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Properties

Label 12.0.300342961072265625.2
Degree $12$
Signature $[0, 6]$
Discriminant $3.003\times 10^{17}$
Root discriminant \(28.61\)
Ramified primes $3,5,29,31$
Class number $2$
Class group [2]
Galois group $(C_3^2\times S_3^2):C_4$ (as 12T209)

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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 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*x + 41)
 
Copy content gp:K = bnfinit(y^12 - 5*y^11 + 19*y^10 - 54*y^9 + 116*y^8 - 217*y^7 + 305*y^6 - 297*y^5 + 258*y^4 - 238*y^3 + 221*y^2 - 109*y + 41, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*x + 41);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*x + 41)
 

\( x^{12} - 5 x^{11} + 19 x^{10} - 54 x^{9} + 116 x^{8} - 217 x^{7} + 305 x^{6} - 297 x^{5} + 258 x^{4} + \cdots + 41 \) 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:   \(300342961072265625\) \(\medspace = 3^{8}\cdot 5^{9}\cdot 29^{3}\cdot 31^{2}\) 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:  \(28.61\)
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:  $3^{25/18}5^{3/4}29^{1/2}31^{2/3}\approx 817.2144566356151$
Ramified primes:   \(3\), \(5\), \(29\), \(31\) 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:  \(\Q(\sqrt{145}) \)
$\Aut(K/\Q)$:   $C_1$
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.
Maximal CM subfield:  4.0.3625.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{12}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{12}a^{5}-\frac{1}{6}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{12}$, $\frac{1}{1410058452}a^{11}-\frac{22687589}{1410058452}a^{10}+\frac{11369208}{117504871}a^{9}+\frac{331148975}{705029226}a^{8}-\frac{24527737}{117504871}a^{7}+\frac{592477481}{1410058452}a^{6}-\frac{558884365}{1410058452}a^{5}-\frac{102998281}{705029226}a^{4}+\frac{133260041}{352514613}a^{3}+\frac{13097343}{235009742}a^{2}+\frac{659987473}{1410058452}a-\frac{311278343}{1410058452}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:   $\frac{139234}{32046783}a^{11}-\frac{319733}{32046783}a^{10}+\frac{336677}{10682261}a^{9}-\frac{984719}{32046783}a^{8}-\frac{507381}{10682261}a^{7}+\frac{7457585}{32046783}a^{6}-\frac{27663640}{32046783}a^{5}+\frac{49331143}{32046783}a^{4}-\frac{54882805}{32046783}a^{3}+\frac{10630860}{10682261}a^{2}-\frac{12331583}{32046783}a+\frac{21221683}{32046783}$, $\frac{1698331}{705029226}a^{11}-\frac{13464349}{705029226}a^{10}+\frac{22866574}{352514613}a^{9}-\frac{22890255}{117504871}a^{8}+\frac{48888429}{117504871}a^{7}-\frac{478796797}{705029226}a^{6}+\frac{569069563}{705029226}a^{5}-\frac{137296409}{352514613}a^{4}-\frac{256156990}{352514613}a^{3}+\frac{442187054}{352514613}a^{2}-\frac{369402985}{705029226}a+\frac{16684747}{235009742}$, $\frac{6209416}{352514613}a^{11}-\frac{59087377}{705029226}a^{10}+\frac{34894429}{117504871}a^{9}-\frac{282262781}{352514613}a^{8}+\frac{186239489}{117504871}a^{7}-\frac{997654705}{352514613}a^{6}+\frac{2667883555}{705029226}a^{5}-\frac{1096512029}{352514613}a^{4}+\frac{904570259}{352514613}a^{3}-\frac{300230696}{117504871}a^{2}+\frac{702149500}{352514613}a-\frac{481733785}{705029226}$, $\frac{10063479}{470019484}a^{11}-\frac{112820809}{1410058452}a^{10}+\frac{100325293}{352514613}a^{9}-\frac{481378243}{705029226}a^{8}+\frac{142874605}{117504871}a^{7}-\frac{944336685}{470019484}a^{6}+\frac{2563367971}{1410058452}a^{5}-\frac{221060369}{705029226}a^{4}+\frac{214686388}{352514613}a^{3}-\frac{812972465}{705029226}a^{2}+\frac{788574989}{1410058452}a+\frac{1931842513}{1410058452}$, $\frac{170348267}{1410058452}a^{11}-\frac{799069151}{1410058452}a^{10}+\frac{704757439}{352514613}a^{9}-\frac{1294936653}{235009742}a^{8}+\frac{1274988646}{117504871}a^{7}-\frac{27365043869}{1410058452}a^{6}+\frac{34929651509}{1410058452}a^{5}-\frac{12195422647}{705029226}a^{4}+\frac{5770124054}{352514613}a^{3}-\frac{13496085233}{705029226}a^{2}+\frac{14893972843}{1410058452}a-\frac{2255893043}{470019484}$ 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:  \( 5193.55523784 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 5193.55523784 \cdot 2}{2\cdot\sqrt{300342961072265625}}\cr\approx \mathstrut & 0.583089520438 \end{aligned}\]

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 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*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 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*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 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*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 = PolynomialRing(QQ); K, a = NumberField(x^12 - 5*x^11 + 19*x^10 - 54*x^9 + 116*x^8 - 217*x^7 + 305*x^6 - 297*x^5 + 258*x^4 - 238*x^3 + 221*x^2 - 109*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

$(C_3^2\times S_3^2):C_4$ (as 12T209):

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 1296
The 36 conjugacy class representatives for $(C_3^2\times S_3^2):C_4$
Character table for $(C_3^2\times S_3^2):C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3625.1

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 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 ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ R R ${\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ R R ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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 3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.2.3.6a5.1$x^{6} + 6 x^{5} + 18 x^{4} + 35 x^{3} + 42 x^{2} + 30 x + 11$$3$$2$$6$$C_3^2: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}$$$[\ ]_{4}^{3}$$
\(29\) Copy content Toggle raw display 29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$
29.3.1.0a1.1$x^{3} + 2 x + 27$$1$$3$$0$$C_3$$$[\ ]^{3}$$
29.3.2.3a1.2$x^{6} + 4 x^{4} + 54 x^{3} + 4 x^{2} + 108 x + 758$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(31\) Copy content Toggle raw display 31.3.1.0a1.1$x^{3} + x + 28$$1$$3$$0$$C_3$$$[\ ]^{3}$$
31.1.3.2a1.3$x^{3} + 279$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.6.1.0a1.1$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$$1$$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)