↑

Properties

Label 11.5.847...000.1
Degree $11$
Signature $[5, 3]$
Discriminant $-8.473\times 10^{19}$
Root discriminant \(64.81\)
Ramified primes $2,3,5$
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 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324)
 
Copy content gp:K = bnfinit(y^11 - y^10 + 10*y^9 - 405*y^3 + 405*y^2 + 324*y - 324, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324)
 

\( x^{11} - x^{10} + 10x^{9} - 405x^{3} + 405x^{2} + 324x - 324 \) 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:   \(-84728860944300000000\) \(\medspace = -\,2^{8}\cdot 3^{25}\cdot 5^{8}\) 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:  \(64.81\)
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:  $2^{8/9}3^{475/162}5^{11/12}\approx 202.89500913181897$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{5}$, $\frac{1}{27}a^{9}-\frac{1}{27}a^{8}+\frac{1}{27}a^{7}-\frac{1}{3}a^{4}$, $\frac{1}{243540}a^{10}+\frac{1073}{243540}a^{9}-\frac{2792}{60885}a^{8}-\frac{571}{20295}a^{7}-\frac{2149}{20295}a^{6}+\frac{1094}{20295}a^{5}-\frac{991}{2255}a^{4}-\frac{2177}{6765}a^{3}+\frac{3441}{9020}a^{2}-\frac{2551}{9020}a+\frac{1159}{4510}$ 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:  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:   $\frac{4579}{12177}a^{10}+\frac{325}{4059}a^{9}+\frac{5180}{1353}a^{8}+\frac{56686}{12177}a^{7}+\frac{23393}{4059}a^{6}+\frac{28187}{4059}a^{5}+\frac{3849}{451}a^{4}+\frac{14312}{1353}a^{3}-\frac{62035}{451}a^{2}-\frac{7796}{451}a+\frac{44035}{451}$, $\frac{97003}{243540}a^{10}-\frac{6581}{243540}a^{9}+\frac{238919}{60885}a^{8}+\frac{214526}{60885}a^{7}+\frac{18631}{6765}a^{6}+\frac{7844}{6765}a^{5}-\frac{9244}{6765}a^{4}-\frac{8311}{6765}a^{3}-\frac{1395877}{9020}a^{2}+\frac{252587}{9020}a+\frac{700247}{4510}$, $\frac{1339}{13530}a^{10}-\frac{21997}{121770}a^{9}+\frac{82106}{60885}a^{8}-\frac{49541}{60885}a^{7}+\frac{10489}{6765}a^{6}+\frac{117133}{20295}a^{5}-\frac{77036}{6765}a^{4}+\frac{31332}{2255}a^{3}-\frac{78629}{4510}a^{2}-\frac{69591}{4510}a+\frac{42484}{2255}$, $\frac{134089}{243540}a^{10}-\frac{18563}{243540}a^{9}+\frac{327842}{60885}a^{8}+\frac{283973}{60885}a^{7}+\frac{7786}{2255}a^{6}+\frac{17657}{6765}a^{5}+\frac{12598}{6765}a^{4}+\frac{4762}{6765}a^{3}-\frac{2020291}{9020}a^{2}+\frac{284041}{9020}a+\frac{1004791}{4510}$, $\frac{8611}{27060}a^{10}+\frac{64187}{243540}a^{9}+\frac{215542}{60885}a^{8}+\frac{374903}{60885}a^{7}+\frac{202879}{20295}a^{6}+\frac{89587}{6765}a^{5}+\frac{44381}{2255}a^{4}+\frac{157117}{6765}a^{3}-\frac{841101}{9020}a^{2}-\frac{630989}{9020}a+\frac{94891}{4510}$, $\frac{2121}{9020}a^{10}-\frac{641}{27060}a^{9}+\frac{47792}{20295}a^{8}+\frac{14381}{6765}a^{7}+\frac{4743}{2255}a^{6}+\frac{42278}{20295}a^{5}+\frac{11239}{6765}a^{4}+\frac{642}{2255}a^{3}-\frac{861093}{9020}a^{2}+\frac{117063}{9020}a+\frac{395663}{4510}$, $\frac{589}{243540}a^{10}-\frac{8423}{243540}a^{9}-\frac{2848}{60885}a^{8}-\frac{12247}{60885}a^{7}-\frac{14236}{20295}a^{6}+\frac{8456}{20295}a^{5}+\frac{16823}{6765}a^{4}+\frac{23392}{6765}a^{3}-\frac{56871}{9020}a^{2}-\frac{32279}{9020}a+\frac{19681}{4510}$ 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:  \( 5378394.89404 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{3}\cdot 5378394.89404 \cdot 1}{2\cdot\sqrt{84728860944300000000}}\cr\approx \mathstrut & 2.31897779257 \end{aligned}\] (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 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324) 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 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324, 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 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324); 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 - x^10 + 10*x^9 - 405*x^3 + 405*x^2 + 324*x - 324); 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 ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.1.9.8a1.1$x^{9} + 2$$9$$1$$8$$(C_9:C_3):C_2$$$[\ ]_{9}^{6}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.9.24a1.27$x^{9} + 18 x^{8} + 9 x^{7} + 18 x^{3} + 54 x + 3$$9$$1$$24$$(C_3^3:C_3):C_2$$$[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}, \frac{19}{6}]_{2}$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.4.3a1.3$x^{4} + 15$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.6.5a1.1$x^{6} + 5$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

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