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Properties

Label 12.0.668168000000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $6.682\times 10^{14}$
Root discriminant \(17.20\)
Ramified primes $2,5,17$
Class number $1$
Class group trivial
Galois group $S_3 \times C_4$ (as 12T11)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25)
 
Copy content gp:K = bnfinit(y^12 - y^11 - 7*y^10 + 15*y^9 + 21*y^8 - 60*y^7 - 23*y^6 + 163*y^5 - 54*y^4 - 240*y^3 + 290*y^2 - 75*y + 25, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25)
 

\( x^{12} - x^{11} - 7 x^{10} + 15 x^{9} + 21 x^{8} - 60 x^{7} - 23 x^{6} + 163 x^{5} - 54 x^{4} + \cdots + 25 \) 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:  $[0, 6]$
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:   \(668168000000000\) \(\medspace = 2^{12}\cdot 5^{9}\cdot 17^{4}\) 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:  \(17.20\)
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:  $2^{3/2}5^{3/4}17^{1/2}\approx 38.99392548461692$
Ramified primes:   \(2\), \(5\), \(17\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
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.
Maximal CM subfield:  \(\Q(\zeta_{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{905}a^{10}-\frac{46}{905}a^{9}-\frac{27}{905}a^{8}-\frac{74}{181}a^{7}-\frac{204}{905}a^{6}-\frac{81}{181}a^{5}+\frac{207}{905}a^{4}+\frac{173}{905}a^{3}+\frac{266}{905}a^{2}-\frac{3}{181}a-\frac{42}{181}$, $\frac{1}{34562855}a^{11}-\frac{59}{6912571}a^{10}-\frac{6825848}{34562855}a^{9}+\frac{9066308}{34562855}a^{8}+\frac{9670446}{34562855}a^{7}-\frac{8957074}{34562855}a^{6}-\frac{309818}{34562855}a^{5}-\frac{2520382}{6912571}a^{4}-\frac{7089141}{34562855}a^{3}-\frac{10301799}{34562855}a^{2}+\frac{2969648}{6912571}a-\frac{2092581}{6912571}$ 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:  Trivial group, which has order $1$
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:  $5$
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:   \( -\frac{17203}{6912571} a^{11} - \frac{60781}{34562855} a^{10} + \frac{980241}{34562855} a^{9} - \frac{1291793}{34562855} a^{8} - \frac{762920}{6912571} a^{7} + \frac{6473079}{34562855} a^{6} + \frac{512341}{6912571} a^{5} - \frac{18503542}{34562855} a^{4} + \frac{3767152}{34562855} a^{3} + \frac{29882134}{34562855} a^{2} - \frac{14506933}{6912571} a + \frac{7573735}{6912571} \)  (order $10$) 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:   $\frac{39259}{34562855}a^{11}+\frac{219614}{34562855}a^{10}+\frac{6989}{34562855}a^{9}-\frac{1656836}{34562855}a^{8}+\frac{2182894}{34562855}a^{7}+\frac{8710578}{34562855}a^{6}-\frac{6758607}{34562855}a^{5}-\frac{16829992}{34562855}a^{4}+\frac{24452638}{34562855}a^{3}+\frac{44616373}{34562855}a^{2}-\frac{9764427}{6912571}a-\frac{1581401}{6912571}$, $\frac{307767}{34562855}a^{11}-\frac{736887}{34562855}a^{10}-\frac{830449}{34562855}a^{9}+\frac{899775}{6912571}a^{8}+\frac{619537}{34562855}a^{7}-\frac{2251784}{6912571}a^{6}-\frac{1050526}{34562855}a^{5}+\frac{30616256}{34562855}a^{4}-\frac{31876483}{34562855}a^{3}+\frac{602183}{6912571}a^{2}-\frac{316756}{6912571}a+\frac{1930762}{6912571}$, $\frac{132688}{34562855}a^{11}+\frac{1797792}{34562855}a^{10}-\frac{5384571}{34562855}a^{9}-\frac{768834}{6912571}a^{8}+\frac{31162078}{34562855}a^{7}-\frac{1121076}{6912571}a^{6}-\frac{73856059}{34562855}a^{5}+\frac{35527009}{34562855}a^{4}+\frac{154336913}{34562855}a^{3}-\frac{47550519}{6912571}a^{2}+\frac{28623417}{6912571}a-\frac{1429776}{6912571}$, $\frac{995111}{34562855}a^{11}-\frac{594584}{34562855}a^{10}-\frac{8142209}{34562855}a^{9}+\frac{11105131}{34562855}a^{8}+\frac{31592696}{34562855}a^{7}-\frac{46312088}{34562855}a^{6}-\frac{67557898}{34562855}a^{5}+\frac{126981442}{34562855}a^{4}+\frac{51937257}{34562855}a^{3}-\frac{205437453}{34562855}a^{2}+\frac{31945746}{6912571}a-\frac{7559833}{6912571}$, $\frac{140486}{6912571}a^{11}-\frac{1634697}{34562855}a^{10}-\frac{4033243}{34562855}a^{9}+\frac{12656229}{34562855}a^{8}+\frac{1624720}{6912571}a^{7}-\frac{47633137}{34562855}a^{6}-\frac{3315386}{6912571}a^{5}+\frac{114598191}{34562855}a^{4}-\frac{50569541}{34562855}a^{3}-\frac{137806972}{34562855}a^{2}+\frac{3241756}{6912571}a-\frac{3223637}{6912571}$ 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:  \( 2014.8871293984223 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 2014.8871293984223 \cdot 1}{10\cdot\sqrt{668168000000000}}\cr\approx \mathstrut & 0.479608748698512 \end{aligned}\]

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 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25) 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 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25, 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 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25); 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 - x^11 - 7*x^10 + 15*x^9 + 21*x^8 - 60*x^7 - 23*x^6 + 163*x^5 - 54*x^4 - 240*x^3 + 290*x^2 - 75*x + 25); 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

$C_4\times S_3$ (as 12T11):

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 24
The 12 conjugacy class representatives for $S_3 \times C_4$
Character table for $S_3 \times C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.680.1, \(\Q(\zeta_{5})\), 6.2.2312000.1

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

Galois closure: deg 24
Degree 12 sibling: 12.4.12358435328000000000.1
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 ${\href{/padicField/3.12.0.1}{12} }$ R ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

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$$$[\ ]^{4}$$
2.4.2.12a1.1$x^{8} + 2 x^{5} + 2 x^{4} + x^{2} + 2 x + 3$$2$$4$$12$$C_4\times C_2$$$[3]^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(17\) Copy content Toggle raw display 17.4.1.0a1.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.680.2t1.b.a$1$ $ 2^{3} \cdot 5 \cdot 17 $ \(\Q(\sqrt{-170}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.136.2t1.b.a$1$ $ 2^{3} \cdot 17 $ \(\Q(\sqrt{-34}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.680.4t1.g.a$1$ $ 2^{3} \cdot 5 \cdot 17 $ 4.4.2312000.1 $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.680.4t1.g.b$1$ $ 2^{3} \cdot 5 \cdot 17 $ 4.4.2312000.1 $C_4$ (as 4T1) $0$ $1$
* 2.680.3t2.a.a$2$ $ 2^{3} \cdot 5 \cdot 17 $ 3.1.680.1 $S_3$ (as 3T2) $1$ $0$
* 2.680.6t3.b.a$2$ $ 2^{3} \cdot 5 \cdot 17 $ 6.0.62886400.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.3400.12t11.b.a$2$ $ 2^{3} \cdot 5^{2} \cdot 17 $ 12.0.668168000000000.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.3400.12t11.b.b$2$ $ 2^{3} \cdot 5^{2} \cdot 17 $ 12.0.668168000000000.1 $S_3 \times C_4$ (as 12T11) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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