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Properties

Label 10.0.322...875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-3.228\times 10^{38}$
Root discriminant \(7093.95\)
Ramified primes $3,5,41,239$
Class number $25$ (GRH)
Class group [5, 5] (GRH)
Galois group $F_{5}\times C_2$ (as 10T5)

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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 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400)
 
Copy content gp:K = bnfinit(y^10 - y^9 - 3919*y^8 + 468000*y^7 + 7306840*y^6 - 1012165680*y^5 + 24905759920*y^4 - 1039715164800*y^3 + 36940306467200*y^2 - 804654756448000*y + 10578056676102400, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400)
 

\( x^{10} - x^{9} - 3919 x^{8} + 468000 x^{7} + 7306840 x^{6} - 1012165680 x^{5} + 24905759920 x^{4} + \cdots + 10\!\cdots\!00 \) 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:   \(-322760486162698979743554510954828796875\) \(\medspace = -\,3^{5}\cdot 5^{6}\cdot 41^{8}\cdot 239^{8}\) 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:  \(7093.95\)
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:  $3^{1/2}5^{3/4}41^{4/5}239^{4/5}\approx 9030.952057415474$
Ramified primes:   \(3\), \(5\), \(41\), \(239\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{4}+\frac{9}{64}a^{3}-\frac{13}{64}a^{2}+\frac{7}{32}a+\frac{1}{16}$, $\frac{1}{128}a^{5}-\frac{1}{128}a^{4}+\frac{25}{128}a^{3}-\frac{3}{8}a^{2}-\frac{1}{16}a-\frac{5}{16}$, $\frac{1}{20480}a^{6}+\frac{59}{20480}a^{5}+\frac{121}{20480}a^{4}-\frac{15}{512}a^{3}-\frac{73}{1024}a^{2}-\frac{7}{32}a-\frac{3}{256}$, $\frac{1}{40960}a^{7}-\frac{1}{40960}a^{6}+\frac{101}{40960}a^{5}+\frac{7}{2048}a^{4}+\frac{47}{2048}a^{3}+\frac{127}{512}a^{2}-\frac{99}{512}a+\frac{5}{128}$, $\frac{1}{1310720}a^{8}+\frac{13}{1310720}a^{7}-\frac{29}{1310720}a^{6}+\frac{175}{131072}a^{5}+\frac{17}{2560}a^{4}-\frac{6685}{32768}a^{3}+\frac{1745}{4096}a^{2}-\frac{2091}{8192}a+\frac{637}{4096}$, $\frac{1}{19\cdots 60}a^{9}-\frac{89\cdots 47}{38\cdots 12}a^{8}-\frac{62\cdots 89}{19\cdots 60}a^{7}+\frac{21\cdots 59}{95\cdots 80}a^{6}+\frac{46\cdots 93}{11\cdots 60}a^{5}-\frac{13\cdots 29}{23\cdots 20}a^{4}+\frac{18\cdots 09}{29\cdots 04}a^{3}-\frac{82\cdots 27}{11\cdots 16}a^{2}-\frac{69\cdots 75}{59\cdots 08}a+\frac{85\cdots 87}{58\cdots 56}$ 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:  $C_{5}\times C_{5}$, which has order $25$ (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:  $C_{5}\times C_{5}$, which has order $25$ (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:   \( -\frac{28793485092456485786789}{5849500124424892225985373482330879920} a^{9} - \frac{1196214365429668449847891}{11699000248849784451970746964661759840} a^{8} + \frac{189570762066143482946898093}{11699000248849784451970746964661759840} a^{7} - \frac{23001826689795980390227042709}{11699000248849784451970746964661759840} a^{6} - \frac{22551522512705522208445414971}{292475006221244611299268674116543996} a^{5} + \frac{17286937899812776757302014433421}{5849500124424892225985373482330879920} a^{4} - \frac{4991998135583734362086925092193}{73118751555311152824817168529135999} a^{3} + \frac{570321546186883086747147397018429}{146237503110622305649634337058271998} a^{2} - \frac{8697349528554892477831302125474386}{73118751555311152824817168529135999} a + \frac{152531683364495495080833652359}{56874266642588408889413885219} \)  (order $6$) 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:   $\frac{11\cdots 17}{19\cdots 60}a^{9}-\frac{20\cdots 63}{38\cdots 12}a^{8}+\frac{38\cdots 91}{19\cdots 60}a^{7}+\frac{13\cdots 89}{95\cdots 80}a^{6}-\frac{22\cdots 37}{23\cdots 20}a^{5}+\frac{64\cdots 19}{23\cdots 20}a^{4}-\frac{12\cdots 23}{11\cdots 16}a^{3}+\frac{37\cdots 21}{11\cdots 16}a^{2}-\frac{37\cdots 05}{59\cdots 08}a+\frac{83\cdots 53}{11\cdots 12}$, $\frac{75\cdots 35}{95\cdots 28}a^{9}-\frac{11\cdots 89}{95\cdots 28}a^{8}-\frac{21\cdots 87}{47\cdots 40}a^{7}+\frac{82\cdots 01}{23\cdots 20}a^{6}+\frac{85\cdots 13}{29\cdots 40}a^{5}-\frac{16\cdots 95}{11\cdots 16}a^{4}-\frac{29\cdots 99}{74\cdots 76}a^{3}-\frac{39\cdots 17}{29\cdots 04}a^{2}+\frac{23\cdots 47}{14\cdots 52}a-\frac{50\cdots 57}{72\cdots 32}$, $\frac{12\cdots 99}{46\cdots 48}a^{9}+\frac{14\cdots 55}{18\cdots 92}a^{8}-\frac{15\cdots 45}{18\cdots 92}a^{7}+\frac{19\cdots 41}{18\cdots 92}a^{6}+\frac{47\cdots 05}{93\cdots 96}a^{5}-\frac{39\cdots 75}{29\cdots 28}a^{4}+\frac{73\cdots 05}{23\cdots 24}a^{3}-\frac{45\cdots 25}{29\cdots 28}a^{2}+\frac{18\cdots 75}{58\cdots 56}a-\frac{21\cdots 13}{29\cdots 28}$, $\frac{61\cdots 89}{23\cdots 32}a^{9}-\frac{44\cdots 89}{59\cdots 80}a^{8}+\frac{23\cdots 29}{29\cdots 40}a^{7}-\frac{11\cdots 29}{11\cdots 60}a^{6}-\frac{28\cdots 87}{59\cdots 80}a^{5}+\frac{17\cdots 89}{14\cdots 20}a^{4}-\frac{83\cdots 27}{29\cdots 04}a^{3}+\frac{13\cdots 81}{74\cdots 76}a^{2}-\frac{29\cdots 55}{74\cdots 76}a+\frac{25\cdots 15}{29\cdots 28}$ 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:  \( 303066841016740.94 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 303066841016740.94 \cdot 25}{6\cdot\sqrt{322760486162698979743554510954828796875}}\cr\approx \mathstrut & 0.688313585987879 \end{aligned}\] (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 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400) 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 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400, 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 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400); 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 - 3919*x^8 + 468000*x^7 + 7306840*x^6 - 1012165680*x^5 + 24905759920*x^4 - 1039715164800*x^3 + 36940306467200*x^2 - 804654756448000*x + 10578056676102400); 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

$C_2\times F_5$ (as 10T5):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 40
The 10 conjugacy class representatives for $F_{5}\times C_2$
Character table for $F_{5}\times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.1.1152489676025100125.1

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

Galois closure: data not computed
Degree 10 sibling: data not computed
Degree 20 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 ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ R R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.2.0.1}{2} }^{5}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ R ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.2.0.1}{2} }^{5}$

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
\(3\) Copy content Toggle raw display 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(41\) Copy content Toggle raw display 41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$
\(239\) Copy content Toggle raw display Deg $10$$5$$2$$8$

Spectrum of ring of integers

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