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Properties

Label 12.4.657366253849018368.9
Degree $12$
Signature $[4, 4]$
Discriminant $6.574\times 10^{17}$
Root discriminant \(30.54\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $S_4^2:C_4$ (as 12T237)

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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^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72)
 
Copy content gp:K = bnfinit(y^12 + 6*y^10 - 39*y^8 - 120*y^6 + 360*y^4 - 864*y^2 + 72, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 6*x^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + 6*x^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72)
 

\( x^{12} + 6x^{10} - 39x^{8} - 120x^{6} + 360x^{4} - 864x^{2} + 72 \) 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:   \(657366253849018368\) \(\medspace = 2^{37}\cdot 3^{14}\) 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:  \(30.54\)
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:  $2^{137/32}3^{25/18}\approx 89.42379121066101$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{42}a^{8}-\frac{1}{14}a^{6}-\frac{1}{7}a^{4}-\frac{2}{7}$, $\frac{1}{84}a^{9}-\frac{1}{84}a^{8}-\frac{1}{28}a^{7}+\frac{1}{28}a^{6}-\frac{1}{14}a^{5}+\frac{1}{14}a^{4}-\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{115836}a^{10}+\frac{643}{57918}a^{8}+\frac{1677}{38612}a^{6}+\frac{1777}{19306}a^{4}-\frac{1744}{9653}a^{2}-\frac{2549}{9653}$, $\frac{1}{115836}a^{11}-\frac{31}{38612}a^{9}-\frac{1}{84}a^{8}+\frac{764}{9653}a^{7}+\frac{1}{28}a^{6}+\frac{1578}{9653}a^{5}+\frac{1}{14}a^{4}-\frac{1744}{9653}a^{3}-\frac{1170}{9653}a+\frac{1}{7}$ 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:   $\frac{2}{591}a^{10}+\frac{11}{591}a^{8}-\frac{167}{1182}a^{6}-\frac{165}{394}a^{4}+\frac{232}{197}a^{2}-\frac{495}{197}$, $\frac{5}{115836}a^{10}+\frac{457}{57918}a^{8}+\frac{3091}{115836}a^{6}-\frac{4905}{19306}a^{4}+\frac{933}{9653}a^{2}-\frac{7229}{9653}$, $\frac{121}{28959}a^{10}+\frac{937}{57918}a^{8}-\frac{13945}{57918}a^{6}-\frac{2972}{9653}a^{4}+\frac{34327}{9653}a^{2}-\frac{24333}{9653}$, $\frac{359}{57918}a^{11}-\frac{3}{9653}a^{10}+\frac{2467}{57918}a^{9}+\frac{295}{57918}a^{8}-\frac{11393}{57918}a^{7}+\frac{1607}{28959}a^{6}-\frac{16237}{19306}a^{5}-\frac{4675}{19306}a^{4}+\frac{12351}{9653}a^{3}-\frac{14440}{9653}a^{2}-\frac{52651}{9653}a+\frac{15919}{9653}$, $\frac{359}{57918}a^{11}+\frac{3}{9653}a^{10}+\frac{2467}{57918}a^{9}-\frac{295}{57918}a^{8}-\frac{11393}{57918}a^{7}-\frac{1607}{28959}a^{6}-\frac{16237}{19306}a^{5}+\frac{4675}{19306}a^{4}+\frac{12351}{9653}a^{3}+\frac{14440}{9653}a^{2}-\frac{52651}{9653}a-\frac{15919}{9653}$, $\frac{2365}{38612}a^{11}+\frac{715}{38612}a^{10}+\frac{14485}{38612}a^{9}+\frac{12881}{115836}a^{8}-\frac{135377}{57918}a^{7}-\frac{14113}{19306}a^{6}-\frac{73278}{9653}a^{5}-\frac{22699}{9653}a^{4}+\frac{204179}{9653}a^{3}+\frac{62402}{9653}a^{2}-\frac{490494}{9653}a-\frac{144665}{9653}$, $\frac{3865}{115836}a^{11}-\frac{3049}{115836}a^{10}+\frac{23917}{115836}a^{9}-\frac{4769}{38612}a^{8}-\frac{78835}{57918}a^{7}+\frac{13485}{9653}a^{6}-\frac{48612}{9653}a^{5}+\frac{28968}{9653}a^{4}+\frac{132376}{9653}a^{3}-\frac{184754}{9653}a^{2}-\frac{59606}{9653}a+\frac{32953}{9653}$ 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:  \( 58908.8441685 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 58908.8441685 \cdot 1}{2\cdot\sqrt{657366253849018368}}\cr\approx \mathstrut & 0.905912514749 \end{aligned}\]

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^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72) 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^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72, 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^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72); 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^10 - 39*x^8 - 120*x^6 + 360*x^4 - 864*x^2 + 72); 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

$S_4^2:C_4$ (as 12T237):

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 2304
The 40 conjugacy class representatives for $S_4^2:C_4$
Character table for $S_4^2:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 6.4.5971968.1

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 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.4.164341563462254592.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.1.4.8b1.6$x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$$[2, 3]$$
2.1.8.29a1.20$x^{8} + 20 x^{6} + 16 x^{5} + 8 x^{4} + 2$$8$$1$$29$$(((C_4 \times C_2): C_2):C_2):C_2$$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$$
\(3\) 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$$$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$

Spectrum of ring of integers

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