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Properties

Label 10.4.102515625000000.1
Degree $10$
Signature $(4, 3)$
Discriminant $-1.025\times 10^{14}$
Root discriminant \(25.18\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $S_5$ (as 10T13)

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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 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2)
 
Copy content gp:K = bnfinit(y^10 - 10*y^8 - 15*y^7 - 35*y^6 - 111*y^5 - 155*y^4 - 120*y^3 - 55*y^2 - 15*y - 2, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2)
 

\( x^{10} - 10x^{8} - 15x^{7} - 35x^{6} - 111x^{5} - 155x^{4} - 120x^{3} - 55x^{2} - 15x - 2 \) 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:  $(4, 3)$
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:   \(-102515625000000\) \(\medspace = -\,2^{6}\cdot 3^{8}\cdot 5^{12}\) 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:  \(25.18\)
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:  $2\cdot 3^{4/5}5^{13/10}\approx 39.029052225185055$
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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1907}a^{9}+\frac{549}{1907}a^{8}+\frac{85}{1907}a^{7}+\frac{882}{1907}a^{6}-\frac{195}{1907}a^{5}-\frac{374}{1907}a^{4}+\frac{475}{1907}a^{3}-\frac{604}{1907}a^{2}+\frac{167}{1907}a+\frac{132}{1907}$ 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:  Trivial group, which has order $1$
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 $1$
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:  $6$
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:   $\frac{6138}{1907}a^{9}-\frac{3721}{1907}a^{8}-\frac{59905}{1907}a^{7}-\frac{55560}{1907}a^{6}-\frac{172851}{1907}a^{5}-\frac{567870}{1907}a^{4}-\frac{585702}{1907}a^{3}-\frac{301450}{1907}a^{2}-\frac{69572}{1907}a-\frac{7887}{1907}$, $\frac{628}{1907}a^{9}+\frac{1512}{1907}a^{8}-\frac{7644}{1907}a^{7}-\frac{23925}{1907}a^{6}-\frac{30924}{1907}a^{5}-\frac{109010}{1907}a^{4}-\frac{226125}{1907}a^{3}-\frac{177170}{1907}a^{2}-\frac{74382}{1907}a-\frac{14361}{1907}$, $\frac{905}{1907}a^{9}-\frac{882}{1907}a^{8}-\frac{8890}{1907}a^{7}-\frac{4637}{1907}a^{6}-\frac{20101}{1907}a^{5}-\frac{73397}{1907}a^{4}-\frac{48782}{1907}a^{3}+\frac{8317}{1907}a^{2}+\frac{21459}{1907}a+\frac{6947}{1907}$, $\frac{7733}{1907}a^{9}+\frac{435}{1907}a^{8}-\frac{80704}{1907}a^{7}-\frac{117160}{1907}a^{6}-\frac{245501}{1907}a^{5}-\frac{853559}{1907}a^{4}-\frac{1157256}{1907}a^{3}-\frac{711800}{1907}a^{2}-\frac{247538}{1907}a-\frac{41443}{1907}$, $\frac{8174}{1907}a^{9}-\frac{5366}{1907}a^{8}-\frac{79452}{1907}a^{7}-\frac{69551}{1907}a^{6}-\frac{228518}{1907}a^{5}-\frac{747699}{1907}a^{4}-\frac{741825}{1907}a^{3}-\frac{383180}{1907}a^{2}-\frac{93797}{1907}a-\frac{6115}{1907}$, $\frac{1745}{1907}a^{9}-\frac{1216}{1907}a^{8}-\frac{17584}{1907}a^{7}-\frac{13208}{1907}a^{6}-\frac{42783}{1907}a^{5}-\frac{154903}{1907}a^{4}-\frac{134160}{1907}a^{3}-\frac{29921}{1907}a^{2}+\frac{16807}{1907}a+\frac{7221}{1907}$ Copy content Toggle raw display
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:  \( 4744.08460033 \)
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:  \( 4 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 4744.08460033 \cdot 1}{2\cdot\sqrt{102515625000000}}\cr\approx \mathstrut & 0.929794526578 \end{aligned}\]

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 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2) 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 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2, 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 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2); 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 - 10*x^8 - 15*x^7 - 35*x^6 - 111*x^5 - 155*x^4 - 120*x^3 - 55*x^2 - 15*x - 2); 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 10T13):

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

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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: 5.3.5062500.1
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: 5.3.5062500.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$$[\ ]$$
2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.3.2.6a1.1$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$$2$$3$$6$$C_6$$$[2]^{3}$$
\(3\) Copy content Toggle raw display 3.2.5.8a1.1$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 720 x^{4} + 640 x^{3} + 400 x^{2} + 160 x + 35$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
\(5\) Copy content Toggle raw display 5.1.5.6a1.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$$[\frac{3}{2}]_{2}$$
5.1.5.6a1.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$$[\frac{3}{2}]_{2}$$

Spectrum of ring of integers

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