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Properties

Label 10.0.186...000.2
Degree $10$
Signature $[0, 5]$
Discriminant $-1.868\times 10^{19}$
Root discriminant \(84.56\)
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 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144)
 
Copy content gp:K = bnfinit(y^10 - 15*y^8 - 30*y^7 + 150*y^6 - 126*y^5 + 600*y^4 - 180*y^3 - 360*y^2 + 144, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144)
 

\( x^{10} - 15x^{8} - 30x^{7} + 150x^{6} - 126x^{5} + 600x^{4} - 180x^{3} - 360x^{2} + 144 \) 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:   \(-18683472656250000000\) \(\medspace = -\,2^{7}\cdot 3^{14}\cdot 5^{15}\) 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:  \(84.56\)
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:  $2^{11/6}3^{23/12}5^{163/100}\approx 403.35907958801755$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{5}-\frac{1}{2}a^{3}+\frac{1}{5}$, $\frac{1}{30}a^{6}-\frac{1}{2}a^{3}+\frac{2}{5}a$, $\frac{1}{60}a^{7}-\frac{1}{20}a^{5}-\frac{3}{10}a^{2}+\frac{2}{5}$, $\frac{1}{300}a^{8}+\frac{1}{300}a^{7}-\frac{1}{100}a^{6}+\frac{1}{100}a^{5}-\frac{1}{10}a^{4}-\frac{13}{50}a^{3}+\frac{7}{50}a^{2}-\frac{8}{25}a-\frac{12}{25}$, $\frac{1}{11400}a^{9}+\frac{1}{5700}a^{8}-\frac{29}{3800}a^{7}+\frac{1}{114}a^{6}+\frac{33}{1900}a^{5}+\frac{7}{1900}a^{4}+\frac{13}{50}a^{3}+\frac{23}{950}a^{2}+\frac{13}{95}a-\frac{231}{475}$ 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:   $\frac{31349}{5700}a^{9}-\frac{10042}{1425}a^{8}-\frac{562579}{5700}a^{7}+\frac{227747}{1425}a^{6}+\frac{38689}{950}a^{5}+\frac{144824}{475}a^{4}+\frac{29228}{25}a^{3}-\frac{542929}{475}a^{2}-\frac{324612}{475}a+\frac{313259}{475}$, $\frac{957281}{228}a^{9}+\frac{2111243}{570}a^{8}-\frac{22756709}{380}a^{7}-\frac{33959247}{190}a^{6}+\frac{18017167}{38}a^{5}-\frac{2055425}{19}a^{4}+\frac{12026074}{5}a^{3}+\frac{137410683}{95}a^{2}-\frac{37108257}{95}a-\frac{10920179}{19}$, $\frac{745576323}{3800}a^{9}+\frac{435811363}{2850}a^{8}-\frac{32292795019}{11400}a^{7}-\frac{46207575031}{5700}a^{6}+\frac{44139652579}{1900}a^{5}-\frac{11727899887}{1900}a^{4}+\frac{2804069954}{25}a^{3}+\frac{49285107081}{950}a^{2}-\frac{16808744981}{475}a-\frac{15845795863}{475}$, $\frac{2039803267}{60}a^{9}+\frac{2608652677}{10}a^{8}-\frac{9139400827}{60}a^{7}-\frac{136897627603}{30}a^{6}-7758394684a^{5}+\frac{94225020527}{5}a^{4}+\frac{126514921272}{5}a^{3}+\frac{734360767968}{5}a^{2}+\frac{737627713209}{5}a+54128251547$ 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:  \( 9401767.20909 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 9401767.20909 \cdot 2}{2\cdot\sqrt{18683472656250000000}}\cr\approx \mathstrut & 21.3000211717 \end{aligned}\] (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 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144) 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 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144, 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 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144); 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 - 15*x^8 - 30*x^7 + 150*x^6 - 126*x^5 + 600*x^4 - 180*x^3 - 360*x^2 + 144); 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 ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$$[\ ]$$
2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.4.3a1.2$x^{4} + 6$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.1.6.11a2.1$x^{6} + 6$$6$$1$$11$$D_{6}$$$[\frac{5}{2}]_{2}^{2}$$
\(5\) 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)