Properties

Label 11.11.143...681.1
Degree $11$
Signature $(11, 0)$
Discriminant $1.439\times 10^{26}$
Root discriminant \(238.78\)
Ramified prime $1861$
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 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992)
 
Copy content gp:K = bnfinit(y^11 - 5*y^10 - 165*y^9 + 1534*y^8 - 269*y^7 - 34679*y^6 + 76430*y^5 + 208536*y^4 - 657992*y^3 - 299504*y^2 + 1268768*y + 356992, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992)
 

\( x^{11} - 5 x^{10} - 165 x^{9} + 1534 x^{8} - 269 x^{7} - 34679 x^{6} + 76430 x^{5} + 208536 x^{4} + \cdots + 356992 \) 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:   \(143870245619672894373619681\) \(\medspace = 1861^{8}\) 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:  \(238.78\)
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:   \(1861\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}+\frac{3}{16}a^{5}+\frac{5}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{17\cdots 56}a^{10}-\frac{48\cdots 63}{17\cdots 56}a^{9}-\frac{69\cdots 31}{17\cdots 56}a^{8}-\frac{36\cdots 39}{42\cdots 64}a^{7}+\frac{92\cdots 71}{17\cdots 56}a^{6}+\frac{99\cdots 79}{17\cdots 56}a^{5}+\frac{17\cdots 77}{53\cdots 33}a^{4}+\frac{79\cdots 37}{42\cdots 64}a^{3}+\frac{43\cdots 35}{21\cdots 32}a^{2}-\frac{39\cdots 19}{10\cdots 66}a+\frac{23\cdots 69}{53\cdots 33}$ 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:  $10$
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:   $\frac{17\cdots 95}{17\cdots 56}a^{10}-\frac{31\cdots 23}{17\cdots 56}a^{9}-\frac{30\cdots 87}{17\cdots 56}a^{8}+\frac{86\cdots 99}{85\cdots 28}a^{7}+\frac{52\cdots 53}{17\cdots 56}a^{6}-\frac{44\cdots 73}{17\cdots 56}a^{5}-\frac{51\cdots 21}{85\cdots 28}a^{4}+\frac{21\cdots 37}{10\cdots 66}a^{3}-\frac{85\cdots 51}{21\cdots 32}a^{2}-\frac{23\cdots 87}{53\cdots 33}a-\frac{61\cdots 29}{53\cdots 33}$, $\frac{11\cdots 45}{17\cdots 56}a^{10}-\frac{21\cdots 47}{17\cdots 56}a^{9}-\frac{20\cdots 07}{17\cdots 56}a^{8}+\frac{73\cdots 33}{10\cdots 66}a^{7}+\frac{34\cdots 27}{17\cdots 56}a^{6}-\frac{30\cdots 25}{17\cdots 56}a^{5}-\frac{16\cdots 15}{42\cdots 64}a^{4}+\frac{57\cdots 33}{42\cdots 64}a^{3}-\frac{57\cdots 41}{21\cdots 32}a^{2}-\frac{15\cdots 72}{53\cdots 33}a-\frac{41\cdots 09}{53\cdots 33}$, $\frac{18\cdots 79}{17\cdots 56}a^{10}+\frac{31\cdots 93}{17\cdots 56}a^{9}+\frac{31\cdots 85}{17\cdots 56}a^{8}-\frac{11\cdots 55}{10\cdots 66}a^{7}-\frac{53\cdots 73}{17\cdots 56}a^{6}+\frac{45\cdots 59}{17\cdots 56}a^{5}+\frac{27\cdots 19}{42\cdots 64}a^{4}-\frac{86\cdots 59}{42\cdots 64}a^{3}+\frac{20\cdots 37}{53\cdots 33}a^{2}+\frac{24\cdots 24}{53\cdots 33}a+\frac{62\cdots 75}{53\cdots 33}$, $\frac{24\cdots 83}{85\cdots 28}a^{10}-\frac{11\cdots 69}{42\cdots 64}a^{9}-\frac{25\cdots 15}{53\cdots 33}a^{8}+\frac{21\cdots 21}{85\cdots 28}a^{7}+\frac{74\cdots 95}{85\cdots 28}a^{6}-\frac{27\cdots 45}{42\cdots 64}a^{5}-\frac{22\cdots 45}{85\cdots 28}a^{4}+\frac{20\cdots 53}{42\cdots 64}a^{3}-\frac{12\cdots 75}{21\cdots 32}a^{2}-\frac{54\cdots 05}{53\cdots 33}a-\frac{13\cdots 59}{53\cdots 33}$, $\frac{12\cdots 49}{17\cdots 56}a^{10}-\frac{27\cdots 39}{17\cdots 56}a^{9}-\frac{21\cdots 15}{17\cdots 56}a^{8}+\frac{33\cdots 65}{42\cdots 64}a^{7}+\frac{35\cdots 95}{17\cdots 56}a^{6}-\frac{35\cdots 57}{17\cdots 56}a^{5}-\frac{38\cdots 71}{21\cdots 32}a^{4}+\frac{88\cdots 88}{53\cdots 33}a^{3}-\frac{44\cdots 35}{10\cdots 66}a^{2}-\frac{45\cdots 19}{10\cdots 66}a-\frac{59\cdots 65}{53\cdots 33}$, $\frac{132798339279841}{53\cdots 33}a^{10}-\frac{47\cdots 17}{85\cdots 28}a^{9}-\frac{36\cdots 01}{85\cdots 28}a^{8}+\frac{22\cdots 47}{85\cdots 28}a^{7}+\frac{13\cdots 35}{21\cdots 32}a^{6}-\frac{52\cdots 27}{85\cdots 28}a^{5}-\frac{52\cdots 03}{85\cdots 28}a^{4}+\frac{47\cdots 69}{10\cdots 66}a^{3}-\frac{20\cdots 01}{21\cdots 32}a^{2}-\frac{51\cdots 19}{53\cdots 33}a-\frac{13\cdots 99}{53\cdots 33}$, $\frac{86599577564739}{85\cdots 28}a^{10}+\frac{14\cdots 21}{21\cdots 32}a^{9}-\frac{30\cdots 11}{10\cdots 66}a^{8}-\frac{91\cdots 55}{85\cdots 28}a^{7}+\frac{59\cdots 45}{85\cdots 28}a^{6}+\frac{84\cdots 69}{53\cdots 33}a^{5}-\frac{14\cdots 41}{85\cdots 28}a^{4}+\frac{12\cdots 04}{53\cdots 33}a^{3}+\frac{25\cdots 03}{21\cdots 32}a^{2}-\frac{66\cdots 27}{10\cdots 66}a-\frac{13\cdots 73}{53\cdots 33}$, $\frac{24\cdots 45}{42\cdots 64}a^{10}-\frac{61\cdots 09}{85\cdots 28}a^{9}-\frac{83\cdots 87}{85\cdots 28}a^{8}+\frac{43\cdots 15}{85\cdots 28}a^{7}+\frac{40\cdots 69}{21\cdots 32}a^{6}-\frac{11\cdots 15}{85\cdots 28}a^{5}-\frac{72\cdots 15}{85\cdots 28}a^{4}+\frac{44\cdots 51}{42\cdots 64}a^{3}+\frac{29\cdots 91}{21\cdots 32}a^{2}-\frac{25\cdots 49}{10\cdots 66}a-\frac{64\cdots 51}{53\cdots 33}$, $\frac{49\cdots 07}{85\cdots 28}a^{10}-\frac{114119856240427}{21\cdots 32}a^{9}-\frac{42\cdots 95}{42\cdots 64}a^{8}+\frac{36\cdots 07}{85\cdots 28}a^{7}+\frac{20\cdots 87}{85\cdots 28}a^{6}-\frac{27\cdots 13}{21\cdots 32}a^{5}-\frac{12\cdots 87}{85\cdots 28}a^{4}+\frac{50\cdots 05}{42\cdots 64}a^{3}+\frac{91\cdots 37}{10\cdots 66}a^{2}-\frac{14\cdots 24}{53\cdots 33}a-\frac{43\cdots 91}{53\cdots 33}$, $\frac{15\cdots 29}{85\cdots 28}a^{10}+\frac{17\cdots 31}{42\cdots 64}a^{9}+\frac{13\cdots 45}{42\cdots 64}a^{8}-\frac{16\cdots 93}{85\cdots 28}a^{7}-\frac{41\cdots 91}{85\cdots 28}a^{6}+\frac{21\cdots 27}{42\cdots 64}a^{5}-\frac{51\cdots 87}{85\cdots 28}a^{4}-\frac{40\cdots 85}{10\cdots 66}a^{3}+\frac{27\cdots 37}{21\cdots 32}a^{2}+\frac{48\cdots 72}{53\cdots 33}a+\frac{12\cdots 69}{53\cdots 33}$ 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:  \( 7352154585.48005 \) (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:  \( 11 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 7352154585.48005 \cdot 1}{2\cdot\sqrt{143870245619672894373619681}}\cr\approx \mathstrut & 0.62766670816733 \end{aligned}\] (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 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992) 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 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992, 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 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992); 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 - 5*x^10 - 165*x^9 + 1534*x^8 - 269*x^7 - 34679*x^6 + 76430*x^5 + 208536*x^4 - 657992*x^3 - 299504*x^2 + 1268768*x + 356992); 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 ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.11.0.1}{11} }$ ${\href{/padicField/7.11.0.1}{11} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.11.0.1}{11} }$ ${\href{/padicField/17.11.0.1}{11} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(1861\) Copy content Toggle raw display $\Q_{1861}$$x$$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)