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Properties

Label 10.10.53339349076992.1
Degree $10$
Signature $[10, 0]$
Discriminant $5.334\times 10^{13}$
Root discriminant \(23.59\)
Ramified primes $2,3,11$
Class number $1$
Class group trivial
Galois group $C_{10}$ (as 10T1)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1)
 
Copy content gp:K = bnfinit(y^10 - 2*y^9 - 22*y^8 + 38*y^7 + 142*y^6 - 194*y^5 - 265*y^4 + 236*y^3 + 66*y^2 - 24*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1)
 

\( x^{10} - 2x^{9} - 22x^{8} + 38x^{7} + 142x^{6} - 194x^{5} - 265x^{4} + 236x^{3} + 66x^{2} - 24x + 1 \) 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:  $10$
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:  $[10, 0]$
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:   \(53339349076992\) \(\medspace = 2^{10}\cdot 3^{5}\cdot 11^{8}\) 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:  \(23.59\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}11^{4/5}\approx 23.588741500303293$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q(\sqrt{3}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{10}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(132=2^{2}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{132}(1,·)$, $\chi_{132}(37,·)$, $\chi_{132}(97,·)$, $\chi_{132}(71,·)$, $\chi_{132}(47,·)$, $\chi_{132}(49,·)$, $\chi_{132}(119,·)$, $\chi_{132}(23,·)$, $\chi_{132}(25,·)$, $\chi_{132}(59,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{114037043}a^{9}-\frac{20513415}{114037043}a^{8}+\frac{44154083}{114037043}a^{7}+\frac{19627688}{114037043}a^{6}-\frac{54384816}{114037043}a^{5}-\frac{42131864}{114037043}a^{4}+\frac{51795533}{114037043}a^{3}+\frac{40166675}{114037043}a^{2}+\frac{16898610}{114037043}a+\frac{40736844}{114037043}$ 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:  $C_{2}$, which has order $2$
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:  $9$
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:   $\frac{9284712}{114037043}a^{9}-\frac{16341213}{114037043}a^{8}-\frac{207676012}{114037043}a^{7}+\frac{305665620}{114037043}a^{6}+\frac{1374300256}{114037043}a^{5}-\frac{1524845483}{114037043}a^{4}-\frac{2656429180}{114037043}a^{3}+\frac{1808020130}{114037043}a^{2}+\frac{633364684}{114037043}a-\frac{103263419}{114037043}$, $\frac{12066792}{114037043}a^{9}-\frac{25738020}{114037043}a^{8}-\frac{263014800}{114037043}a^{7}+\frac{492189497}{114037043}a^{6}+\frac{1673684946}{114037043}a^{5}-\frac{2538320967}{114037043}a^{4}-\frac{3056330286}{114037043}a^{3}+\frac{3128546172}{114037043}a^{2}+\frac{778852218}{114037043}a-\frac{264715546}{114037043}$, $\frac{2300302}{114037043}a^{9}-\frac{3639489}{114037043}a^{8}-\frac{51137142}{114037043}a^{7}+\frac{63897216}{114037043}a^{6}+\frac{328193772}{114037043}a^{5}-\frac{261584862}{114037043}a^{4}-\frac{534375434}{114037043}a^{3}+\frac{47745261}{114037043}a^{2}-\frac{128541276}{114037043}a+\frac{68604756}{114037043}$, $\frac{5458558}{114037043}a^{9}-\frac{8811612}{114037043}a^{8}-\frac{123536358}{114037043}a^{7}+\frac{159159060}{114037043}a^{6}+\frac{838623132}{114037043}a^{5}-\frac{718674012}{114037043}a^{4}-\frac{1751243970}{114037043}a^{3}+\frac{514986345}{114037043}a^{2}+\frac{671758884}{114037043}a+\frac{60416919}{114037043}$, $\frac{7310593}{114037043}a^{9}-\frac{16448644}{114037043}a^{8}-\frac{153629366}{114037043}a^{7}+\frac{306292245}{114037043}a^{6}+\frac{903967107}{114037043}a^{5}-\frac{1471040846}{114037043}a^{4}-\frac{1317448442}{114037043}a^{3}+\frac{1388844995}{114037043}a^{2}+\frac{136515927}{114037043}a-\frac{216169126}{114037043}$, $\frac{15419846}{114037043}a^{9}-\frac{31101550}{114037043}a^{8}-\frac{338285014}{114037043}a^{7}+\frac{587840661}{114037043}a^{6}+\frac{2172021838}{114037043}a^{5}-\frac{2952268879}{114037043}a^{4}-\frac{4034340913}{114037043}a^{3}+\frac{3332038461}{114037043}a^{2}+\frac{1259188791}{114037043}a-\frac{312877112}{114037043}$, $\frac{4690345}{114037043}a^{9}-\frac{1669344}{114037043}a^{8}-\frac{115844902}{114037043}a^{7}+\frac{5940019}{114037043}a^{6}+\frac{894055288}{114037043}a^{5}+\frac{217643018}{114037043}a^{4}-\frac{2290104068}{114037043}a^{3}-\frac{930153877}{114037043}a^{2}+\frac{1258961203}{114037043}a-\frac{125271664}{114037043}$, $\frac{2729279}{114037043}a^{9}-\frac{4405806}{114037043}a^{8}-\frac{61768179}{114037043}a^{7}+\frac{79579530}{114037043}a^{6}+\frac{419311566}{114037043}a^{5}-\frac{359337006}{114037043}a^{4}-\frac{875621985}{114037043}a^{3}+\frac{314511694}{114037043}a^{2}+\frac{335879442}{114037043}a-\frac{26810062}{114037043}$, $\frac{26810062}{114037043}a^{9}-\frac{50890845}{114037043}a^{8}-\frac{594227170}{114037043}a^{7}+\frac{957014177}{114037043}a^{6}+\frac{3886608334}{114037043}a^{5}-\frac{4781840462}{114037043}a^{4}-\frac{7464003436}{114037043}a^{3}+\frac{5451552647}{114037043}a^{2}+\frac{1969938743}{114037043}a-\frac{79487960}{114037043}$ 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:  \( 3410.56648978 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 3410.56648978 \cdot 1}{2\cdot\sqrt{53339349076992}}\cr\approx \mathstrut & 0.239096173572 \end{aligned}\]

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^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1) 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^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1, 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^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1); 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 = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 22*x^8 + 38*x^7 + 142*x^6 - 194*x^5 - 265*x^4 + 236*x^3 + 66*x^2 - 24*x + 1); 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

$C_{10}$ (as 10T1):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{11})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
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 R R ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.10.0.1}{10} }$ R ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.1.0.1}{1} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(2\) Copy content Toggle raw display 2.5.2.10a1.2$x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 9$$2$$5$$10$$C_{10}$$$[2]^{5}$$
\(3\) Copy content Toggle raw display 3.5.2.5a1.1$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 7 x + 1$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(11\) Copy content Toggle raw display 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)