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Properties

Label 12.4.103...144.1
Degree $12$
Signature $[4, 4]$
Discriminant $1.039\times 10^{37}$
Root discriminant \(1215.43\)
Ramified primes $2,3,7,181$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $S_3\times D_6$ (as 12T37)

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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^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 - 377*y^10 + 492*y^9 + 173850*y^8 - 694074*y^7 - 21012689*y^6 + 443688*y^5 + 4848871342*y^4 - 24924809490*y^3 - 46044538197*y^2 + 258630537564*y - 208255070079, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079)
 

\( x^{12} - 6 x^{11} - 377 x^{10} + 492 x^{9} + 173850 x^{8} - 694074 x^{7} - 21012689 x^{6} + \cdots - 208255070079 \) 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:   \(10393875914015693226294553808347398144\) \(\medspace = 2^{16}\cdot 3^{6}\cdot 7^{8}\cdot 181^{10}\) 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:  \(1215.43\)
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:  $2^{4/3}3^{1/2}7^{5/6}181^{5/6}\approx 1681.0519469631765$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{114}a^{8}-\frac{1}{114}a^{7}-\frac{4}{57}a^{6}+\frac{37}{114}a^{5}-\frac{11}{57}a^{4}+\frac{41}{114}a^{3}+\frac{29}{114}a^{2}-\frac{4}{19}a+\frac{2}{19}$, $\frac{1}{798}a^{9}+\frac{1}{798}a^{8}-\frac{8}{133}a^{7}+\frac{1}{399}a^{6}+\frac{11}{266}a^{5}-\frac{269}{798}a^{4}+\frac{8}{399}a^{3}-\frac{1}{19}a^{2}+\frac{1}{38}a-\frac{7}{38}$, $\frac{1}{4784444910}a^{10}-\frac{144216}{265802495}a^{9}+\frac{1975046}{2392222455}a^{8}+\frac{8409349}{227830710}a^{7}+\frac{73555571}{1594814970}a^{6}-\frac{27327911}{1594814970}a^{5}-\frac{30928589}{4784444910}a^{4}-\frac{52499131}{1594814970}a^{3}+\frac{114369419}{341746065}a^{2}+\frac{43656697}{113915355}a+\frac{90456743}{227830710}$, $\frac{1}{18\cdots 10}a^{11}-\frac{22\cdots 44}{31\cdots 85}a^{10}-\frac{53\cdots 28}{90\cdots 19}a^{9}+\frac{61\cdots 76}{44\cdots 55}a^{8}+\frac{15\cdots 34}{28\cdots 35}a^{7}-\frac{78\cdots 72}{10\cdots 95}a^{6}+\frac{69\cdots 19}{18\cdots 10}a^{5}+\frac{23\cdots 57}{62\cdots 70}a^{4}-\frac{30\cdots 63}{94\cdots 55}a^{3}-\frac{37\cdots 67}{89\cdots 10}a^{2}+\frac{34\cdots 72}{44\cdots 55}a+\frac{14\cdots 79}{29\cdots 70}$ 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:  $C_{6}$, which has order $6$ (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:  $C_{2}\times C_{12}$, which has order $24$ (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:  $7$
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:   $\frac{52\cdots 64}{82\cdots 01}a^{11}+\frac{22\cdots 74}{82\cdots 01}a^{10}+\frac{18\cdots 84}{75\cdots 91}a^{9}+\frac{43\cdots 43}{39\cdots 81}a^{8}-\frac{27\cdots 84}{25\cdots 97}a^{7}+\frac{70\cdots 22}{27\cdots 67}a^{6}+\frac{11\cdots 64}{82\cdots 01}a^{5}+\frac{19\cdots 10}{82\cdots 01}a^{4}-\frac{35\cdots 56}{11\cdots 43}a^{3}+\frac{42\cdots 74}{39\cdots 81}a^{2}+\frac{16\cdots 00}{39\cdots 81}a-\frac{41\cdots 34}{39\cdots 81}$, $\frac{27\cdots 76}{26\cdots 55}a^{11}+\frac{13\cdots 29}{52\cdots 10}a^{10}+\frac{94\cdots 56}{23\cdots 05}a^{9}+\frac{68\cdots 99}{74\cdots 30}a^{8}-\frac{14\cdots 34}{79\cdots 35}a^{7}+\frac{12\cdots 57}{11\cdots 58}a^{6}+\frac{56\cdots 07}{26\cdots 55}a^{5}+\frac{40\cdots 39}{52\cdots 10}a^{4}-\frac{17\cdots 14}{37\cdots 65}a^{3}+\frac{14\cdots 33}{14\cdots 46}a^{2}+\frac{27\cdots 76}{41\cdots 85}a-\frac{20\cdots 11}{24\cdots 10}$, $\frac{974857603955534}{16\cdots 05}a^{11}-\frac{51\cdots 38}{49\cdots 15}a^{10}+\frac{43\cdots 61}{16\cdots 05}a^{9}+\frac{72\cdots 09}{14\cdots 90}a^{8}-\frac{15\cdots 14}{16\cdots 05}a^{7}-\frac{33\cdots 97}{16\cdots 05}a^{6}+\frac{49\cdots 43}{33\cdots 81}a^{5}+\frac{30\cdots 99}{99\cdots 30}a^{4}-\frac{91\cdots 58}{47\cdots 83}a^{3}-\frac{37\cdots 01}{70\cdots 45}a^{2}+\frac{42\cdots 48}{23\cdots 15}a-\frac{62\cdots 39}{47\cdots 30}$, $\frac{75\cdots 90}{29\cdots 17}a^{11}-\frac{21\cdots 29}{17\cdots 02}a^{10}-\frac{54\cdots 70}{57\cdots 87}a^{9}+\frac{73\cdots 90}{62\cdots 57}a^{8}+\frac{24\cdots 91}{57\cdots 87}a^{7}-\frac{15\cdots 33}{12\cdots 14}a^{6}-\frac{34\cdots 82}{62\cdots 57}a^{5}-\frac{38\cdots 08}{62\cdots 57}a^{4}+\frac{76\cdots 67}{62\cdots 57}a^{3}-\frac{29\cdots 33}{59\cdots 34}a^{2}-\frac{51\cdots 23}{29\cdots 17}a+\frac{13\cdots 24}{29\cdots 17}$, $\frac{80\cdots 51}{20\cdots 19}a^{11}-\frac{11\cdots 95}{41\cdots 38}a^{10}+\frac{49\cdots 55}{38\cdots 58}a^{9}+\frac{38\cdots 25}{20\cdots 19}a^{8}-\frac{91\cdots 78}{19\cdots 29}a^{7}-\frac{11\cdots 49}{22\cdots 02}a^{6}+\frac{12\cdots 65}{41\cdots 38}a^{5}+\frac{26\cdots 03}{29\cdots 17}a^{4}-\frac{14\cdots 57}{29\cdots 17}a^{3}-\frac{48\cdots 09}{59\cdots 34}a^{2}+\frac{28\cdots 17}{59\cdots 34}a-\frac{11\cdots 52}{29\cdots 17}$, $\frac{21\cdots 63}{18\cdots 10}a^{11}+\frac{44\cdots 21}{53\cdots 06}a^{10}-\frac{16\cdots 02}{85\cdots 05}a^{9}+\frac{63\cdots 17}{37\cdots 42}a^{8}-\frac{47\cdots 83}{16\cdots 82}a^{7}+\frac{86\cdots 01}{27\cdots 30}a^{6}-\frac{86\cdots 43}{94\cdots 55}a^{5}+\frac{48\cdots 37}{53\cdots 06}a^{4}-\frac{39\cdots 21}{18\cdots 10}a^{3}-\frac{25\cdots 83}{26\cdots 30}a^{2}+\frac{30\cdots 04}{89\cdots 51}a-\frac{22\cdots 77}{89\cdots 10}$, $\frac{35\cdots 21}{95\cdots 45}a^{11}+\frac{12\cdots 59}{57\cdots 70}a^{10}-\frac{26\cdots 83}{19\cdots 90}a^{9}-\frac{20\cdots 26}{28\cdots 35}a^{8}+\frac{16\cdots 93}{28\cdots 35}a^{7}+\frac{77\cdots 33}{57\cdots 70}a^{6}-\frac{11\cdots 93}{16\cdots 82}a^{5}-\frac{12\cdots 98}{28\cdots 35}a^{4}+\frac{84\cdots 98}{57\cdots 87}a^{3}+\frac{17\cdots 41}{27\cdots 70}a^{2}-\frac{34\cdots 39}{27\cdots 70}a+\frac{15\cdots 63}{13\cdots 35}$ 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:  \( 56411363733056.39 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 56411363733056.39 \cdot 6}{2\cdot\sqrt{10393875914015693226294553808347398144}}\cr\approx \mathstrut & 1.30899668214625 \end{aligned}\] (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 - 6*x^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079) 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^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079, 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^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079); 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 - 6*x^11 - 377*x^10 + 492*x^9 + 173850*x^8 - 694074*x^7 - 21012689*x^6 + 443688*x^5 + 4848871342*x^4 - 24924809490*x^3 - 46044538197*x^2 + 258630537564*x - 208255070079); 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

\(\Q(\sqrt{543}) \), \(\Q(\sqrt{1267}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{21}, \sqrt{543})\), 6.2.65794968577029888.1

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(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 ${\href{/padicField/5.2.0.1}{2} }^{6}$ R ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.2.6.16a1.5$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 143 x^{6} + 132 x^{5} + 102 x^{4} + 64 x^{3} + 33 x^{2} + 12 x + 5$$6$$2$$16$$D_6$$$[2]_{3}^{2}$$
\(3\) 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 7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
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$$$[\ ]_{2}^{3}$$
\(181\) 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)