Properties

Label 11.11.448...625.1
Degree $11$
Signature $(11, 0)$
Discriminant $4.487\times 10^{23}$
Root discriminant \(141.31\)
Ramified primes $5,11$
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 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675)
 
Copy content gp:K = bnfinit(y^11 - 55*y^9 + 1100*y^7 - 9625*y^5 + 34375*y^3 - 34375*y - 12675, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^11 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^11 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675)
 

\( x^{11} - 55x^{9} + 1100x^{7} - 9625x^{5} + 34375x^{3} - 34375x - 12675 \) 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:   \(448727830698946884765625\) \(\medspace = 5^{10}\cdot 11^{16}\) 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:  \(141.31\)
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:  $5^{10/11}11^{84/55}\approx 168.23560841235079$
Ramified primes:   \(5\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{355}a^{6}-\frac{25}{71}a^{5}-\frac{6}{71}a^{4}-\frac{14}{71}a^{3}-\frac{26}{71}a^{2}-\frac{1}{71}a+\frac{21}{71}$, $\frac{1}{355}a^{7}-\frac{7}{71}a^{5}+\frac{17}{71}a^{4}-\frac{1}{71}a^{3}+\frac{15}{71}a^{2}-\frac{33}{71}a-\frac{2}{71}$, $\frac{1}{355}a^{8}-\frac{6}{71}a^{5}+\frac{2}{71}a^{4}+\frac{22}{71}a^{3}-\frac{20}{71}a^{2}+\frac{34}{71}a+\frac{25}{71}$, $\frac{1}{355}a^{9}+\frac{33}{71}a^{5}-\frac{16}{71}a^{4}-\frac{14}{71}a^{3}+\frac{35}{71}a^{2}-\frac{5}{71}a-\frac{9}{71}$, $\frac{1}{355}a^{10}-\frac{9}{71}a^{5}-\frac{18}{71}a^{4}+\frac{2}{71}a^{3}+\frac{25}{71}a^{2}+\frac{14}{71}a+\frac{14}{71}$ 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:   $\frac{2}{355}a^{9}+\frac{3}{355}a^{8}-\frac{83}{355}a^{7}-\frac{114}{355}a^{6}+3a^{5}+\frac{241}{71}a^{4}-\frac{839}{71}a^{3}-\frac{543}{71}a^{2}+\frac{531}{71}a+\frac{314}{71}$, $\frac{1}{355}a^{10}-\frac{3}{355}a^{9}-\frac{43}{355}a^{8}+\frac{132}{355}a^{7}+\frac{598}{355}a^{6}-\frac{388}{71}a^{5}-\frac{548}{71}a^{4}+\frac{2025}{71}a^{3}+\frac{276}{71}a^{2}-\frac{1914}{71}a-\frac{739}{71}$, $\frac{1}{355}a^{10}-\frac{2}{355}a^{9}-\frac{10}{71}a^{8}+\frac{98}{355}a^{7}+\frac{878}{355}a^{6}-\frac{330}{71}a^{5}-\frac{1274}{71}a^{4}+\frac{2160}{71}a^{3}+\frac{3453}{71}a^{2}-\frac{4368}{71}a-\frac{2501}{71}$, $\frac{1}{355}a^{10}-\frac{1}{355}a^{9}-\frac{43}{355}a^{8}+\frac{26}{355}a^{7}+\frac{624}{355}a^{6}-\frac{17}{71}a^{5}-\frac{692}{71}a^{4}-\frac{249}{71}a^{3}+\frac{991}{71}a^{2}+\frac{909}{71}a+\frac{214}{71}$, $\frac{2}{355}a^{10}+\frac{1}{71}a^{9}-\frac{82}{355}a^{8}-\frac{187}{355}a^{7}+\frac{1064}{355}a^{6}+\frac{411}{71}a^{5}-\frac{968}{71}a^{4}-\frac{1314}{71}a^{3}+\frac{1500}{71}a^{2}+\frac{973}{71}a+\frac{61}{71}$, $\frac{1}{355}a^{10}-\frac{2}{355}a^{9}-\frac{44}{355}a^{8}+\frac{96}{355}a^{7}+\frac{643}{355}a^{6}-\frac{299}{71}a^{5}-\frac{667}{71}a^{4}+\frac{1608}{71}a^{3}+\frac{751}{71}a^{2}-\frac{1591}{71}a-\frac{821}{71}$, $\frac{3}{355}a^{10}-\frac{7}{355}a^{9}-\frac{129}{355}a^{8}+\frac{266}{355}a^{7}+\frac{1947}{355}a^{6}-\frac{676}{71}a^{5}-\frac{2532}{71}a^{4}+\frac{3538}{71}a^{3}+\frac{6543}{71}a^{2}-\frac{6656}{71}a-\frac{4016}{71}$, $\frac{2}{355}a^{9}+\frac{12}{355}a^{8}-\frac{54}{355}a^{7}-\frac{388}{355}a^{6}+\frac{61}{71}a^{5}+\frac{763}{71}a^{4}+\frac{255}{71}a^{3}-\frac{2110}{71}a^{2}-\frac{1266}{71}a+\frac{194}{71}$, $\frac{2}{355}a^{9}-\frac{3}{355}a^{8}-\frac{19}{71}a^{7}+\frac{143}{355}a^{6}+\frac{298}{71}a^{5}-\frac{452}{71}a^{4}-\frac{1575}{71}a^{3}+\frac{2371}{71}a^{2}+\frac{537}{71}a-\frac{379}{71}$, $\frac{4}{355}a^{10}-\frac{12}{355}a^{9}-\frac{212}{355}a^{8}+\frac{504}{355}a^{7}+\frac{762}{71}a^{6}-\frac{1307}{71}a^{5}-\frac{5082}{71}a^{4}+\frac{4929}{71}a^{3}+\frac{8341}{71}a^{2}+\frac{582}{71}a-\frac{826}{71}$ 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:  \( 677575079.592 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 677575079.592 \cdot 1}{2\cdot\sqrt{448727830698946884765625}}\cr\approx \mathstrut & 1.0357760828 \end{aligned}\] (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 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675) 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 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675, 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 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675); 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 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675); 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 ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ 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
\(5\) 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 11.1.11.16a2.1$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$$[\frac{8}{5}]_{5}$$

Spectrum of ring of integers

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