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Properties

Label 12.0.140...176.3
Degree $12$
Signature $[0, 6]$
Discriminant $1.405\times 10^{26}$
Root discriminant \(151.00\)
Ramified primes $2,3,11$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

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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 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089)
 
Copy content gp:K = bnfinit(y^12 - 4*y^11 + 66*y^10 - 308*y^9 + 1331*y^8 - 2904*y^7 + 6248*y^6 + 37840*y^5 + 141999*y^4 + 173756*y^3 + 537526*y^2 + 334932*y + 259089, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089)
 

\( x^{12} - 4 x^{11} + 66 x^{10} - 308 x^{9} + 1331 x^{8} - 2904 x^{7} + 6248 x^{6} + 37840 x^{5} + \cdots + 259089 \) 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:   \(140498081973195423325618176\) \(\medspace = 2^{22}\cdot 3^{6}\cdot 11^{16}\) 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:  \(151.00\)
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:  $2^{11/6}3^{1/2}11^{84/55}\approx 240.40284291526194$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{24}a^{6}+\frac{1}{4}a^{5}+\frac{7}{24}a^{4}-\frac{11}{24}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{24}a^{7}-\frac{5}{24}a^{5}+\frac{1}{4}a^{4}-\frac{11}{24}a^{3}-\frac{1}{2}a^{2}-\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{72}a^{8}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}-\frac{1}{6}a^{3}-\frac{5}{36}a^{2}-\frac{1}{6}a+\frac{7}{24}$, $\frac{1}{72}a^{9}+\frac{1}{3}a^{5}-\frac{5}{36}a^{3}+\frac{7}{24}a-\frac{1}{2}$, $\frac{1}{216}a^{10}-\frac{1}{216}a^{8}-\frac{1}{72}a^{7}+\frac{1}{72}a^{6}-\frac{11}{24}a^{5}+\frac{17}{216}a^{4}-\frac{1}{8}a^{3}+\frac{41}{108}a^{2}-\frac{11}{72}a+\frac{4}{9}$, $\frac{1}{69\cdots 68}a^{11}-\frac{12\cdots 09}{23\cdots 56}a^{10}+\frac{24\cdots 63}{69\cdots 68}a^{9}-\frac{75\cdots 63}{23\cdots 56}a^{8}-\frac{13\cdots 43}{11\cdots 28}a^{7}+\frac{33\cdots 63}{19\cdots 88}a^{6}+\frac{69\cdots 85}{17\cdots 92}a^{5}+\frac{18\cdots 29}{11\cdots 28}a^{4}+\frac{76\cdots 85}{69\cdots 68}a^{3}-\frac{92\cdots 59}{23\cdots 56}a^{2}+\frac{14\cdots 75}{23\cdots 56}a-\frac{80\cdots 99}{25\cdots 84}$ 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:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Ideal class group:  $C_{2}\times C_{2}$, which has order $4$ (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:  $C_{2}\times C_{2}$, which has order $4$ (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:   $\frac{34\cdots 02}{14\cdots 41}a^{11}-\frac{12\cdots 11}{11\cdots 28}a^{10}+\frac{87\cdots 27}{58\cdots 64}a^{9}-\frac{83\cdots 67}{11\cdots 28}a^{8}+\frac{16\cdots 79}{64\cdots 96}a^{7}-\frac{46\cdots 93}{19\cdots 88}a^{6}-\frac{71\cdots 15}{58\cdots 64}a^{5}+\frac{28\cdots 71}{14\cdots 41}a^{4}-\frac{19\cdots 25}{58\cdots 64}a^{3}+\frac{77\cdots 85}{11\cdots 28}a^{2}+\frac{56\cdots 65}{16\cdots 49}a+\frac{15\cdots 57}{38\cdots 76}$, $\frac{22\cdots 39}{69\cdots 68}a^{11}+\frac{38\cdots 61}{23\cdots 56}a^{10}+\frac{49\cdots 75}{69\cdots 68}a^{9}+\frac{24\cdots 57}{23\cdots 56}a^{8}-\frac{36\cdots 81}{58\cdots 64}a^{7}+\frac{47\cdots 69}{12\cdots 92}a^{6}-\frac{32\cdots 57}{34\cdots 84}a^{5}+\frac{20\cdots 99}{58\cdots 64}a^{4}+\frac{10\cdots 57}{69\cdots 68}a^{3}+\frac{87\cdots 97}{23\cdots 56}a^{2}+\frac{85\cdots 17}{23\cdots 56}a+\frac{10\cdots 69}{77\cdots 52}$, $\frac{32\cdots 49}{34\cdots 84}a^{11}-\frac{20\cdots 15}{34\cdots 84}a^{10}+\frac{19\cdots 72}{43\cdots 23}a^{9}-\frac{11\cdots 47}{34\cdots 84}a^{8}+\frac{96\cdots 89}{11\cdots 28}a^{7}+\frac{60\cdots 53}{58\cdots 64}a^{6}-\frac{32\cdots 93}{34\cdots 84}a^{5}-\frac{23\cdots 99}{17\cdots 92}a^{4}-\frac{84\cdots 49}{87\cdots 46}a^{3}-\frac{26\cdots 35}{34\cdots 84}a^{2}-\frac{54\cdots 63}{11\cdots 28}a-\frac{48\cdots 85}{11\cdots 28}$, $\frac{16\cdots 53}{34\cdots 84}a^{11}-\frac{86\cdots 43}{34\cdots 84}a^{10}+\frac{11\cdots 97}{34\cdots 84}a^{9}-\frac{32\cdots 35}{17\cdots 92}a^{8}+\frac{95\cdots 11}{11\cdots 28}a^{7}-\frac{24\cdots 79}{11\cdots 28}a^{6}+\frac{16\cdots 35}{34\cdots 84}a^{5}+\frac{52\cdots 05}{34\cdots 84}a^{4}+\frac{20\cdots 12}{43\cdots 23}a^{3}+\frac{10\cdots 89}{87\cdots 46}a^{2}+\frac{26\cdots 06}{14\cdots 41}a-\frac{10\cdots 43}{11\cdots 28}$, $\frac{82\cdots 73}{23\cdots 56}a^{11}+\frac{93\cdots 95}{69\cdots 68}a^{10}+\frac{82\cdots 15}{23\cdots 56}a^{9}+\frac{10\cdots 75}{69\cdots 68}a^{8}-\frac{15\cdots 05}{11\cdots 28}a^{7}+\frac{10\cdots 19}{11\cdots 28}a^{6}-\frac{50\cdots 61}{14\cdots 41}a^{5}+\frac{21\cdots 43}{17\cdots 92}a^{4}-\frac{19\cdots 19}{23\cdots 56}a^{3}+\frac{22\cdots 19}{69\cdots 68}a^{2}+\frac{31\cdots 61}{23\cdots 56}a+\frac{48\cdots 95}{23\cdots 56}$ 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:  \( 665279714.869 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 665279714.869 \cdot 4}{2\cdot\sqrt{140498081973195423325618176}}\cr\approx \mathstrut & 6.90682241229 \end{aligned}\] (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^11 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089) 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 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089, 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 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089); 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^11 + 66*x^10 - 308*x^9 + 1331*x^8 - 2904*x^7 + 6248*x^6 + 37840*x^5 + 141999*x^4 + 173756*x^3 + 537526*x^2 + 334932*x + 259089); 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: 11.3.243920281203464276606976.1, 11.3.243920281203464276606976.6
Minimal sibling: 11.3.243920281203464276606976.1

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 ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 2.2.6.22a1.1$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 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(11\) 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)