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Properties

Label 10.0.26439622160671.1
Degree $10$
Signature $[0, 5]$
Discriminant $-2.644\times 10^{13}$
Root discriminant \(21.99\)
Ramified prime $31$
Class number $3$
Class group [3]
Galois group $C_{10}$ (as 10T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25)
 
Copy content gp:K = bnfinit(y^10 - y^9 + 2*y^8 + 16*y^7 - 9*y^6 + 11*y^5 + 43*y^4 - 6*y^3 + 63*y^2 - 20*y + 25, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25)
 

\( x^{10} - x^{9} + 2x^{8} + 16x^{7} - 9x^{6} + 11x^{5} + 43x^{4} - 6x^{3} + 63x^{2} - 20x + 25 \) 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:  $[0, 5]$
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:   \(-26439622160671\) \(\medspace = -\,31^{9}\) 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:  \(21.99\)
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:  $31^{9/10}\approx 21.990014941549536$
Ramified primes:   \(31\) 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:  \(\Q(\sqrt{-31}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{10}$
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:  \(31\)
Dirichlet character group:    $\lbrace$$\chi_{31}(1,·)$, $\chi_{31}(2,·)$, $\chi_{31}(4,·)$, $\chi_{31}(8,·)$, $\chi_{31}(15,·)$, $\chi_{31}(16,·)$, $\chi_{31}(23,·)$, $\chi_{31}(27,·)$, $\chi_{31}(29,·)$, $\chi_{31}(30,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-31}) \), 10.0.26439622160671.1$^{15}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{4}$, $\frac{1}{8375}a^{9}+\frac{628}{8375}a^{8}-\frac{286}{8375}a^{7}-\frac{653}{8375}a^{6}-\frac{371}{8375}a^{5}-\frac{3873}{8375}a^{4}+\frac{2726}{8375}a^{3}+\frac{2798}{8375}a^{2}+\frac{251}{1675}a-\frac{49}{335}$ 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:  No
Index:  Not computed
Inessential primes:  $5$

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$
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:  $C_{3}$, which has order $3$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $3$

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:  $4$
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:   $\frac{93}{8375}a^{9}-\frac{221}{8375}a^{8}+\frac{202}{8375}a^{7}+\frac{1246}{8375}a^{6}-\frac{2678}{8375}a^{5}-\frac{64}{8375}a^{4}+\frac{593}{8375}a^{3}-\frac{2761}{8375}a^{2}+\frac{228}{1675}a-\frac{202}{335}$, $\frac{133}{8375}a^{9}-\frac{226}{8375}a^{8}+\frac{487}{8375}a^{7}+\frac{1926}{8375}a^{6}-\frac{2443}{8375}a^{5}+\frac{4141}{8375}a^{4}+\frac{5783}{8375}a^{3}-\frac{1391}{8375}a^{2}+\frac{553}{1675}a+\frac{518}{335}$, $\frac{617}{8375}a^{9}-\frac{1124}{8375}a^{8}+\frac{1088}{8375}a^{7}+\frac{9149}{8375}a^{6}-\frac{14507}{8375}a^{5}+\frac{584}{8375}a^{4}+\frac{22017}{8375}a^{3}-\frac{25684}{8375}a^{2}+\frac{3112}{1675}a-\frac{2093}{335}$, $\frac{681}{8375}a^{9}-\frac{1132}{8375}a^{8}+\frac{1209}{8375}a^{7}+\frac{10907}{8375}a^{6}-\frac{14801}{8375}a^{5}+\frac{2287}{8375}a^{4}+\frac{35681}{8375}a^{3}-\frac{32537}{8375}a^{2}+\frac{4436}{1675}a-\frac{204}{335}$ 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:  \( 485.913224212 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 485.913224212 \cdot 3}{2\cdot\sqrt{26439622160671}}\cr\approx \mathstrut & 1.38810301402 \end{aligned}\]

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 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25) 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 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25, 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 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25); 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 - x^9 + 2*x^8 + 16*x^7 - 9*x^6 + 11*x^5 + 43*x^4 - 6*x^3 + 63*x^2 - 20*x + 25); 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

\(\Q(\sqrt{-31}) \), 5.5.923521.1

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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }$ ${\href{/padicField/5.1.0.1}{1} }^{10}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.10.0.1}{10} }$ R ${\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(31\) Copy content Toggle raw display 31.1.10.9a1.1$x^{10} + 31$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*10 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*10 1.31.2t1.a.a$1$ $ 31 $ \(\Q(\sqrt{-31}) \) $C_2$ (as 2T1) $1$ $-1$
*10 1.31.5t1.a.a$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
*10 1.31.10t1.a.a$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
*10 1.31.5t1.a.b$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
*10 1.31.10t1.a.b$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
*10 1.31.5t1.a.c$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
*10 1.31.10t1.a.c$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$
*10 1.31.5t1.a.d$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
*10 1.31.10t1.a.d$1$ $ 31 $ 10.0.26439622160671.1 $C_{10}$ (as 10T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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