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Properties

Label 12.0.814...896.1
Degree $12$
Signature $[0, 6]$
Discriminant $8.149\times 10^{39}$
Root discriminant \(2117.99\)
Ramified primes $2,3,7,181$
Class number $145800$ (GRH)
Class group [3, 3, 3, 30, 180] (GRH)
Galois group $S_3\times D_6$ (as 12T37)

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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^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400)
 
Copy content gp:K = bnfinit(y^12 + 4*y^10 - 3372*y^8 + 277432*y^6 + 2870368*y^4 - 325351584*y^2 + 9308390400, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 4*x^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 + 4*x^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400)
 

\( x^{12} + 4x^{10} - 3372x^{8} + 277432x^{6} + 2870368x^{4} - 325351584x^{2} + 9308390400 \) 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:   \(8148798716588303489414930185744360144896\) \(\medspace = 2^{20}\cdot 3^{6}\cdot 7^{10}\cdot 181^{10}\) 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:  \(2117.99\)
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:  $2^{11/6}3^{1/2}7^{5/6}181^{5/6}\approx 2377.366462449021$
Ramified primes:   \(2\), \(3\), \(7\), \(181\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-6}, \sqrt{-2534})\)

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{12}a^{5}+\frac{1}{6}a^{3}+\frac{1}{3}a$, $\frac{1}{24}a^{6}+\frac{1}{12}a^{4}+\frac{1}{6}a^{2}$, $\frac{1}{96}a^{7}-\frac{1}{12}a$, $\frac{1}{1152}a^{8}-\frac{1}{48}a^{6}-\frac{1}{24}a^{4}-\frac{13}{144}a^{2}$, $\frac{1}{771840}a^{9}+\frac{77}{16080}a^{7}-\frac{19}{1072}a^{5}-\frac{3421}{96480}a^{3}+\frac{1571}{4020}a$, $\frac{1}{16371640258560}a^{10}+\frac{2558892983}{8185820129280}a^{8}-\frac{586607507}{34107583872}a^{6}+\frac{242278253039}{2046455032320}a^{4}+\frac{197111645377}{1023227516160}a^{2}-\frac{3229235}{10605592}$, $\frac{1}{49114920775680}a^{11}+\frac{13550903}{24557460387840}a^{9}-\frac{347981687}{102322751616}a^{7}+\frac{249914279279}{6139365096960}a^{5}+\frac{718906771777}{3069682548480}a^{3}-\frac{245524123}{2131723992}a$ 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:  $C_{3}\times C_{3}\times C_{3}\times C_{30}\times C_{180}$, which has order $145800$ (assuming GRH)
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:  $C_{3}\times C_{3}\times C_{3}\times C_{30}\times C_{180}$, which has order $145800$ (assuming GRH)
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:   $\frac{920945675}{3274328051712}a^{10}-\frac{30598388093}{1637164025856}a^{8}+\frac{13758536437}{34107583872}a^{6}+\frac{2745076669883}{409291006464}a^{4}-\frac{182384184684331}{204645503232}a^{2}+\frac{53733587797}{10605592}$, $\frac{3632246820125}{363814227968}a^{10}-\frac{21\cdots 53}{1637164025856}a^{8}+\frac{892296989008111}{11369194624}a^{6}+\frac{81\cdots 07}{45476778496}a^{4}-\frac{27\cdots 51}{204645503232}a^{2}+\frac{36\cdots 01}{10605592}$, $\frac{68\cdots 01}{4911492077568}a^{11}+\frac{25\cdots 57}{1637164025856}a^{10}-\frac{34\cdots 87}{2455746038784}a^{9}+\frac{92\cdots 95}{90953556992}a^{8}-\frac{65\cdots 77}{51161375808}a^{7}-\frac{48\cdots 83}{17053791936}a^{6}+\frac{44\cdots 69}{613936509696}a^{5}-\frac{76\cdots 77}{204645503232}a^{4}-\frac{90\cdots 57}{306968254848}a^{3}+\frac{17\cdots 95}{34107583872}a^{2}-\frac{93\cdots 31}{1065861996}a-\frac{30\cdots 79}{5302796}$, $\frac{92\cdots 79}{113691946240}a^{10}+\frac{64\cdots 73}{511613758080}a^{8}-\frac{33\cdots 51}{710574664}a^{6}+\frac{17\cdots 21}{14211493280}a^{4}+\frac{26\cdots 67}{63951719760}a^{2}-\frac{19\cdots 51}{1325699}$, $\frac{21\cdots 63}{19428370560}a^{11}+\frac{10\cdots 25}{16290189312}a^{10}-\frac{13\cdots 97}{38856741120}a^{9}+\frac{39\cdots 65}{2715031552}a^{8}+\frac{52\cdots 91}{1619030880}a^{7}-\frac{10\cdots 41}{509068416}a^{6}-\frac{22\cdots 71}{1214273160}a^{5}+\frac{24\cdots 63}{2036273664}a^{4}-\frac{46\cdots 43}{4857092640}a^{3}+\frac{48\cdots 85}{1018136832}a^{2}+\frac{27\cdots 17}{16864905}a-\frac{15\cdots 71}{10605592}$ Copy content Toggle raw display (assuming GRH)
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:  \( 337066535001.9145 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 337066535001.9145 \cdot 145800}{2\cdot\sqrt{8148798716588303489414930185744360144896}}\cr\approx \mathstrut & 16.7484869742379 \end{aligned}\] (assuming GRH)

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^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400) 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^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400, 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^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400); 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^10 - 3372*x^8 + 277432*x^6 + 2870368*x^4 - 325351584*x^2 + 9308390400); 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

$S_3\times D_6$ (as 12T37):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 72
The 18 conjugacy class representatives for $S_3\times D_6$
Character table for $S_3\times D_6$

Intermediate fields

\(\Q(\sqrt{-2534}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3801}) \), \(\Q(\sqrt{-6}, \sqrt{-2534})\), 6.2.352619909717519556.2

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

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 sibling: 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 R ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.1.0.1}{1} }^{12}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.1.2.3a1.4$x^{2} + 4 x + 10$$2$$1$$3$$C_2$$$[3]$$
2.2.2.6a1.5$x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.1.6.11a1.6$x^{6} + 4 x^{3} + 10$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(7\) Copy content Toggle raw display 7.1.6.5a1.3$x^{6} + 21$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.1.6.5a1.3$x^{6} + 21$$6$$1$$5$$C_6$$$[\ ]_{6}$$
\(181\) Copy content Toggle raw display 181.2.6.10a1.4$x^{12} + 1062 x^{11} + 469947 x^{10} + 110915280 x^{9} + 14726353155 x^{8} + 1043025098382 x^{7} + 30808510677349 x^{6} + 2086050196764 x^{5} + 58905412620 x^{4} + 887322240 x^{3} + 7519152 x^{2} + 42672 x + 27757$$6$$2$$10$$C_6\times C_2$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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