Properties

Label 11.11.123...625.1
Degree $11$
Signature $(11, 0)$
Discriminant $1.236\times 10^{20}$
Root discriminant \(67.07\)
Ramified primes $3,5,11$
Class number $1$
Class group trivial
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 - x^10 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4)
 
Copy content gp:K = bnfinit(y^11 - y^10 - 20*y^9 + 30*y^8 + 105*y^7 - 207*y^6 - 63*y^5 + 240*y^4 - 15*y^3 - 80*y^2 + 11*y + 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - x^10 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - x^10 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4)
 

\( x^{11} - x^{10} - 20x^{9} + 30x^{8} + 105x^{7} - 207x^{6} - 63x^{5} + 240x^{4} - 15x^{3} - 80x^{2} + 11x + 4 \) 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:   \(123610132462587890625\) \(\medspace = 3^{10}\cdot 5^{10}\cdot 11^{8}\) 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:  \(67.07\)
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:  $3^{10/11}5^{10/11}11^{4/5}\approx 79.85240853282978$
Ramified primes:   \(3\), \(5\), \(11\) 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: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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{9461}a^{10}-\frac{1132}{9461}a^{9}+\frac{3037}{9461}a^{8}-\frac{474}{9461}a^{7}-\frac{3078}{9461}a^{6}-\frac{637}{9461}a^{5}+\frac{1348}{9461}a^{4}-\frac{1127}{9461}a^{3}-\frac{2613}{9461}a^{2}+\frac{3391}{9461}a-\frac{3505}{9461}$ 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 $1$
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 $1$
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:   $\frac{34}{9461}a^{10}-\frac{644}{9461}a^{9}-\frac{813}{9461}a^{8}+\frac{12267}{9461}a^{7}-\frac{581}{9461}a^{6}-\frac{59502}{9461}a^{5}+\frac{45832}{9461}a^{4}+\frac{18448}{9461}a^{3}-\frac{32076}{9461}a^{2}+\frac{11223}{9461}a+\frac{3823}{9461}$, $\frac{588}{9461}a^{10}-\frac{3346}{9461}a^{9}-\frac{11834}{9461}a^{8}+\frac{71345}{9461}a^{7}+\frac{35031}{9461}a^{6}-\frac{412400}{9461}a^{5}+\frac{206042}{9461}a^{4}+\frac{434800}{9461}a^{3}-\frac{174060}{9461}a^{2}-\frac{125356}{9461}a+\frac{29941}{9461}$, $\frac{9427}{9461}a^{10}+\frac{644}{9461}a^{9}-\frac{188407}{9461}a^{8}+\frac{82343}{9461}a^{7}+\frac{1088596}{9461}a^{6}-\frac{810910}{9461}a^{5}-\frac{1512287}{9461}a^{4}+\frac{795198}{9461}a^{3}+\frac{713268}{9461}a^{2}-\frac{143677}{9461}a-\frac{60589}{9461}$, $\frac{174}{9461}a^{10}+\frac{1713}{9461}a^{9}-\frac{1378}{9461}a^{8}-\frac{25710}{9461}a^{7}+\frac{13166}{9461}a^{6}+\frac{116226}{9461}a^{5}-\frac{96583}{9461}a^{4}-\frac{101488}{9461}a^{3}+\frac{75154}{9461}a^{2}+\frac{22374}{9461}a-\frac{13827}{9461}$, $\frac{1741}{9461}a^{10}-\frac{2924}{9461}a^{9}-\frac{29665}{9461}a^{8}+\frac{73561}{9461}a^{7}+\frac{71816}{9461}a^{6}-\frac{399442}{9461}a^{5}+\frac{492512}{9461}a^{4}-\frac{79368}{9461}a^{3}-\frac{329627}{9461}a^{2}+\frac{113599}{9461}a+\frac{28523}{9461}$, $\frac{10198}{9461}a^{10}-\frac{11177}{9461}a^{9}-\frac{202669}{9461}a^{8}+\frac{322393}{9461}a^{7}+\frac{1033403}{9461}a^{6}-\frac{2153527}{9461}a^{5}-\frac{406752}{9461}a^{4}+\frac{2196921}{9461}a^{3}-\frac{289028}{9461}a^{2}-\frac{575658}{9461}a+\frac{160505}{9461}$, $\frac{2758}{9461}a^{10}+\frac{74}{9461}a^{9}-\frac{53705}{9461}a^{8}+\frac{26709}{9461}a^{7}+\frac{290684}{9461}a^{6}-\frac{243086}{9461}a^{5}-\frac{293680}{9461}a^{4}+\frac{165240}{9461}a^{3}+\frac{78316}{9461}a^{2}-\frac{14012}{9461}a-\frac{7109}{9461}$, $\frac{3221}{9461}a^{10}+\frac{5774}{9461}a^{9}-\frac{66724}{9461}a^{8}-\frac{79221}{9461}a^{7}+\frac{473940}{9461}a^{6}+\frac{266168}{9461}a^{5}-\frac{1249543}{9461}a^{4}+\frac{2957}{9461}a^{3}+\frac{694470}{9461}a^{2}-\frac{137498}{9461}a-\frac{31015}{9461}$, $\frac{786}{9461}a^{10}-\frac{418}{9461}a^{9}-\frac{16012}{9461}a^{8}+\frac{15337}{9461}a^{7}+\frac{97318}{9461}a^{6}-\frac{122242}{9461}a^{5}-\frac{179863}{9461}a^{4}+\frac{296803}{9461}a^{3}-\frac{10242}{9461}a^{2}-\frac{116208}{9461}a+\frac{36065}{9461}$, $\frac{2044}{9461}a^{10}+\frac{4137}{9461}a^{9}-\frac{46093}{9461}a^{8}-\frac{60600}{9461}a^{7}+\frac{378573}{9461}a^{6}+\frac{202271}{9461}a^{5}-\frac{1237230}{9461}a^{4}+\frac{175194}{9461}a^{3}+\frac{912749}{9461}a^{2}-\frac{192929}{9461}a-\frac{40087}{9461}$ Copy content Toggle raw display
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:  \( 12183336.2677 \)
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:  \( 11 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 12183336.2677 \cdot 1}{2\cdot\sqrt{123610132462587890625}}\cr\approx \mathstrut & 1.1221196095 \end{aligned}\]

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 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4) 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 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4, 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 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4); 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 - x^10 - 20*x^9 + 30*x^8 + 105*x^7 - 207*x^6 - 63*x^5 + 240*x^4 - 15*x^3 - 80*x^2 + 11*x + 4); 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 ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ R R ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.11.0.1}{11} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.11.0.1}{11} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3\) Copy content Toggle raw display 3.1.11.10a1.1$x^{11} + 3$$11$$1$$10$$C_{11}:C_5$$$[\ ]_{11}^{5}$$
\(5\) Copy content Toggle raw display 5.1.11.10a1.1$x^{11} + 5$$11$$1$$10$$C_{11}:C_5$$$[\ ]_{11}^{5}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.1.5.4a1.1$x^{5} + 11$$5$$1$$4$$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)