↑

Properties

Label 10.10.23050420428800000.1
Degree $10$
Signature $(10, 0)$
Discriminant $2.305\times 10^{16}$
Root discriminant \(43.28\)
Ramified primes $2,5,103$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_5$ (as 10T12)

Related objects

Downloads

Learn more

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 - x^9 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55)
 
Copy content gp:K = bnfinit(y^10 - y^9 - 31*y^8 + 6*y^7 + 321*y^6 + 133*y^5 - 1141*y^4 - 918*y^3 + 619*y^2 + 615*y + 55, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55)
 

\( x^{10} - x^{9} - 31x^{8} + 6x^{7} + 321x^{6} + 133x^{5} - 1141x^{4} - 918x^{3} + 619x^{2} + 615x + 55 \) 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:   \(23050420428800000\) \(\medspace = 2^{16}\cdot 5^{5}\cdot 103^{4}\) 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:  \(43.28\)
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:  $2^{13/6}5^{1/2}103^{1/2}\approx 101.89087030315206$
Ramified primes:   \(2\), \(5\), \(103\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{33244}a^{9}-\frac{951}{16622}a^{8}+\frac{193}{16622}a^{7}-\frac{603}{8311}a^{6}-\frac{2139}{33244}a^{5}-\frac{3009}{16622}a^{4}+\frac{2863}{8311}a^{3}-\frac{3243}{8311}a^{2}-\frac{6657}{33244}a+\frac{3115}{16622}$ 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:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (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:  Trivial group, which has order $1$ (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:  $9$
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:   $\frac{1749}{33244}a^{9}-\frac{1099}{16622}a^{8}-\frac{28127}{16622}a^{7}+\frac{9161}{8311}a^{6}+\frac{597231}{33244}a^{5}-\frac{60055}{16622}a^{4}-\frac{1097033}{16622}a^{3}-\frac{37149}{8311}a^{2}+\frac{1371933}{33244}a+\frac{79229}{16622}$, $\frac{865}{16622}a^{9}+\frac{7615}{33244}a^{8}+\frac{7586}{8311}a^{7}-\frac{124027}{33244}a^{6}-\frac{180767}{33244}a^{5}+\frac{545983}{33244}a^{4}+\frac{392081}{33244}a^{3}-\frac{190223}{16622}a^{2}-\frac{226833}{33244}a+\frac{4889}{16622}$, $\frac{865}{16622}a^{9}-\frac{7615}{33244}a^{8}-\frac{7586}{8311}a^{7}+\frac{124027}{33244}a^{6}+\frac{180767}{33244}a^{5}-\frac{545983}{33244}a^{4}-\frac{392081}{33244}a^{3}+\frac{190223}{16622}a^{2}+\frac{226833}{33244}a-\frac{21511}{16622}$, $\frac{458}{8311}a^{9}+\frac{2155}{33244}a^{8}+\frac{14365}{8311}a^{7}-\frac{27601}{33244}a^{6}-\frac{594225}{33244}a^{5}-\frac{20343}{33244}a^{4}+\frac{2049717}{33244}a^{3}+\frac{173342}{8311}a^{2}-\frac{744603}{33244}a-\frac{46889}{16622}$, $\frac{199}{16622}a^{9}-\frac{695}{33244}a^{8}-\frac{3148}{8311}a^{7}+\frac{12411}{33244}a^{6}+\frac{137687}{33244}a^{5}-\frac{43151}{33244}a^{4}-\frac{553369}{33244}a^{3}-\frac{63195}{16622}a^{2}+\frac{384029}{33244}a+\frac{67919}{16622}$, $\frac{3317}{8311}a^{9}-\frac{45095}{33244}a^{8}-\frac{74331}{8311}a^{7}+\frac{767857}{33244}a^{6}+\frac{2312271}{33244}a^{5}-\frac{3360691}{33244}a^{4}-\frac{6504079}{33244}a^{3}+\frac{785391}{16622}a^{2}+\frac{3652761}{33244}a+\frac{86874}{8311}$, $\frac{2715}{16622}a^{9}-\frac{13909}{33244}a^{8}-\frac{73995}{16622}a^{7}+\frac{258617}{33244}a^{6}+\frac{1408567}{33244}a^{5}-\frac{1370203}{33244}a^{4}-\frac{4926965}{33244}a^{3}+\frac{475246}{8311}a^{2}+\frac{3637247}{33244}a-\frac{86483}{8311}$, $\frac{3299}{16622}a^{9}+\frac{8097}{33244}a^{8}+\frac{53106}{8311}a^{7}-\frac{134165}{33244}a^{6}-\frac{2251237}{33244}a^{5}+\frac{437289}{33244}a^{4}+\frac{8222895}{33244}a^{3}+\frac{233997}{16622}a^{2}-\frac{5037215}{33244}a-\frac{265619}{16622}$, $\frac{16925}{33244}a^{9}+\frac{6945}{8311}a^{8}+\frac{257339}{16622}a^{7}-\frac{116487}{8311}a^{6}-\frac{5269315}{33244}a^{5}+\frac{426775}{8311}a^{4}+\frac{9443317}{16622}a^{3}+\frac{201395}{8311}a^{2}-\frac{11679457}{33244}a-\frac{293236}{8311}$ 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:  \( 212552.42166224215 \) (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:  \( 10 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 212552.42166224215 \cdot 1}{2\cdot\sqrt{23050420428800000}}\cr\approx \mathstrut & 0.716797682901645 \end{aligned}\] (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 - x^9 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55) 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 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55, 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 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55); 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 - x^9 - 31*x^8 + 6*x^7 + 321*x^6 + 133*x^5 - 1141*x^4 - 918*x^3 + 619*x^2 + 615*x + 55); 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

\(\Q(\sqrt{5}) \), 5.5.13579520.1

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: data not computed
Degree 6 sibling: data not computed
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: 5.5.13579520.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{5}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.4.16a1.1$x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 14 x^{2} + 8 x + 7$$4$$2$$16$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
\(5\) 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$$$[\ ]_{2}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(103\) Copy content Toggle raw display 103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
103.2.2.2a1.2$x^{4} + 204 x^{3} + 10414 x^{2} + 1020 x + 128$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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