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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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 10*y^10 - 16*y^9 + 28*y^8 - 36*y^7 + 58*y^6 - 52*y^5 + 79*y^4 - 60*y^3 + 92*y^2 - 52*y + 41, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41)
 

\( x^{12} - 4 x^{11} + 10 x^{10} - 16 x^{9} + 28 x^{8} - 36 x^{7} + 58 x^{6} - 52 x^{5} + 79 x^{4} + \cdots + 41 \) 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:   \(34162868224000000\) \(\medspace = 2^{24}\cdot 5^{6}\cdot 19^{4}\) 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:  \(23.87\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{2}5^{1/2}19^{2/3}\approx 63.686501757102$
Ramified primes:   \(2\), \(5\), \(19\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_6$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{2}, \sqrt{-5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3993007}a^{11}-\frac{369155}{3993007}a^{10}+\frac{594519}{3993007}a^{9}+\frac{360356}{3993007}a^{8}+\frac{1250477}{3993007}a^{7}+\frac{732179}{3993007}a^{6}-\frac{1959148}{3993007}a^{5}+\frac{29442}{3993007}a^{4}+\frac{421391}{3993007}a^{3}-\frac{1335402}{3993007}a^{2}+\frac{318595}{3993007}a+\frac{365281}{3993007}$ 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:  $C_{2}$, which has order $2$
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:  $5$
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:   \( -1 \)  (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:   $\frac{550170}{3993007}a^{11}-\frac{1691309}{3993007}a^{10}+\frac{3342832}{3993007}a^{9}-\frac{3723044}{3993007}a^{8}+\frac{7776039}{3993007}a^{7}-\frac{7597758}{3993007}a^{6}+\frac{15840434}{3993007}a^{5}-\frac{5517266}{3993007}a^{4}+\frac{22665085}{3993007}a^{3}-\frac{8788382}{3993007}a^{2}+\frac{24340913}{3993007}a-\frac{5387547}{3993007}$, $\frac{286945}{3993007}a^{11}-\frac{691779}{3993007}a^{10}+\frac{1016394}{3993007}a^{9}-\frac{556852}{3993007}a^{8}+\frac{2520738}{3993007}a^{7}-\frac{953157}{3993007}a^{6}+\frac{3739663}{3993007}a^{5}+\frac{3024885}{3993007}a^{4}+\frac{7788535}{3993007}a^{3}+\frac{1989865}{3993007}a^{2}+\frac{3340017}{3993007}a+\frac{3115802}{3993007}$, $\frac{270776}{3993007}a^{11}-\frac{1370049}{3993007}a^{10}+\frac{3399539}{3993007}a^{9}-\frac{5348810}{3993007}a^{8}+\frac{8138580}{3993007}a^{7}-\frac{12268674}{3993007}a^{6}+\frac{16658165}{3993007}a^{5}-\frac{17820015}{3993007}a^{4}+\frac{18366419}{3993007}a^{3}-\frac{24035095}{3993007}a^{2}+\frac{18928520}{3993007}a-\frac{13427362}{3993007}$, $\frac{217406}{3993007}a^{11}-\frac{1064237}{3993007}a^{10}+\frac{2354131}{3993007}a^{9}-\frac{3233811}{3993007}a^{8}+\frac{5307081}{3993007}a^{7}-\frac{9102395}{3993007}a^{6}+\frac{11512616}{3993007}a^{5}-\frac{11901790}{3993007}a^{4}+\frac{13351166}{3993007}a^{3}-\frac{16826284}{3993007}a^{2}+\frac{13744169}{3993007}a-\frac{6628144}{3993007}$, $\frac{354281}{3993007}a^{11}-\frac{1644284}{3993007}a^{10}+\frac{3652603}{3993007}a^{9}-\frac{5121782}{3993007}a^{8}+\frac{8094408}{3993007}a^{7}-\frac{12584463}{3993007}a^{6}+\frac{15494222}{3993007}a^{5}-\frac{14965110}{3993007}a^{4}+\frac{16251183}{3993007}a^{3}-\frac{24072616}{3993007}a^{2}+\frac{13805347}{3993007}a-\frac{17210937}{3993007}$ 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:  \( 597.4033513730107 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 597.4033513730107 \cdot 2}{2\cdot\sqrt{34162868224000000}}\cr\approx \mathstrut & 0.198870196058062 \end{aligned}\]

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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41) 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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41, 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 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41); 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 = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41); 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

$C_6\times S_3$ (as 12T18):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), 6.0.23104000.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 36
Degree 18 siblings: 18.6.297066099785564011102208000000.1, 18.0.4641657809149437673472000000000.1
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 ${\href{/padicField/3.6.0.1}{6} }^{2}$ R ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ R ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.3.4.24b1.10$x^{12} + 4 x^{10} + 8 x^{9} + 6 x^{8} + 24 x^{7} + 24 x^{6} + 24 x^{5} + 41 x^{4} + 32 x^{3} + 20 x^{2} + 24 x + 25$$4$$3$$24$$C_6\times C_2$$$[2, 3]^{3}$$
\(5\) Copy content Toggle raw display 5.6.2.6a1.2$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(19\) Copy content Toggle raw display 19.6.1.0a1.1$x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
19.2.3.4a1.3$x^{6} + 54 x^{5} + 978 x^{4} + 6048 x^{3} + 1956 x^{2} + 235 x + 331$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*36 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*36 1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
*36 1.40.2t1.b.a$1$ $ 2^{3} \cdot 5 $ \(\Q(\sqrt{-10}) \) $C_2$ (as 2T1) $1$ $-1$
*36 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
1.760.6t1.b.a$1$ $ 2^{3} \cdot 5 \cdot 19 $ 6.0.8340544000.3 $C_6$ (as 6T1) $0$ $-1$
1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.152.6t1.b.a$1$ $ 2^{3} \cdot 19 $ 6.6.66724352.1 $C_6$ (as 6T1) $0$ $1$
1.380.6t1.b.a$1$ $ 2^{2} \cdot 5 \cdot 19 $ 6.0.1042568000.1 $C_6$ (as 6T1) $0$ $-1$
1.760.6t1.b.b$1$ $ 2^{3} \cdot 5 \cdot 19 $ 6.0.8340544000.3 $C_6$ (as 6T1) $0$ $-1$
1.380.6t1.b.b$1$ $ 2^{2} \cdot 5 \cdot 19 $ 6.0.1042568000.1 $C_6$ (as 6T1) $0$ $-1$
1.152.6t1.b.b$1$ $ 2^{3} \cdot 19 $ 6.6.66724352.1 $C_6$ (as 6T1) $0$ $1$
2.14440.3t2.a.a$2$ $ 2^{3} \cdot 5 \cdot 19^{2}$ 3.1.14440.1 $S_3$ (as 3T2) $1$ $0$
2.57760.6t3.a.a$2$ $ 2^{5} \cdot 5 \cdot 19^{2}$ 6.0.16681088000.5 $D_{6}$ (as 6T3) $1$ $0$
*36 2.760.6t5.a.a$2$ $ 2^{3} \cdot 5 \cdot 19 $ 6.0.23104000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
*36 2.3040.12t18.c.a$2$ $ 2^{5} \cdot 5 \cdot 19 $ 12.0.34162868224000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
*36 2.760.6t5.a.b$2$ $ 2^{3} \cdot 5 \cdot 19 $ 6.0.23104000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
*36 2.3040.12t18.c.b$2$ $ 2^{5} \cdot 5 \cdot 19 $ 12.0.34162868224000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$

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)