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Properties

Label 10.2.10784008474947049.1
Degree $10$
Signature $(2, 4)$
Discriminant $1.078\times 10^{16}$
Root discriminant \(40.11\)
Ramified prime $1609$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $S_5$ (as 10T12)

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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 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849)
 
Copy content gp:K = bnfinit(y^10 - 2*y^9 + 29*y^8 - 15*y^7 + 205*y^6 - 1592*y^5 - 1518*y^4 - 14731*y^3 - 283188*y^2 + 334708*y - 4099849, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 2*x^9 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849)
 

\( x^{10} - 2 x^{9} + 29 x^{8} - 15 x^{7} + 205 x^{6} - 1592 x^{5} - 1518 x^{4} - 14731 x^{3} + \cdots - 4099849 \) 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:  $(2, 4)$
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:   \(10784008474947049\) \(\medspace = 1609^{5}\) 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:  \(40.11\)
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:  $1609^{1/2}\approx 40.11234224026316$
Ramified primes:   \(1609\) 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:  \(\Q(\sqrt{1609}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8}a^{8}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{18\cdots 84}a^{9}-\frac{10\cdots 81}{18\cdots 84}a^{8}-\frac{20\cdots 63}{42\cdots 36}a^{7}+\frac{50\cdots 53}{18\cdots 84}a^{6}-\frac{13\cdots 91}{94\cdots 92}a^{5}-\frac{24\cdots 35}{94\cdots 92}a^{4}-\frac{16\cdots 01}{47\cdots 96}a^{3}-\frac{37\cdots 07}{18\cdots 84}a^{2}+\frac{48\cdots 85}{18\cdots 84}a+\frac{73\cdots 85}{18\cdots 84}$ 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:  $C_{3}$, which has order $3$ (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:  $C_{3}$, which has order $3$ (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:  $5$
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:   $\frac{15\cdots 95}{18\cdots 84}a^{9}+\frac{18\cdots 57}{18\cdots 84}a^{8}+\frac{25\cdots 97}{21\cdots 18}a^{7}+\frac{31\cdots 55}{18\cdots 84}a^{6}-\frac{43\cdots 65}{94\cdots 92}a^{5}-\frac{22\cdots 79}{94\cdots 92}a^{4}-\frac{18\cdots 23}{47\cdots 96}a^{3}+\frac{66\cdots 31}{18\cdots 84}a^{2}-\frac{35\cdots 25}{18\cdots 84}a+\frac{52\cdots 75}{18\cdots 84}$, $\frac{59\cdots 85}{18\cdots 84}a^{9}+\frac{29\cdots 09}{18\cdots 84}a^{8}+\frac{24\cdots 98}{10\cdots 59}a^{7}-\frac{13\cdots 79}{18\cdots 84}a^{6}+\frac{53\cdots 62}{11\cdots 49}a^{5}-\frac{27\cdots 76}{11\cdots 49}a^{4}-\frac{33\cdots 03}{94\cdots 92}a^{3}-\frac{22\cdots 97}{18\cdots 84}a^{2}+\frac{94\cdots 01}{18\cdots 84}a+\frac{69\cdots 71}{18\cdots 84}$, $\frac{17\cdots 75}{18\cdots 84}a^{9}+\frac{17\cdots 95}{18\cdots 84}a^{8}+\frac{56\cdots 04}{10\cdots 59}a^{7}+\frac{50\cdots 87}{18\cdots 84}a^{6}+\frac{34\cdots 29}{47\cdots 96}a^{5}-\frac{11\cdots 51}{47\cdots 96}a^{4}-\frac{18\cdots 55}{94\cdots 92}a^{3}-\frac{17\cdots 31}{18\cdots 84}a^{2}-\frac{65\cdots 13}{18\cdots 84}a-\frac{19\cdots 43}{18\cdots 84}$, $\frac{13\cdots 89}{23\cdots 98}a^{9}+\frac{28\cdots 35}{47\cdots 96}a^{8}+\frac{42\cdots 05}{42\cdots 36}a^{7}+\frac{22\cdots 33}{47\cdots 96}a^{6}+\frac{23\cdots 45}{11\cdots 49}a^{5}-\frac{71\cdots 51}{47\cdots 96}a^{4}+\frac{12\cdots 61}{11\cdots 49}a^{3}+\frac{79\cdots 65}{47\cdots 96}a^{2}-\frac{34\cdots 07}{47\cdots 96}a-\frac{12\cdots 30}{11\cdots 49}$, $\frac{50\cdots 23}{47\cdots 96}a^{9}+\frac{79\cdots 03}{47\cdots 96}a^{8}+\frac{23\cdots 37}{21\cdots 18}a^{7}+\frac{28\cdots 75}{47\cdots 96}a^{6}+\frac{16\cdots 85}{23\cdots 98}a^{5}-\frac{49\cdots 81}{23\cdots 98}a^{4}-\frac{23\cdots 57}{23\cdots 98}a^{3}-\frac{22\cdots 11}{47\cdots 96}a^{2}-\frac{10\cdots 63}{47\cdots 96}a-\frac{31\cdots 87}{47\cdots 96}$ 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:  \( 3525.875783 \) (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:  \( 2 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 3525.875783 \cdot 3}{2\cdot\sqrt{10784008474947049}}\cr\approx \mathstrut & 0.3175028453 \end{aligned}\] (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^10 - 2*x^9 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849) 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 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849, 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 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849); 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^10 - 2*x^9 + 29*x^8 - 15*x^7 + 205*x^6 - 1592*x^5 - 1518*x^4 - 14731*x^3 - 283188*x^2 + 334708*x - 4099849); 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

$S_5$ (as 10T12):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{1609}) \), 5.1.1609.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 5 sibling: 5.1.1609.1
Degree 6 sibling: 6.2.4165509529.1
Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.1.1609.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(1609\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$

Spectrum of ring of integers

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