Properties

Label 11.11.270...841.1
Degree $11$
Signature $(11, 0)$
Discriminant $2.704\times 10^{28}$
Root discriminant \(384.35\)
Ramified prime $3581$
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 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691)
 
Copy content gp:K = bnfinit(y^11 - 178*y^9 - 36*y^8 + 9850*y^7 + 3981*y^6 - 187211*y^5 - 23214*y^4 + 1031588*y^3 - 171775*y^2 - 261279*y - 12691, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691)
 

\( x^{11} - 178 x^{9} - 36 x^{8} + 9850 x^{7} + 3981 x^{6} - 187211 x^{5} - 23214 x^{4} + 1031588 x^{3} + \cdots - 12691 \) 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:   \(27041735968924735397910759841\) \(\medspace = 3581^{8}\) 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:  \(384.35\)
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:   \(3581\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{77\cdots 77}a^{10}-\frac{39\cdots 50}{77\cdots 77}a^{9}-\frac{31\cdots 61}{77\cdots 77}a^{8}-\frac{10\cdots 12}{77\cdots 77}a^{7}+\frac{32\cdots 03}{77\cdots 77}a^{6}+\frac{31\cdots 43}{77\cdots 77}a^{5}+\frac{10\cdots 08}{77\cdots 77}a^{4}-\frac{21\cdots 56}{77\cdots 77}a^{3}-\frac{37\cdots 44}{77\cdots 77}a^{2}-\frac{21\cdots 88}{77\cdots 77}a-\frac{28\cdots 85}{77\cdots 77}$ 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:   $\frac{57\cdots 44}{77\cdots 77}a^{10}+\frac{30\cdots 65}{77\cdots 77}a^{9}+\frac{85\cdots 33}{77\cdots 77}a^{8}-\frac{40\cdots 23}{77\cdots 77}a^{7}-\frac{36\cdots 34}{77\cdots 77}a^{6}+\frac{13\cdots 57}{77\cdots 77}a^{5}+\frac{47\cdots 48}{77\cdots 77}a^{4}-\frac{92\cdots 05}{77\cdots 77}a^{3}-\frac{18\cdots 10}{77\cdots 77}a^{2}+\frac{66\cdots 40}{77\cdots 77}a+\frac{31\cdots 52}{77\cdots 77}$, $\frac{38\cdots 31}{77\cdots 77}a^{10}+\frac{18\cdots 03}{77\cdots 77}a^{9}+\frac{57\cdots 26}{77\cdots 77}a^{8}-\frac{23\cdots 59}{77\cdots 77}a^{7}-\frac{22\cdots 66}{77\cdots 77}a^{6}+\frac{67\cdots 96}{77\cdots 77}a^{5}+\frac{26\cdots 43}{77\cdots 77}a^{4}-\frac{42\cdots 43}{77\cdots 77}a^{3}-\frac{85\cdots 47}{77\cdots 77}a^{2}+\frac{19\cdots 12}{77\cdots 77}a+\frac{14\cdots 13}{77\cdots 77}$, $\frac{10\cdots 48}{77\cdots 77}a^{10}+\frac{23\cdots 70}{77\cdots 77}a^{9}-\frac{19\cdots 33}{77\cdots 77}a^{8}-\frac{67\cdots 22}{77\cdots 77}a^{7}+\frac{10\cdots 37}{77\cdots 77}a^{6}+\frac{49\cdots 81}{77\cdots 77}a^{5}-\frac{21\cdots 08}{77\cdots 77}a^{4}-\frac{18\cdots 89}{77\cdots 77}a^{3}+\frac{11\cdots 14}{77\cdots 77}a^{2}-\frac{28\cdots 50}{77\cdots 77}a-\frac{26\cdots 59}{77\cdots 77}$, $\frac{33\cdots 69}{77\cdots 77}a^{10}-\frac{16\cdots 53}{77\cdots 77}a^{9}+\frac{11\cdots 54}{77\cdots 77}a^{8}+\frac{23\cdots 47}{77\cdots 77}a^{7}-\frac{96\cdots 14}{77\cdots 77}a^{6}-\frac{89\cdots 77}{77\cdots 77}a^{5}+\frac{22\cdots 19}{77\cdots 77}a^{4}+\frac{50\cdots 22}{77\cdots 77}a^{3}-\frac{16\cdots 38}{77\cdots 77}a^{2}+\frac{68\cdots 48}{77\cdots 77}a+\frac{65\cdots 50}{77\cdots 77}$, $\frac{78\cdots 70}{77\cdots 77}a^{10}-\frac{59\cdots 24}{77\cdots 77}a^{9}-\frac{13\cdots 44}{77\cdots 77}a^{8}-\frac{14\cdots 58}{77\cdots 77}a^{7}+\frac{75\cdots 00}{77\cdots 77}a^{6}+\frac{19\cdots 87}{77\cdots 77}a^{5}-\frac{13\cdots 24}{77\cdots 77}a^{4}+\frac{23\cdots 68}{77\cdots 77}a^{3}+\frac{73\cdots 74}{77\cdots 77}a^{2}-\frac{41\cdots 98}{77\cdots 77}a-\frac{22\cdots 32}{77\cdots 77}$, $\frac{16\cdots 82}{77\cdots 77}a^{10}-\frac{97\cdots 64}{77\cdots 77}a^{9}-\frac{24\cdots 26}{77\cdots 77}a^{8}+\frac{13\cdots 13}{77\cdots 77}a^{7}+\frac{87\cdots 76}{77\cdots 77}a^{6}-\frac{44\cdots 09}{77\cdots 77}a^{5}-\frac{58\cdots 22}{77\cdots 77}a^{4}+\frac{30\cdots 36}{77\cdots 77}a^{3}-\frac{36\cdots 93}{77\cdots 77}a^{2}-\frac{70\cdots 55}{77\cdots 77}a-\frac{71\cdots 89}{77\cdots 77}$, $\frac{40\cdots 09}{77\cdots 77}a^{10}-\frac{26\cdots 25}{77\cdots 77}a^{9}-\frac{56\cdots 36}{77\cdots 77}a^{8}+\frac{36\cdots 29}{77\cdots 77}a^{7}+\frac{19\cdots 49}{77\cdots 77}a^{6}-\frac{12\cdots 88}{77\cdots 77}a^{5}-\frac{80\cdots 40}{77\cdots 77}a^{4}+\frac{87\cdots 30}{77\cdots 77}a^{3}-\frac{49\cdots 59}{77\cdots 77}a^{2}-\frac{39\cdots 60}{77\cdots 77}a-\frac{21\cdots 50}{77\cdots 77}$, $\frac{63\cdots 43}{77\cdots 77}a^{10}+\frac{42\cdots 17}{77\cdots 77}a^{9}+\frac{23\cdots 29}{77\cdots 77}a^{8}-\frac{72\cdots 06}{77\cdots 77}a^{7}-\frac{28\cdots 38}{77\cdots 77}a^{6}+\frac{36\cdots 12}{77\cdots 77}a^{5}+\frac{12\cdots 71}{77\cdots 77}a^{4}-\frac{50\cdots 21}{77\cdots 77}a^{3}-\frac{13\cdots 70}{77\cdots 77}a^{2}+\frac{79\cdots 64}{77\cdots 77}a+\frac{43\cdots 40}{77\cdots 77}$, $\frac{35\cdots 90}{77\cdots 77}a^{10}+\frac{10\cdots 54}{77\cdots 77}a^{9}+\frac{58\cdots 66}{77\cdots 77}a^{8}-\frac{16\cdots 59}{77\cdots 77}a^{7}-\frac{29\cdots 35}{77\cdots 77}a^{6}+\frac{75\cdots 27}{77\cdots 77}a^{5}+\frac{42\cdots 19}{77\cdots 77}a^{4}-\frac{11\cdots 65}{77\cdots 77}a^{3}+\frac{98\cdots 29}{77\cdots 77}a^{2}+\frac{29\cdots 17}{77\cdots 77}a+\frac{14\cdots 56}{77\cdots 77}$, $\frac{66\cdots 87}{77\cdots 77}a^{10}-\frac{58\cdots 32}{77\cdots 77}a^{9}+\frac{11\cdots 24}{77\cdots 77}a^{8}+\frac{34\cdots 14}{77\cdots 77}a^{7}-\frac{65\cdots 44}{77\cdots 77}a^{6}-\frac{32\cdots 67}{77\cdots 77}a^{5}+\frac{12\cdots 42}{77\cdots 77}a^{4}+\frac{25\cdots 30}{77\cdots 77}a^{3}-\frac{67\cdots 88}{77\cdots 77}a^{2}+\frac{59\cdots 17}{77\cdots 77}a+\frac{13\cdots 32}{77\cdots 77}$ 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:  \( 123956548381.318 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 123956548381.318 \cdot 1}{2\cdot\sqrt{27041735968924735397910759841}}\cr\approx \mathstrut & 0.77188419280282 \end{aligned}\] (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 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691) 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 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691, 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 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691); 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 - 178*x^9 - 36*x^8 + 9850*x^7 + 3981*x^6 - 187211*x^5 - 23214*x^4 + 1031588*x^3 - 171775*x^2 - 261279*x - 12691); 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 ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3581\) Copy content Toggle raw display $\Q_{3581}$$x$$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)