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Properties

Label 12.0.746...616.2
Degree $12$
Signature $[0, 6]$
Discriminant $7.467\times 10^{31}$
Root discriminant \(453.00\)
Ramified primes $2,3,11$
Class number $12$ (GRH)
Class group [2, 6] (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

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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 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723)
 
Copy content gp:K = bnfinit(y^12 - 6*y^11 + 33*y^10 - 110*y^9 + 2112*y^8 - 7854*y^7 + 24013*y^6 - 35442*y^5 + 1081080*y^4 - 2029698*y^3 + 1589841*y^2 + 6368382*y + 214386723, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723)
 

\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 2112 x^{8} - 7854 x^{7} + 24013 x^{6} - 35442 x^{5} + \cdots + 214386723 \) 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:   \(74666441181916948967589849071616\) \(\medspace = 2^{22}\cdot 3^{18}\cdot 11^{16}\) 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:  \(453.00\)
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:  $2^{11/6}3^{3/2}11^{84/55}\approx 721.2085287457858$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{12}a^{4}-\frac{1}{6}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{12}a^{5}-\frac{1}{12}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{36}a^{6}+\frac{1}{36}a^{5}+\frac{1}{36}a^{4}+\frac{1}{12}a^{3}-\frac{5}{12}a^{2}+\frac{1}{4}a$, $\frac{1}{36}a^{7}-\frac{1}{36}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{432}a^{8}-\frac{1}{72}a^{6}-\frac{1}{108}a^{5}+\frac{1}{144}a^{4}+\frac{5}{36}a^{3}-\frac{7}{24}a^{2}-\frac{1}{12}a+\frac{7}{16}$, $\frac{1}{432}a^{9}-\frac{1}{72}a^{7}-\frac{1}{108}a^{6}+\frac{1}{144}a^{5}-\frac{1}{36}a^{4}+\frac{1}{24}a^{3}-\frac{1}{4}a^{2}+\frac{7}{16}a-\frac{1}{2}$, $\frac{1}{2592}a^{10}-\frac{1}{2592}a^{9}-\frac{1}{864}a^{8}-\frac{11}{1296}a^{7}-\frac{11}{2592}a^{6}+\frac{5}{288}a^{5}-\frac{1}{288}a^{4}-\frac{1}{16}a^{3}+\frac{15}{32}a^{2}-\frac{35}{96}a-\frac{19}{96}$, $\frac{1}{15\cdots 16}a^{11}-\frac{888239361157447}{76\cdots 08}a^{10}-\frac{18719020370717}{38\cdots 04}a^{9}+\frac{10\cdots 19}{15\cdots 16}a^{8}-\frac{33\cdots 41}{41\cdots 68}a^{7}+\frac{15\cdots 65}{19\cdots 52}a^{6}+\frac{13\cdots 03}{42\cdots 56}a^{5}+\frac{805951915828951}{62\cdots 12}a^{4}+\frac{91\cdots 27}{56\cdots 08}a^{3}-\frac{228450352024571}{70\cdots 76}a^{2}-\frac{36\cdots 87}{28\cdots 04}a+\frac{13\cdots 53}{56\cdots 08}$ 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:  $C_{2}\times C_{6}$, which has order $12$ (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:  $C_{2}\times C_{6}$, which has order $12$ (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:   $\frac{18\cdots 57}{25\cdots 36}a^{11}-\frac{17\cdots 09}{84\cdots 12}a^{10}+\frac{77\cdots 47}{63\cdots 84}a^{9}-\frac{72\cdots 47}{12\cdots 68}a^{8}+\frac{27\cdots 45}{22\cdots 76}a^{7}-\frac{60\cdots 49}{25\cdots 36}a^{6}-\frac{22\cdots 31}{21\cdots 28}a^{5}-\frac{10\cdots 57}{15\cdots 28}a^{4}+\frac{10\cdots 29}{28\cdots 04}a^{3}-\frac{66\cdots 61}{94\cdots 68}a^{2}-\frac{28\cdots 65}{47\cdots 84}a-\frac{79\cdots 69}{23\cdots 92}$, $\frac{23\cdots 81}{76\cdots 08}a^{11}-\frac{18\cdots 57}{76\cdots 08}a^{10}+\frac{98\cdots 61}{76\cdots 08}a^{9}-\frac{13\cdots 33}{38\cdots 04}a^{8}+\frac{11\cdots 33}{20\cdots 84}a^{7}-\frac{19\cdots 83}{76\cdots 08}a^{6}+\frac{16\cdots 45}{28\cdots 04}a^{5}+\frac{26\cdots 33}{14\cdots 52}a^{4}+\frac{93\cdots 43}{94\cdots 68}a^{3}-\frac{69\cdots 99}{28\cdots 04}a^{2}-\frac{49\cdots 59}{28\cdots 04}a+\frac{29\cdots 33}{17\cdots 19}$, $\frac{30\cdots 21}{95\cdots 26}a^{11}-\frac{48\cdots 93}{76\cdots 08}a^{10}+\frac{22\cdots 21}{38\cdots 04}a^{9}-\frac{13\cdots 69}{38\cdots 04}a^{8}+\frac{57\cdots 17}{25\cdots 98}a^{7}-\frac{11\cdots 87}{76\cdots 08}a^{6}+\frac{36\cdots 51}{42\cdots 56}a^{5}-\frac{48\cdots 67}{14\cdots 52}a^{4}+\frac{84\cdots 29}{70\cdots 76}a^{3}-\frac{15\cdots 83}{28\cdots 04}a^{2}+\frac{28\cdots 09}{14\cdots 52}a-\frac{14\cdots 59}{35\cdots 38}$, $\frac{71\cdots 35}{12\cdots 68}a^{11}-\frac{20\cdots 27}{31\cdots 42}a^{10}+\frac{47\cdots 71}{12\cdots 68}a^{9}-\frac{21\cdots 39}{25\cdots 36}a^{8}+\frac{17\cdots 25}{34\cdots 64}a^{7}-\frac{53\cdots 71}{12\cdots 68}a^{6}+\frac{87\cdots 69}{42\cdots 56}a^{5}-\frac{21\cdots 57}{31\cdots 56}a^{4}+\frac{24\cdots 35}{14\cdots 52}a^{3}-\frac{10\cdots 11}{15\cdots 28}a^{2}+\frac{56\cdots 61}{15\cdots 28}a+\frac{23\cdots 83}{94\cdots 68}$, $\frac{21\cdots 27}{76\cdots 08}a^{11}-\frac{53\cdots 57}{19\cdots 52}a^{10}-\frac{19\cdots 21}{38\cdots 04}a^{9}-\frac{55\cdots 97}{76\cdots 08}a^{8}-\frac{19\cdots 19}{20\cdots 84}a^{7}-\frac{20\cdots 15}{38\cdots 04}a^{6}-\frac{73\cdots 13}{42\cdots 56}a^{5}-\frac{80\cdots 95}{94\cdots 68}a^{4}-\frac{41\cdots 99}{28\cdots 04}a^{3}-\frac{32\cdots 29}{14\cdots 52}a^{2}-\frac{66\cdots 37}{70\cdots 76}a-\frac{73\cdots 87}{28\cdots 04}$ 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:  \( 39976757877.1 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 39976757877.1 \cdot 12}{2\cdot\sqrt{74666441181916948967589849071616}}\cr\approx \mathstrut & 1.70795059659 \end{aligned}\] (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 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723) 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 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723, 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 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723); 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 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723); 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

$\PSL(2,11)$ (as 12T179):

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 non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 11 siblings: 11.3.14403248684783362069365325824.2, 11.3.14403248684783362069365325824.4
Minimal sibling: 11.3.14403248684783362069365325824.2

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 ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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 2.2.6.22a1.1$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$$6$$2$$22$$D_6$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.6.9a1.7$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$$[2]_{2}$$
3.1.6.9a1.7$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$$[2]_{2}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$

Spectrum of ring of integers

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