305}{119719}a^{10}+\frac{21631}{119719}a^{9}+\frac{242895}{119719}a^{8}-\frac{241994}{119719}a^{7}-\frac{40442}{119719}a^{6}+\frac{207460}{119719}a^{5}-\frac{727481}{119719}a^{4}+\frac{97070}{119719}a^{3}+\frac{799937}{119719}a^{2}-\frac{28723}{119719}a-\frac{238917}{119719}$, $\frac{1263}{6301}a^{11}+\frac{8214}{6301}a^{10}-\frac{15623}{6301}a^{9}-\frac{2976}{6301}a^{8}+\frac{42549}{6301}a^{7}-\frac{59361}{6301}a^{6}+\frac{42522}{6301}a^{5}+\frac{39885}{6301}a^{4}-\frac{117072}{6301}a^{3}+\frac{51168}{6301}a^{2}+\frac{20086}{6301}a-\frac{12881}{6301}$ 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:  \( 1589.3897461008255 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{2}\cdot 1589.3897461008255 \cdot 1}{2\cdot\sqrt{1130350095703125}}\cr\approx \mathstrut & 0.238887610015920 \end{aligned}\]

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 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^12 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 9*x^10 + 9*x^9 - 33*x^8 + 30*x^7 - 10*x^6 - 48*x^5 + 78*x^4 + 3*x^3 - 30*x^2 - 3*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

$S_3^2:C_4$ (as 12T79):

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 144
The 18 conjugacy class representatives for $S_3^2:C_4$
Character table for $S_3^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 6.4.5011875.1

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 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 ${\href{/padicField/2.12.0.1}{12} }$ R R ${\href{/padicField/7.12.0.1}{12} }$ R ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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

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
\(3\) 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 5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(11\) Copy content Toggle raw display 11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.6.1.0a1.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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