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Properties

Label 10.10.304358957700017.1
Degree $10$
Signature $[10, 0]$
Discriminant $3.044\times 10^{14}$
Root discriminant \(28.08\)
Ramified primes $11,17$
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 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43)
 
Copy content gp:K = bnfinit(y^10 - 3*y^9 - 26*y^8 + 68*y^7 + 210*y^6 - 464*y^5 - 526*y^4 + 987*y^3 + 166*y^2 - 389*y + 43, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43)
 

\( x^{10} - 3x^{9} - 26x^{8} + 68x^{7} + 210x^{6} - 464x^{5} - 526x^{4} + 987x^{3} + 166x^{2} - 389x + 43 \) 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:   \(304358957700017\) \(\medspace = 11^{8}\cdot 17^{5}\) 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:  \(28.08\)
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:  $11^{4/5}17^{1/2}\approx 28.076218190635743$
Ramified primes:   \(11\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{10}$
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:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(135,·)$, $\chi_{187}(137,·)$, $\chi_{187}(103,·)$, $\chi_{187}(16,·)$, $\chi_{187}(86,·)$, $\chi_{187}(169,·)$, $\chi_{187}(152,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{48621749287}a^{9}-\frac{13674562081}{48621749287}a^{8}-\frac{22080750492}{48621749287}a^{7}-\frac{19937438765}{48621749287}a^{6}+\frac{8084525792}{48621749287}a^{5}+\frac{18976718707}{48621749287}a^{4}+\frac{20987082499}{48621749287}a^{3}+\frac{233579219}{48621749287}a^{2}+\frac{21444104401}{48621749287}a+\frac{3415815615}{48621749287}$ 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:  $9$
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:   $\frac{223499010}{48621749287}a^{9}-\frac{238326345}{48621749287}a^{8}-\frac{6927315080}{48621749287}a^{7}+\frac{4266897445}{48621749287}a^{6}+\frac{70336251153}{48621749287}a^{5}-\frac{20438240715}{48621749287}a^{4}-\frac{260782547535}{48621749287}a^{3}+\frac{38211114160}{48621749287}a^{2}+\frac{277882622773}{48621749287}a-\frac{70829832130}{48621749287}$, $\frac{439929706}{48621749287}a^{9}-\frac{1096515009}{48621749287}a^{8}-\frac{11875821044}{48621749287}a^{7}+\frac{23490878475}{48621749287}a^{6}+\frac{100226826543}{48621749287}a^{5}-\frac{141411691658}{48621749287}a^{4}-\frac{263568024079}{48621749287}a^{3}+\frac{211142498787}{48621749287}a^{2}+\frac{90528742997}{48621749287}a+\frac{294356794}{48621749287}$, $\frac{143108798}{48621749287}a^{9}-\frac{254434914}{48621749287}a^{8}-\frac{4168223744}{48621749287}a^{7}+\frac{4832708853}{48621749287}a^{6}+\frac{40149881979}{48621749287}a^{5}-\frac{20197475672}{48621749287}a^{4}-\frac{145387974957}{48621749287}a^{3}-\frac{16878946590}{48621749287}a^{2}+\frac{163613510789}{48621749287}a+\frac{21114138744}{48621749287}$, $\frac{143108798}{48621749287}a^{9}-\frac{254434914}{48621749287}a^{8}-\frac{4168223744}{48621749287}a^{7}+\frac{4832708853}{48621749287}a^{6}+\frac{40149881979}{48621749287}a^{5}-\frac{20197475672}{48621749287}a^{4}-\frac{145387974957}{48621749287}a^{3}-\frac{16878946590}{48621749287}a^{2}+\frac{163613510789}{48621749287}a+\frac{69735888031}{48621749287}$, $\frac{114292607}{48621749287}a^{9}+\frac{378686506}{48621749287}a^{8}-\frac{5597562415}{48621749287}a^{7}-\frac{7932094203}{48621749287}a^{6}+\frac{77881929767}{48621749287}a^{5}+\frac{33647154402}{48621749287}a^{4}-\frac{346391110709}{48621749287}a^{3}+\frac{71595264426}{48621749287}a^{2}+\frac{213690033723}{48621749287}a-\frac{29620393755}{48621749287}$, $\frac{243714579}{48621749287}a^{9}-\frac{716828360}{48621749287}a^{8}-\frac{6814115021}{48621749287}a^{7}+\frac{19665400144}{48621749287}a^{6}+\frac{52381578674}{48621749287}a^{5}-\frac{164167225710}{48621749287}a^{4}-\frac{51213796684}{48621749287}a^{3}+\frac{365578263488}{48621749287}a^{2}-\frac{194392992035}{48621749287}a-\frac{22910919595}{48621749287}$, $\frac{31782682}{48621749287}a^{9}-\frac{192960082}{48621749287}a^{8}-\frac{730465667}{48621749287}a^{7}+\frac{5264325546}{48621749287}a^{6}+\frac{6055124621}{48621749287}a^{5}-\frac{49915271537}{48621749287}a^{4}-\frac{12739355259}{48621749287}a^{3}+\frac{156736396911}{48621749287}a^{2}-\frac{38208300336}{48621749287}a-\frac{27267817058}{48621749287}$, $\frac{179721433}{48621749287}a^{9}-\frac{93242111}{48621749287}a^{8}-\frac{4980200433}{48621749287}a^{7}-\frac{341162249}{48621749287}a^{6}+\frac{42064011111}{48621749287}a^{5}+\frac{21485855584}{48621749287}a^{4}-\frac{110341825982}{48621749287}a^{3}-\frac{97050455668}{48621749287}a^{2}+\frac{42836216919}{48621749287}a+\frac{36620620802}{48621749287}$, $\frac{200322127}{48621749287}a^{9}-\frac{140917133}{48621749287}a^{8}-\frac{7193434148}{48621749287}a^{7}+\frac{4222122152}{48621749287}a^{6}+\frac{83389389587}{48621749287}a^{5}-\frac{48586425006}{48621749287}a^{4}-\frac{324714115784}{48621749287}a^{3}+\frac{237277230085}{48621749287}a^{2}+\frac{167503885295}{48621749287}a-\frac{104007446008}{48621749287}$ 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:  \( 6521.45238479 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 6521.45238479 \cdot 1}{2\cdot\sqrt{304358957700017}}\cr\approx \mathstrut & 0.191390882340 \end{aligned}\]

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 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43) 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 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43, 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 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43); 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 - 3*x^9 - 26*x^8 + 68*x^7 + 210*x^6 - 464*x^5 - 526*x^4 + 987*x^3 + 166*x^2 - 389*x + 43); 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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }$ ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.10.0.1}{10} }$ R ${\href{/padicField/13.5.0.1}{5} }^{2}$ R ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{5}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.1.0.1}{1} }^{10}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(11\) Copy content Toggle raw display 11.2.5.8a1.2$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
\(17\) Copy content Toggle raw display 17.5.2.5a1.2$x^{10} + 2 x^{6} + 28 x^{5} + x^{2} + 28 x + 213$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$

Spectrum of ring of integers

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