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Properties

Label 10.0.388...000.76
Degree $10$
Signature $[0, 5]$
Discriminant $-3.888\times 10^{21}$
Root discriminant \(144.20\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (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 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600)
 
Copy content gp:K = bnfinit(y^10 - 100*y^7 - 300*y^6 + 1080*y^5 + 7500*y^4 - 45000*y^3 + 58500*y^2 + 378000*y + 696600, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600)
 

\( x^{10} - 100x^{7} - 300x^{6} + 1080x^{5} + 7500x^{4} - 45000x^{3} + 58500x^{2} + 378000x + 696600 \) 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:   \(-3888000000000000000000\) \(\medspace = -\,2^{22}\cdot 3^{5}\cdot 5^{18}\) 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:  \(144.20\)
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:  $2^{3}3^{7/6}5^{203/100}\approx 756.206767448359$
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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_1$
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:  \(\Q(\sqrt{-2}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{150}a^{5}+\frac{1}{6}a^{4}-\frac{1}{3}a^{3}-\frac{2}{5}$, $\frac{1}{300}a^{6}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{5}a$, $\frac{1}{300}a^{7}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{900}a^{8}+\frac{1}{450}a^{5}-\frac{1}{15}a^{3}+\frac{1}{3}a^{2}+\frac{1}{5}$, $\frac{1}{2201859150300}a^{9}-\frac{178595861}{733953050100}a^{8}+\frac{3206582}{12232550835}a^{7}-\frac{3452894047}{2201859150300}a^{6}-\frac{256763923}{366976525050}a^{5}+\frac{3583492831}{24465101670}a^{4}-\frac{3053695177}{36697652505}a^{3}+\frac{1769880223}{4893020334}a^{2}-\frac{201998021}{12232550835}a-\frac{916344688}{4077516945}$ 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:  $2$, $3$

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:  $4$
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:   $\frac{71\cdots 67}{2201859150300}a^{9}+\frac{12\cdots 31}{733953050100}a^{8}+\frac{70\cdots 21}{244651016700}a^{7}-\frac{94\cdots 33}{220185915030}a^{6}-\frac{12\cdots 94}{36697652505}a^{5}-\frac{16\cdots 43}{24465101670}a^{4}+\frac{32\cdots 69}{73395305010}a^{3}+\frac{20\cdots 37}{12232550835}a^{2}+\frac{56\cdots 32}{2446510167}a+\frac{67\cdots 81}{815503389}$, $\frac{61\cdots 28}{550464787575}a^{9}+\frac{63\cdots 76}{36697652505}a^{8}-\frac{35\cdots 11}{244651016700}a^{7}+\frac{17\cdots 45}{22018591503}a^{6}-\frac{38\cdots 01}{73395305010}a^{5}+\frac{25\cdots 01}{12232550835}a^{4}-\frac{54\cdots 11}{14679061002}a^{3}-\frac{16\cdots 82}{12232550835}a^{2}+\frac{24\cdots 15}{2446510167}a+\frac{48\cdots 65}{815503389}$, $\frac{12\cdots 68}{61162754175}a^{9}+\frac{21\cdots 61}{20387584725}a^{8}+\frac{72\cdots 91}{13591723150}a^{7}+\frac{62\cdots 58}{12232550835}a^{6}-\frac{20\cdots 93}{4077516945}a^{5}-\frac{38\cdots 42}{4077516945}a^{4}+\frac{36\cdots 53}{4077516945}a^{3}-\frac{72\cdots 41}{1359172315}a^{2}-\frac{51\cdots 89}{271834463}a-\frac{71\cdots 15}{271834463}$, $\frac{14\cdots 18}{550464787575}a^{9}+\frac{35\cdots 37}{733953050100}a^{8}-\frac{53\cdots 43}{244651016700}a^{7}-\frac{23\cdots 18}{110092957515}a^{6}-\frac{95\cdots 89}{73395305010}a^{5}+\frac{84\cdots 27}{24465101670}a^{4}+\frac{19\cdots 73}{73395305010}a^{3}-\frac{13\cdots 41}{12232550835}a^{2}-\frac{41\cdots 41}{2446510167}a+\frac{20\cdots 27}{815503389}$ 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:  \( 279210594.529 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 279210594.529 \cdot 1}{2\cdot\sqrt{3888000000000000000000}}\cr\approx \mathstrut & 21.9249247458 \end{aligned}\] (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 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600) 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 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600, 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 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600); 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 - 100*x^7 - 300*x^6 + 1080*x^5 + 7500*x^4 - 45000*x^3 + 58500*x^2 + 378000*x + 696600); 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

\(\Q(\sqrt{-2}) \)

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 ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.1.2.3a1.1$x^{2} + 2$$2$$1$$3$$C_2$$$[3]$$
2.1.4.11a1.4$x^{4} + 8 x^{3} + 8 x + 2$$4$$1$$11$$D_{4}$$$[3, 4]^{2}$$
2.1.4.8b1.5$x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$$[2, 3]$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.3.3a1.1$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$$[\frac{3}{2}]_{2}$$
\(5\) Copy content Toggle raw display 5.2.5.18a1.15$x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2280 x^{6} + 4064 x^{5} + 4560 x^{4} + 3250 x^{3} + 1660 x^{2} + 820 x + 237$$5$$2$$18$$(C_5^2 : C_4) : C_2$$$[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$$

Spectrum of ring of integers

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