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Properties

Label 12.2.72185515343872.1
Degree $12$
Signature $[2, 5]$
Discriminant $-7.219\times 10^{13}$
Root discriminant \(14.28\)
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 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 4*y^10 + 4*y^9 - 17*y^8 + 16*y^7 - 12*y^6 + 16*y^5 - 12*y^4 + 8*y^3 - 4*y^2 - 2, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 4*x^10 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 4*x^10 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2)
 

\( x^{12} - 4x^{11} + 4x^{10} + 4x^{9} - 17x^{8} + 16x^{7} - 12x^{6} + 16x^{5} - 12x^{4} + 8x^{3} - 4x^{2} - 2 \) 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:  $[2, 5]$
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:   \(-72185515343872\) \(\medspace = -\,2^{32}\cdot 7^{5}\) 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:  \(14.28\)
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:  $2^{3}7^{5/6}\approx 40.48912147837109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{38391}a^{11}-\frac{4370}{38391}a^{10}-\frac{13700}{38391}a^{9}+\frac{13823}{38391}a^{8}-\frac{583}{38391}a^{7}-\frac{14006}{38391}a^{6}+\frac{6305}{12797}a^{5}+\frac{2988}{12797}a^{4}-\frac{3610}{38391}a^{3}-\frac{4636}{38391}a^{2}-\frac{4082}{38391}a+\frac{8588}{38391}$ 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:  $6$
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:   $\frac{2897}{38391}a^{11}-\frac{1219}{12797}a^{10}-\frac{18200}{38391}a^{9}+\frac{41809}{38391}a^{8}-\frac{8447}{12797}a^{7}-\frac{28558}{12797}a^{6}+\frac{102377}{38391}a^{5}-\frac{85984}{38391}a^{4}+\frac{11790}{12797}a^{3}-\frac{44830}{38391}a^{2}+\frac{37265}{38391}a-\frac{10729}{38391}$, $\frac{10127}{38391}a^{11}-\frac{28558}{38391}a^{10}+\frac{5174}{38391}a^{9}+\frac{50326}{38391}a^{8}-\frac{107000}{38391}a^{7}+\frac{15983}{38391}a^{6}-\frac{31889}{12797}a^{5}+\frac{32962}{12797}a^{4}-\frac{48629}{38391}a^{3}+\frac{3421}{38391}a^{2}+\frac{8693}{38391}a+\frac{15061}{38391}$, $\frac{1606}{38391}a^{11}+\frac{7333}{38391}a^{10}-\frac{42548}{38391}a^{9}+\frac{48131}{38391}a^{8}+\frac{23477}{38391}a^{7}-\frac{188465}{38391}a^{6}+\frac{54591}{12797}a^{5}-\frac{51335}{12797}a^{4}+\frac{114554}{38391}a^{3}-\frac{112735}{38391}a^{2}+\frac{85951}{38391}a+\frac{9959}{38391}$, $\frac{5894}{38391}a^{11}-\frac{22013}{38391}a^{10}+\frac{4689}{12797}a^{9}+\frac{45451}{38391}a^{8}-\frac{108982}{38391}a^{7}+\frac{40474}{38391}a^{6}+\frac{23140}{38391}a^{5}+\frac{33394}{38391}a^{4}-\frac{59914}{38391}a^{3}-\frac{18059}{12797}a^{2}+\frac{50240}{38391}a+\frac{10377}{12797}$, $\frac{1210}{12797}a^{11}-\frac{20414}{38391}a^{10}+\frac{36533}{38391}a^{9}+\frac{151}{12797}a^{8}-\frac{107161}{38391}a^{7}+\frac{167062}{38391}a^{6}-\frac{84034}{38391}a^{5}+\frac{34940}{38391}a^{4}-\frac{76954}{38391}a^{3}+\frac{88954}{38391}a^{2}+\frac{13219}{12797}a-\frac{11849}{38391}$, $\frac{2395}{12797}a^{11}-\frac{11001}{12797}a^{10}+\frac{12805}{12797}a^{9}+\frac{13043}{12797}a^{8}-\frac{52600}{12797}a^{7}+\frac{47755}{12797}a^{6}-\frac{12752}{12797}a^{5}+\frac{33805}{12797}a^{4}-\frac{46366}{12797}a^{3}+\frac{4576}{12797}a^{2}+\frac{518}{12797}a+\frac{3481}{12797}$ 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:  \( 339.925030123 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 339.925030123 \cdot 1}{2\cdot\sqrt{72185515343872}}\cr\approx \mathstrut & 0.783587413588 \end{aligned}\]

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 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2) 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 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2, 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 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2); 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 + 4*x^9 - 17*x^8 + 16*x^7 - 12*x^6 + 16*x^5 - 12*x^4 + 8*x^3 - 4*x^2 - 2); 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

\(\Q(\sqrt{2}) \), 4.2.1792.1, 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: 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 ${\href{/padicField/3.4.0.1}{4} }^{3}$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ R ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}$

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$$$[2, 3]$$
2.1.8.24c1.61$x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$$[2, 3, 4]$$
\(7\) Copy content Toggle raw display 7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.1.6.5a1.2$x^{6} + 14$$6$$1$$5$$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)