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Properties

Label 12.4.56593444029595648.1
Degree $12$
Signature $[4, 4]$
Discriminant $5.659\times 10^{16}$
Root discriminant \(24.89\)
Ramified primes $2,7$
Class number $1$
Class group trivial
Galois group $C_6^2:C_4$ (as 12T82)

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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 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 4*y^10 - 16*y^9 + 84*y^8 - 152*y^7 + 192*y^6 - 384*y^5 + 283*y^4 + 764*y^3 - 1548*y^2 + 936*y - 158, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158)
 

\( x^{12} - 4 x^{11} + 4 x^{10} - 16 x^{9} + 84 x^{8} - 152 x^{7} + 192 x^{6} - 384 x^{5} + 283 x^{4} + \cdots - 158 \) 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:  $[4, 4]$
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:   \(56593444029595648\) \(\medspace = 2^{36}\cdot 7^{7}\) 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:  \(24.89\)
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:  $2^{27/8}7^{5/6}\approx 52.5078942662214$
Ramified primes:   \(2\), \(7\) 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:  \(\Q(\sqrt{7}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3459270819}a^{11}-\frac{1065858398}{3459270819}a^{10}-\frac{106650424}{1153090273}a^{9}-\frac{193977064}{3459270819}a^{8}-\frac{922644227}{3459270819}a^{7}-\frac{284456030}{1153090273}a^{6}+\frac{408204125}{1153090273}a^{5}+\frac{469967605}{1153090273}a^{4}-\frac{693688118}{3459270819}a^{3}+\frac{1165465006}{3459270819}a^{2}+\frac{1140281717}{3459270819}a+\frac{87034114}{3459270819}$ 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:  $7$
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:   $\frac{138196}{437163}a^{11}-\frac{379703}{437163}a^{10}+\frac{22316}{145721}a^{9}-\frac{2127187}{437163}a^{8}+\frac{8959774}{437163}a^{7}-\frac{3196478}{145721}a^{6}+\frac{4788138}{145721}a^{5}-\frac{11730443}{145721}a^{4}-\frac{5830790}{437163}a^{3}+\frac{98440759}{437163}a^{2}-\frac{87872614}{437163}a+\frac{17301211}{437163}$, $\frac{369963094}{1153090273}a^{11}-\frac{881275450}{1153090273}a^{10}+\frac{50453811}{1153090273}a^{9}-\frac{5863300596}{1153090273}a^{8}+\frac{21590232848}{1153090273}a^{7}-\frac{21206531164}{1153090273}a^{6}+\frac{37157615097}{1153090273}a^{5}-\frac{82334201613}{1153090273}a^{4}-\frac{29045591934}{1153090273}a^{3}+\frac{233558218311}{1153090273}a^{2}-\frac{193064288248}{1153090273}a+\frac{38783273421}{1153090273}$, $\frac{14513351}{28124153}a^{11}-\frac{38986407}{28124153}a^{10}+\frac{7562558}{28124153}a^{9}-\frac{223238191}{28124153}a^{8}+\frac{924851953}{28124153}a^{7}-\frac{1002923344}{28124153}a^{6}+\frac{1497815871}{28124153}a^{5}-\frac{3617495544}{28124153}a^{4}-\frac{589484264}{28124153}a^{3}+\frac{10236508838}{28124153}a^{2}-\frac{9182881481}{28124153}a+\frac{1796586689}{28124153}$, $\frac{472059604}{3459270819}a^{11}-\frac{1182436742}{3459270819}a^{10}-\frac{3302211}{1153090273}a^{9}-\frac{7483729405}{3459270819}a^{8}+\frac{28595882290}{3459270819}a^{7}-\frac{8902832178}{1153090273}a^{6}+\frac{15687339523}{1153090273}a^{5}-\frac{36664342970}{1153090273}a^{4}-\frac{42101284766}{3459270819}a^{3}+\frac{308763337519}{3459270819}a^{2}-\frac{242212112704}{3459270819}a+\frac{45303419887}{3459270819}$, $\frac{412871086}{1153090273}a^{11}-\frac{1149871726}{1153090273}a^{10}+\frac{215732479}{1153090273}a^{9}-\frac{6303534137}{1153090273}a^{8}+\frac{27051034791}{1153090273}a^{7}-\frac{29176330008}{1153090273}a^{6}+\frac{42505144983}{1153090273}a^{5}-\frac{106066949499}{1153090273}a^{4}-\frac{16391768340}{1153090273}a^{3}+\frac{300373109241}{1153090273}a^{2}-\frac{268088347466}{1153090273}a+\frac{53568763789}{1153090273}$, $\frac{16868225}{84372459}a^{11}-\frac{43676542}{84372459}a^{10}+\frac{3255570}{28124153}a^{9}-\frac{259167482}{84372459}a^{8}+\frac{1045319477}{84372459}a^{7}-\frac{386003090}{28124153}a^{6}+\frac{573705237}{28124153}a^{5}-\frac{1353520045}{28124153}a^{4}-\frac{643110322}{84372459}a^{3}+\frac{11710460027}{84372459}a^{2}-\frac{10589229941}{84372459}a+\frac{2134998803}{84372459}$, $\frac{620101583}{1153090273}a^{11}-\frac{2092776786}{1153090273}a^{10}+\frac{702900161}{1153090273}a^{9}-\frac{8959482835}{1153090273}a^{8}+\frac{47166748516}{1153090273}a^{7}-\frac{56449513969}{1153090273}a^{6}+\frac{66846335381}{1153090273}a^{5}-\frac{192438818611}{1153090273}a^{4}+\frac{13753058628}{1153090273}a^{3}+\frac{534489086973}{1153090273}a^{2}-\frac{515561083787}{1153090273}a+\frac{103232906217}{1153090273}$ 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:  \( 15543.9677994 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 15543.9677994 \cdot 1}{2\cdot\sqrt{56593444029595648}}\cr\approx \mathstrut & 0.814682310357 \end{aligned}\]

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 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158) 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 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158, 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 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158); 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 - 4*x^11 + 4*x^10 - 16*x^9 + 84*x^8 - 152*x^7 + 192*x^6 - 384*x^5 + 283*x^4 + 764*x^3 - 1548*x^2 + 936*x - 158); 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

$C_6^2:C_4$ (as 12T82):

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 $C_6^2:C_4$
Character table for $C_6^2:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, 6.2.802816.1

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: 12.4.1154968245501952.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.4.0.1}{4} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.1.4.10a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 2$$4$$1$$10$$D_{4}$$$[2, 3, \frac{7}{2}]$$
2.1.8.26c1.11$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$$[2, 3, \frac{7}{2}, 4]$$
\(7\) Copy content Toggle raw display 7.3.2.3a1.2$x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 23$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
7.2.3.4a1.1$x^{6} + 18 x^{5} + 117 x^{4} + 324 x^{3} + 351 x^{2} + 169 x + 27$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

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