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Properties

Label 10.2.136...000.41
Degree $10$
Signature $[2, 4]$
Discriminant $1.367\times 10^{20}$
Root discriminant \(103.17\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{10}$ (as 10T45)

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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 - 15*x^8 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277)
 
Copy content gp:K = bnfinit(y^10 - 15*y^8 + 90*y^6 - 840*y^5 + 1530*y^4 - 3600*y^3 + 7605*y^2 - 1800*y + 20277, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 15*x^8 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 15*x^8 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277)
 

\( x^{10} - 15x^{8} + 90x^{6} - 840x^{5} + 1530x^{4} - 3600x^{3} + 7605x^{2} - 1800x + 20277 \) 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:  $[2, 4]$
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:   \(136687500000000000000\) \(\medspace = 2^{14}\cdot 3^{7}\cdot 5^{18}\) 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:  \(103.17\)
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:  $2^{71/40}3^{19/20}5^{203/100}\approx 254.97790184447035$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{12}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{12}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{12}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{24}a^{8}-\frac{1}{4}a^{4}-\frac{1}{8}$, $\frac{1}{122182204824}a^{9}+\frac{84798653}{13575800536}a^{8}+\frac{2966222}{5090925201}a^{7}-\frac{31777373}{20363700804}a^{6}-\frac{121333762}{5090925201}a^{5}-\frac{426926765}{20363700804}a^{4}+\frac{61771996}{1696975067}a^{3}+\frac{111340855}{6787900268}a^{2}-\frac{5506158953}{13575800536}a+\frac{3965805849}{13575800536}$ 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 $1$ (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:  $5$
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:   $\frac{250707655}{20363700804}a^{9}+\frac{32954241}{1696975067}a^{8}-\frac{862751325}{3393950134}a^{7}-\frac{1092443807}{3393950134}a^{6}+\frac{3605084850}{1696975067}a^{5}-\frac{27949926449}{3393950134}a^{4}-\frac{6763482519}{3393950134}a^{3}+\frac{153353087433}{3393950134}a^{2}-\frac{577528758201}{6787900268}a+\frac{919675398533}{3393950134}$, $\frac{64647331}{20363700804}a^{9}+\frac{38355293}{40727401608}a^{8}-\frac{473915211}{6787900268}a^{7}-\frac{1523021087}{20363700804}a^{6}+\frac{3023298707}{5090925201}a^{5}-\frac{9345072003}{6787900268}a^{4}+\frac{21445085351}{6787900268}a^{3}-\frac{20689800907}{6787900268}a^{2}+\frac{42869865663}{6787900268}a-\frac{76742049497}{13575800536}$, $\frac{28169581}{10181850402}a^{9}+\frac{41441323}{20363700804}a^{8}-\frac{293921429}{6787900268}a^{7}-\frac{172721359}{3393950134}a^{6}+\frac{5742105091}{20363700804}a^{5}-\frac{2879186088}{1696975067}a^{4}+\frac{15321240995}{6787900268}a^{3}-\frac{16620302583}{1696975067}a^{2}+\frac{197587113027}{6787900268}a+\frac{26294928757}{6787900268}$, $\frac{11477028947}{61091102412}a^{9}+\frac{4447607399}{40727401608}a^{8}-\frac{75171874103}{20363700804}a^{7}-\frac{2106043616}{1696975067}a^{6}+\frac{54200214684}{1696975067}a^{5}-\frac{1603808631115}{10181850402}a^{4}+\frac{528226040385}{6787900268}a^{3}+\frac{499485807613}{1696975067}a^{2}-\frac{2612535926839}{6787900268}a+\frac{29948518048847}{13575800536}$, $\frac{589744554379}{122182204824}a^{9}-\frac{128732562295}{3393950134}a^{8}+\frac{248137765866}{1696975067}a^{7}-\frac{4799072586653}{10181850402}a^{6}+\frac{1994593790412}{1696975067}a^{5}-\frac{63079786458869}{20363700804}a^{4}+\frac{21473811415437}{3393950134}a^{3}-\frac{16074959885844}{1696975067}a^{2}+\frac{195482634796833}{13575800536}a-\frac{40854544747409}{6787900268}$ 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:  \( 6493755.58396 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 6493755.58396 \cdot 1}{2\cdot\sqrt{136687500000000000000}}\cr\approx \mathstrut & 1.73133459580 \end{aligned}\] (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 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277) 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 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277, 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 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277); 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 + 90*x^6 - 840*x^5 + 1530*x^4 - 3600*x^3 + 7605*x^2 - 1800*x + 20277); 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

$S_{10}$ (as 10T45):

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 3628800
The 42 conjugacy class representatives for $S_{10}$
Character table for $S_{10}$

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 sibling: data not computed
Degree 45 sibling: 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 ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.1.10.14a1.6$x^{10} + 2 x^{7} + 2 x^{5} + 6$$10$$1$$14$$((C_2^4 : C_5):C_4)\times C_2$$$[\frac{8}{5}, \frac{8}{5}, \frac{8}{5}, \frac{8}{5}, 2]_{5}^{4}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.1.4.3a1.2$x^{4} + 6$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.1.5.4a1.1$x^{5} + 3$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$
\(5\) 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)