↑

Properties

Label 11.5.967...000.1
Degree $11$
Signature $[5, 3]$
Discriminant $-9.679\times 10^{18}$
Root discriminant \(53.21\)
Ramified primes $2,3,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{11}$ (as 11T8)

Related objects

Downloads

Learn more

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^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168)
 
Copy content gp:K = bnfinit(y^11 - 4*y^10 + 6*y^9 - 8*y^8 - 31*y^7 + 168*y^6 + 106*y^5 - 748*y^4 - 396*y^3 + 1008*y^2 + 882*y + 168, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168)
 

\( x^{11} - 4 x^{10} + 6 x^{9} - 8 x^{8} - 31 x^{7} + 168 x^{6} + 106 x^{5} - 748 x^{4} - 396 x^{3} + \cdots + 168 \) 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:  $11$
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:  $[5, 3]$
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:   \(-9678619238400000000\) \(\medspace = -\,2^{19}\cdot 3^{9}\cdot 5^{8}\cdot 7^{4}\) 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:  \(53.21\)
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:  $2^{115/48}3^{5/4}5^{11/12}7^{2/3}\approx 332.46081383675426$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{-6}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}$, $\frac{1}{24}a^{9}-\frac{1}{12}a^{8}+\frac{1}{24}a^{7}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{1}{6}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{182960160}a^{10}-\frac{2833111}{182960160}a^{9}+\frac{13893323}{182960160}a^{8}+\frac{5281277}{60986720}a^{7}-\frac{9078017}{45740040}a^{6}+\frac{17000761}{45740040}a^{5}+\frac{7439119}{91480080}a^{4}-\frac{9805169}{30493360}a^{3}+\frac{7890337}{30493360}a^{2}-\frac{5158011}{30493360}a-\frac{3467259}{7623340}$ 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:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (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:  Trivial group, which has order $1$ (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:   $\frac{8963491}{182960160}a^{10}-\frac{10056487}{60986720}a^{9}+\frac{16646411}{60986720}a^{8}-\frac{79086779}{182960160}a^{7}-\frac{23067149}{15246680}a^{6}+\frac{104280257}{15246680}a^{5}+\frac{584905709}{91480080}a^{4}-\frac{703866419}{30493360}a^{3}-\frac{480683453}{30493360}a^{2}+\frac{369229039}{30493360}a+\frac{34307451}{7623340}$, $\frac{731981}{36592032}a^{10}-\frac{7767359}{36592032}a^{9}+\frac{18373447}{36592032}a^{8}-\frac{8917603}{12197344}a^{7}+\frac{3237371}{9148008}a^{6}+\frac{70683029}{9148008}a^{5}-\frac{250105429}{18296016}a^{4}-\frac{245132825}{6098672}a^{3}+\frac{247209029}{6098672}a^{2}+\frac{516674421}{6098672}a+\frac{31234753}{1524668}$, $\frac{3116037}{30493360}a^{10}-\frac{28726021}{91480080}a^{9}+\frac{58671773}{91480080}a^{8}-\frac{116973379}{91480080}a^{7}-\frac{52029887}{22870020}a^{6}+\frac{256144951}{22870020}a^{5}+\frac{626886689}{45740040}a^{4}-\frac{373865579}{15246680}a^{3}-\frac{398453713}{15246680}a^{2}-\frac{437904241}{15246680}a-\frac{26287959}{3811670}$, $\frac{12287}{1143501}a^{10}-\frac{66829}{3049336}a^{9}-\frac{4835}{4574004}a^{8}-\frac{1019803}{9148008}a^{7}-\frac{457505}{762334}a^{6}+\frac{651431}{1143501}a^{5}+\frac{2717239}{762334}a^{4}+\frac{1217335}{1524668}a^{3}-\frac{3879423}{762334}a^{2}-\frac{5994709}{1524668}a-\frac{266950}{381167}$, $\frac{28686697}{60986720}a^{10}-\frac{123344327}{60986720}a^{9}+\frac{148152931}{60986720}a^{8}+\frac{156926487}{60986720}a^{7}-\frac{614725989}{15246680}a^{6}+\frac{2382925057}{15246680}a^{5}-\frac{2959992457}{30493360}a^{4}-\frac{11095501299}{30493360}a^{3}+\frac{6794978427}{30493360}a^{2}+\frac{10402929599}{30493360}a+\frac{522763091}{7623340}$, $\frac{10590127}{36592032}a^{10}-\frac{30362609}{36592032}a^{9}+\frac{31761989}{36592032}a^{8}-\frac{21522143}{36592032}a^{7}-\frac{115070107}{9148008}a^{6}+\frac{367532479}{9148008}a^{5}+\frac{1162017713}{18296016}a^{4}-\frac{1004460263}{6098672}a^{3}-\frac{880992849}{6098672}a^{2}+\frac{895696467}{6098672}a+\frac{163602367}{1524668}$, $\frac{26388401}{60986720}a^{10}-\frac{134306351}{60986720}a^{9}+\frac{301399323}{60986720}a^{8}-\frac{529028609}{60986720}a^{7}-\frac{64258837}{15246680}a^{6}+\frac{1186243521}{15246680}a^{5}-\frac{1136307041}{30493360}a^{4}-\frac{8844651467}{30493360}a^{3}+\frac{4111787971}{30493360}a^{2}+\frac{9386922247}{30493360}a+\frac{548395483}{7623340}$ 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:  \( 8123279.63873 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{3}\cdot 8123279.63873 \cdot 1}{2\cdot\sqrt{9678619238400000000}}\cr\approx \mathstrut & 10.3629711503 \end{aligned}\] (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^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168) 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^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168, 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^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168); 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^11 - 4*x^10 + 6*x^9 - 8*x^8 - 31*x^7 + 168*x^6 + 106*x^5 - 748*x^4 - 396*x^3 + 1008*x^2 + 882*x + 168); 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

$S_{11}$ (as 11T8):

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 non-solvable group of order 39916800
The 56 conjugacy class representatives for $S_{11}$
Character table for $S_{11}$

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 22 sibling: 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 R R ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.11.0.1}{11} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.4.8b1.4$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$$[2, 3]^{2}$$
2.1.6.11a1.7$x^{6} + 4 x^{5} + 4 x^{3} + 2$$6$$1$$11$$S_4\times C_2$$$[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.3.3a1.2$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.4.3a1.1$x^{4} + 3$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.1.4.3a1.1$x^{4} + 3$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.6.5a1.1$x^{6} + 5$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$
\(7\) Copy content Toggle raw display 7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.1.3.2a1.1$x^{3} + 7$$3$$1$$2$$C_3$$$[\ ]_{3}$$
7.5.1.0a1.1$x^{5} + x + 4$$1$$5$$0$$C_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)