Properties

Label 11.11.933...801.1
Degree $11$
Signature $(11, 0)$
Discriminant $9.333\times 10^{26}$
Root discriminant \(283.02\)
Ramified prime $2351$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{11}:C_5$ (as 11T3)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^11 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031)
 
Copy content gp:K = bnfinit(y^11 - 2*y^10 - 171*y^9 + 712*y^8 + 8054*y^7 - 51764*y^6 - 26840*y^5 + 682397*y^4 - 753338*y^3 - 2482765*y^2 + 4197766*y + 83031, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031)
 

\( x^{11} - 2 x^{10} - 171 x^{9} + 712 x^{8} + 8054 x^{7} - 51764 x^{6} - 26840 x^{5} + 682397 x^{4} + \cdots + 83031 \) 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:  $(11, 0)$
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:   \(933299499650312459853148801\) \(\medspace = 2351^{8}\) 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:  \(283.02\)
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:  not computed
Ramified primes:   \(2351\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10\cdots 91}a^{10}+\frac{35\cdots 87}{10\cdots 91}a^{9}+\frac{45\cdots 20}{10\cdots 91}a^{8}-\frac{40\cdots 18}{10\cdots 91}a^{7}+\frac{14\cdots 05}{10\cdots 91}a^{6}+\frac{50\cdots 08}{10\cdots 91}a^{5}+\frac{41\cdots 90}{10\cdots 91}a^{4}+\frac{37\cdots 85}{10\cdots 91}a^{3}+\frac{20\cdots 49}{10\cdots 91}a^{2}-\frac{74\cdots 65}{10\cdots 91}a+\frac{34\cdots 87}{10\cdots 91}$ 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:  $C_{2}\times C_{2}$, 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:  $10$
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:   $\frac{77\cdots 52}{10\cdots 91}a^{10}+\frac{10\cdots 16}{10\cdots 91}a^{9}-\frac{12\cdots 06}{10\cdots 91}a^{8}+\frac{11\cdots 93}{10\cdots 91}a^{7}+\frac{66\cdots 70}{10\cdots 91}a^{6}-\frac{17\cdots 02}{10\cdots 91}a^{5}-\frac{80\cdots 52}{10\cdots 91}a^{4}+\frac{25\cdots 06}{10\cdots 91}a^{3}+\frac{28\cdots 76}{10\cdots 91}a^{2}-\frac{96\cdots 92}{10\cdots 91}a-\frac{18\cdots 50}{10\cdots 91}$, $\frac{27\cdots 97}{10\cdots 91}a^{10}-\frac{73\cdots 06}{10\cdots 91}a^{9}+\frac{76\cdots 77}{10\cdots 91}a^{8}+\frac{85\cdots 43}{10\cdots 91}a^{7}-\frac{24\cdots 17}{10\cdots 91}a^{6}-\frac{30\cdots 55}{10\cdots 91}a^{5}+\frac{61\cdots 73}{10\cdots 91}a^{4}+\frac{19\cdots 54}{10\cdots 91}a^{3}-\frac{49\cdots 13}{10\cdots 91}a^{2}+\frac{39\cdots 66}{10\cdots 91}a-\frac{20\cdots 92}{10\cdots 91}$, $\frac{15\cdots 81}{10\cdots 91}a^{10}+\frac{11\cdots 76}{10\cdots 91}a^{9}-\frac{26\cdots 50}{10\cdots 91}a^{8}+\frac{39\cdots 22}{10\cdots 91}a^{7}+\frac{13\cdots 39}{10\cdots 91}a^{6}-\frac{43\cdots 41}{10\cdots 91}a^{5}-\frac{16\cdots 51}{10\cdots 91}a^{4}+\frac{63\cdots 44}{10\cdots 91}a^{3}+\frac{54\cdots 60}{10\cdots 91}a^{2}-\frac{24\cdots 34}{10\cdots 91}a-\frac{43\cdots 35}{10\cdots 91}$, $\frac{10\cdots 49}{10\cdots 91}a^{10}-\frac{21\cdots 36}{10\cdots 91}a^{9}+\frac{17\cdots 25}{10\cdots 91}a^{8}-\frac{71\cdots 66}{10\cdots 91}a^{7}-\frac{90\cdots 12}{10\cdots 91}a^{6}+\frac{20\cdots 35}{10\cdots 91}a^{5}+\frac{11\cdots 23}{10\cdots 91}a^{4}-\frac{30\cdots 72}{10\cdots 91}a^{3}-\frac{38\cdots 09}{10\cdots 91}a^{2}+\frac{11\cdots 78}{10\cdots 91}a+\frac{22\cdots 44}{10\cdots 91}$, $\frac{53\cdots 69}{10\cdots 91}a^{10}+\frac{27\cdots 56}{10\cdots 91}a^{9}-\frac{74\cdots 42}{10\cdots 91}a^{8}-\frac{18\cdots 04}{10\cdots 91}a^{7}+\frac{34\cdots 46}{10\cdots 91}a^{6}-\frac{12\cdots 59}{10\cdots 91}a^{5}-\frac{38\cdots 38}{10\cdots 91}a^{4}+\frac{89\cdots 62}{10\cdots 91}a^{3}+\frac{13\cdots 07}{10\cdots 91}a^{2}-\frac{11\cdots 68}{10\cdots 91}a-\frac{14\cdots 95}{10\cdots 91}$, $\frac{27\cdots 61}{10\cdots 91}a^{10}+\frac{36\cdots 30}{10\cdots 91}a^{9}-\frac{45\cdots 97}{10\cdots 91}a^{8}+\frac{43\cdots 45}{10\cdots 91}a^{7}+\frac{23\cdots 51}{10\cdots 91}a^{6}-\frac{63\cdots 17}{10\cdots 91}a^{5}-\frac{28\cdots 70}{10\cdots 91}a^{4}+\frac{92\cdots 82}{10\cdots 91}a^{3}+\frac{10\cdots 52}{10\cdots 91}a^{2}-\frac{34\cdots 30}{10\cdots 91}a-\frac{68\cdots 01}{10\cdots 91}$, $\frac{52\cdots 18}{10\cdots 91}a^{10}-\frac{82\cdots 98}{10\cdots 91}a^{9}+\frac{86\cdots 36}{10\cdots 91}a^{8}-\frac{63\cdots 01}{10\cdots 91}a^{7}-\frac{44\cdots 24}{10\cdots 91}a^{6}+\frac{11\cdots 64}{10\cdots 91}a^{5}+\frac{53\cdots 33}{10\cdots 91}a^{4}-\frac{16\cdots 83}{10\cdots 91}a^{3}-\frac{18\cdots 67}{10\cdots 91}a^{2}+\frac{61\cdots 02}{10\cdots 91}a+\frac{14\cdots 53}{10\cdots 91}$, $\frac{10\cdots 03}{10\cdots 91}a^{10}+\frac{19\cdots 94}{10\cdots 91}a^{9}-\frac{16\cdots 10}{10\cdots 91}a^{8}+\frac{71\cdots 39}{10\cdots 91}a^{7}+\frac{83\cdots 30}{10\cdots 91}a^{6}-\frac{19\cdots 52}{10\cdots 91}a^{5}-\frac{10\cdots 44}{10\cdots 91}a^{4}+\frac{28\cdots 40}{10\cdots 91}a^{3}+\frac{35\cdots 18}{10\cdots 91}a^{2}-\frac{10\cdots 64}{10\cdots 91}a-\frac{21\cdots 38}{10\cdots 91}$, $\frac{56\cdots 49}{10\cdots 91}a^{10}+\frac{70\cdots 69}{10\cdots 91}a^{9}-\frac{94\cdots 48}{10\cdots 91}a^{8}+\frac{99\cdots 34}{10\cdots 91}a^{7}+\frac{48\cdots 98}{10\cdots 91}a^{6}-\frac{13\cdots 69}{10\cdots 91}a^{5}-\frac{57\cdots 75}{10\cdots 91}a^{4}+\frac{20\cdots 90}{10\cdots 91}a^{3}+\frac{19\cdots 93}{10\cdots 91}a^{2}-\frac{79\cdots 45}{10\cdots 91}a-\frac{15\cdots 68}{10\cdots 91}$, $\frac{31\cdots 92}{10\cdots 91}a^{10}+\frac{27\cdots 33}{10\cdots 91}a^{9}-\frac{53\cdots 32}{10\cdots 91}a^{8}+\frac{61\cdots 04}{10\cdots 91}a^{7}+\frac{27\cdots 62}{10\cdots 91}a^{6}-\frac{75\cdots 01}{10\cdots 91}a^{5}-\frac{33\cdots 08}{10\cdots 91}a^{4}+\frac{10\cdots 85}{10\cdots 91}a^{3}+\frac{11\cdots 59}{10\cdots 91}a^{2}-\frac{39\cdots 70}{10\cdots 91}a+\frac{47\cdots 47}{10\cdots 91}$ 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:  \( 11828783188.0315 \) (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)
 
Unit signature rank:  \( 9 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 11828783188.0315 \cdot 1}{2\cdot\sqrt{933299499650312459853148801}}\cr\approx \mathstrut & 0.39648751077908 \end{aligned}\] (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 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031) 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 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031, 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 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031); 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^11 - 2*x^10 - 171*x^9 + 712*x^8 + 8054*x^7 - 51764*x^6 - 26840*x^5 + 682397*x^4 - 753338*x^3 - 2482765*x^2 + 4197766*x + 83031); 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

$C_{11}:C_5$ (as 11T3):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 55
The 7 conjugacy class representatives for $C_{11}:C_5$
Character table for $C_{11}:C_5$

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.11.0.1}{11} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.11.0.1}{11} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2351\) Copy content Toggle raw display $\Q_{2351}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $5$$5$$1$$4$
Deg $5$$5$$1$$4$

Spectrum of ring of integers

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