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Properties

Label 10.0.57665039062500000.1
Degree $10$
Signature $[0, 5]$
Discriminant $-5.767\times 10^{16}$
Root discriminant \(47.43\)
Ramified primes $2,3,5$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $S_5^2 \wr C_2$ (as 10T43)

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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 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540)
 
Copy content gp:K = bnfinit(y^10 - 20*y^7 + 30*y^6 + 100*y^4 - 300*y^3 + 225*y^2 + 540, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540)
 

\( x^{10} - 20x^{7} + 30x^{6} + 100x^{4} - 300x^{3} + 225x^{2} + 540 \) 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:   \(-57665039062500000\) \(\medspace = -\,2^{5}\cdot 3^{10}\cdot 5^{15}\) 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:  \(47.43\)
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:  $2^{2}3^{5/4}5^{163/100}\approx 217.6620257611859$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q(\sqrt{-10}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-15}) \)

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{5}+\frac{1}{6}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{6}+\frac{1}{6}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{24}a^{7}-\frac{1}{24}a^{6}+\frac{1}{12}a^{4}+\frac{1}{24}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{144}a^{8}-\frac{1}{72}a^{7}-\frac{5}{144}a^{6}-\frac{1}{36}a^{5}-\frac{1}{144}a^{4}-\frac{17}{72}a^{3}-\frac{7}{16}a^{2}-\frac{1}{4}$, $\frac{1}{288}a^{9}-\frac{1}{288}a^{8}+\frac{5}{288}a^{7}-\frac{1}{32}a^{6}-\frac{5}{288}a^{5}-\frac{11}{288}a^{4}+\frac{11}{288}a^{3}+\frac{1}{32}a^{2}-\frac{3}{8}a+\frac{3}{8}$ 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:  $C_{2}$, which has order $2$ (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:  $C_{2}$, which has order $2$ (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:  $4$
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:   $\frac{1}{12}a^{9}+\frac{7}{144}a^{8}-\frac{2}{9}a^{7}-\frac{233}{144}a^{6}+\frac{31}{18}a^{5}+\frac{857}{144}a^{4}+\frac{59}{36}a^{3}-\frac{953}{48}a^{2}-\frac{15}{2}a+\frac{311}{4}$, $\frac{95}{288}a^{9}-\frac{17}{96}a^{8}-\frac{83}{96}a^{7}-\frac{511}{288}a^{6}+\frac{799}{96}a^{5}+\frac{701}{96}a^{4}-\frac{15055}{288}a^{3}+\frac{3709}{96}a^{2}+\frac{95}{8}a+\frac{917}{8}$, $\frac{989}{32}a^{9}+\frac{9773}{288}a^{8}-\frac{1915}{288}a^{7}-\frac{215407}{288}a^{6}-\frac{22697}{288}a^{5}+\frac{200431}{288}a^{4}+\frac{1532731}{288}a^{3}-\frac{210827}{96}a^{2}-\frac{10495}{8}a-\frac{65563}{8}$, $\frac{317}{16}a^{9}-\frac{5275}{144}a^{8}+\frac{1451}{144}a^{7}-\frac{51949}{144}a^{6}+\frac{183427}{144}a^{5}-\frac{182717}{144}a^{4}+\frac{228937}{144}a^{3}-\frac{367237}{48}a^{2}+\frac{59455}{4}a-\frac{32729}{4}$ 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:  \( 240128.203156 \) (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)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 240128.203156 \cdot 2}{2\cdot\sqrt{57665039062500000}}\cr\approx \mathstrut & 9.79233398925 \end{aligned}\] (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 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540) 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 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540, 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 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540); 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 - 20*x^7 + 30*x^6 + 100*x^4 - 300*x^3 + 225*x^2 + 540); 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

$S_5\wr C_2$ (as 10T43):

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 non-solvable group of order 28800
The 35 conjugacy class representatives for $S_5^2 \wr C_2$
Character table for $S_5^2 \wr C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \)

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]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 25 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(3\) Copy content Toggle raw display 3.1.4.3a1.2$x^{4} + 6$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.1.6.7a1.4$x^{6} + 3 x^{3} + 6 x^{2} + 6$$6$$1$$7$$S_3\times C_3$$$[\frac{3}{2}]_{2}^{3}$$
\(5\) Copy content Toggle raw display 5.1.10.15a2.9$x^{10} + 10 x^{7} + 5 x^{6} + 10$$10$$1$$15$$C_5^2 : C_4$$$[\frac{5}{4}, \frac{7}{4}]_{4}$$

Spectrum of ring of integers

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