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Properties

Label 10.10.2665284492003125.1
Degree $10$
Signature $[10, 0]$
Discriminant $2.665\times 10^{15}$
Root discriminant \(34.88\)
Ramified primes $5,31$
Class number $1$
Class group trivial
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 - 3*x^9 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*x + 25)
 
Copy content gp:K = bnfinit(y^10 - 3*y^9 - 27*y^8 + 44*y^7 + 228*y^6 - 138*y^5 - 634*y^4 - 95*y^3 + 435*y^2 + 225*y + 25, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*x + 25);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 3*x^9 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*x + 25)
 

\( x^{10} - 3x^{9} - 27x^{8} + 44x^{7} + 228x^{6} - 138x^{5} - 634x^{4} - 95x^{3} + 435x^{2} + 225x + 25 \) 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:  $[10, 0]$
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:   \(2665284492003125\) \(\medspace = 5^{5}\cdot 31^{8}\) 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:  \(34.88\)
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:  $5^{1/2}31^{4/5}\approx 34.87982981267706$
Ramified primes:   \(5\), \(31\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{10}$
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:  \(155=5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(66,·)$, $\chi_{155}(4,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(109,·)$, $\chi_{155}(16,·)$, $\chi_{155}(94,·)$, $\chi_{155}(126,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{125573275}a^{9}+\frac{6490312}{125573275}a^{8}-\frac{38751527}{125573275}a^{7}+\frac{40721029}{125573275}a^{6}+\frac{41697023}{125573275}a^{5}-\frac{18777263}{125573275}a^{4}-\frac{56785644}{125573275}a^{3}+\frac{8283617}{25114655}a^{2}+\frac{10197496}{25114655}a+\frac{1910785}{5022931}$ 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:  $C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$
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:   $\frac{193122}{25114655}a^{9}-\frac{167676}{25114655}a^{8}-\frac{7041774}{25114655}a^{7}-\frac{242957}{25114655}a^{6}+\frac{75528501}{25114655}a^{5}+\frac{27565019}{25114655}a^{4}-\frac{268149473}{25114655}a^{3}-\frac{19680709}{5022931}a^{2}+\frac{40357066}{5022931}a+\frac{9905482}{5022931}$, $\frac{4342372}{125573275}a^{9}-\frac{15365076}{125573275}a^{8}-\frac{108339149}{125573275}a^{7}+\frac{250784018}{125573275}a^{6}+\frac{834292946}{125573275}a^{5}-\frac{1099343281}{125573275}a^{4}-\frac{2077448648}{125573275}a^{3}+\frac{192400256}{25114655}a^{2}+\frac{316678642}{25114655}a+\frac{9792430}{5022931}$, $\frac{2631066}{25114655}a^{9}-\frac{9242246}{25114655}a^{8}-\frac{66174453}{25114655}a^{7}+\frac{29923211}{5022931}a^{6}+\frac{517766906}{25114655}a^{5}-\frac{622856517}{25114655}a^{4}-\frac{261396471}{5022931}a^{3}+\frac{383331202}{25114655}a^{2}+\frac{178523892}{5022931}a+\frac{30328686}{5022931}$, $\frac{216381}{25114655}a^{9}-\frac{1169142}{25114655}a^{8}-\frac{804753}{5022931}a^{7}+\frac{21369987}{25114655}a^{6}+\frac{23816737}{25114655}a^{5}-\frac{20912757}{5022931}a^{4}-\frac{54773717}{25114655}a^{3}+\frac{77724496}{25114655}a^{2}+\frac{5954193}{5022931}a+\frac{133755}{5022931}$, $\frac{416507}{25114655}a^{9}-\frac{143424}{5022931}a^{8}-\frac{12684062}{25114655}a^{7}+\frac{4077311}{25114655}a^{6}+\frac{22417198}{5022931}a^{5}+\frac{58072692}{25114655}a^{4}-\frac{283667876}{25114655}a^{3}-\frac{304052819}{25114655}a^{2}+\frac{8333906}{5022931}a+\frac{20289017}{5022931}$, $\frac{19815374}{125573275}a^{9}-\frac{69794892}{125573275}a^{8}-\frac{497111408}{125573275}a^{7}+\frac{1127857381}{125573275}a^{6}+\frac{3885830507}{125573275}a^{5}-\frac{4699413777}{125573275}a^{4}-\frac{9778278491}{125573275}a^{3}+\frac{585813572}{25114655}a^{2}+\frac{1313242424}{25114655}a+\frac{40415935}{5022931}$, $\frac{5156686}{125573275}a^{9}-\frac{16551963}{125573275}a^{8}-\frac{133384812}{125573275}a^{7}+\frac{247013009}{125573275}a^{6}+\frac{1068874473}{125573275}a^{5}-\frac{830706353}{125573275}a^{4}-\frac{2746519999}{125573275}a^{3}-\frac{43203317}{25114655}a^{2}+\frac{370907186}{25114655}a+\frac{40455981}{5022931}$, $\frac{3175356}{125573275}a^{9}-\frac{9627273}{125573275}a^{8}-\frac{89073702}{125573275}a^{7}+\frac{158824864}{125573275}a^{6}+\frac{791782758}{125573275}a^{5}-\frac{721537363}{125573275}a^{4}-\frac{2412708154}{125573275}a^{3}+\frac{156573743}{25114655}a^{2}+\frac{380531731}{25114655}a+\frac{18319389}{5022931}$, $\frac{193122}{25114655}a^{9}-\frac{167676}{25114655}a^{8}-\frac{7041774}{25114655}a^{7}-\frac{242957}{25114655}a^{6}+\frac{75528501}{25114655}a^{5}+\frac{27565019}{25114655}a^{4}-\frac{268149473}{25114655}a^{3}-\frac{19680709}{5022931}a^{2}+\frac{45379997}{5022931}a+\frac{19951344}{5022931}$ 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:  \( 47214.2711571 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 47214.2711571 \cdot 1}{2\cdot\sqrt{2665284492003125}}\cr\approx \mathstrut & 0.468243184469 \end{aligned}\]

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 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*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 - 3*x^9 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*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 - 3*x^9 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*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 - 3*x^9 - 27*x^8 + 44*x^7 + 228*x^6 - 138*x^5 - 634*x^4 - 95*x^3 + 435*x^2 + 225*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

\(\Q(\sqrt{5}) \), 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 ${\href{/padicField/2.10.0.1}{10} }$ ${\href{/padicField/3.10.0.1}{10} }$ R ${\href{/padicField/7.10.0.1}{10} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ R ${\href{/padicField/37.2.0.1}{2} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(31\) Copy content Toggle raw display 31.1.5.4a1.1$x^{5} + 31$$5$$1$$4$$C_5$$$[\ ]_{5}$$
31.1.5.4a1.1$x^{5} + 31$$5$$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)