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Properties

Label 10.0.405...000.44
Degree $10$
Signature $[0, 5]$
Discriminant $-4.050\times 10^{19}$
Root discriminant \(91.36\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(A_5^2 : C_2):C_2$ (as 10T41)

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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 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225)
 
Copy content gp:K = bnfinit(y^10 + 50*y^8 - 150*y^7 + 725*y^6 - 4440*y^5 + 8125*y^4 - 24750*y^3 + 74250*y^2 - 130500*y + 234225, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225)
 

\( x^{10} + 50x^{8} - 150x^{7} + 725x^{6} - 4440x^{5} + 8125x^{4} - 24750x^{3} + 74250x^{2} - 130500x + 234225 \) 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:   \(-40500000000000000000\) \(\medspace = -\,2^{17}\cdot 3^{4}\cdot 5^{18}\) 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:  \(91.36\)
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:  $2^{11/6}3^{7/6}5^{203/100}\approx 336.8518198843963$
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:  \(\Q(\sqrt{-2}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{20}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{60}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{6}a^{2}+\frac{1}{4}a$, $\frac{1}{60}a^{7}-\frac{1}{4}a^{4}+\frac{1}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{360}a^{8}+\frac{1}{180}a^{7}-\frac{1}{120}a^{6}-\frac{1}{60}a^{5}+\frac{1}{18}a^{4}-\frac{17}{36}a^{3}+\frac{11}{24}a^{2}-\frac{1}{4}a-\frac{3}{8}$, $\frac{1}{539279596080}a^{9}-\frac{295688591}{539279596080}a^{8}+\frac{343574351}{107855919216}a^{7}-\frac{159507089}{35951973072}a^{6}+\frac{952560053}{53927959608}a^{5}-\frac{13228421617}{53927959608}a^{4}-\frac{7555668377}{107855919216}a^{3}-\frac{6198585425}{35951973072}a^{2}-\frac{2589543333}{11983991024}a-\frac{972264925}{11983991024}$ 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:  $4$
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:   $\frac{17059708766}{33704974755}a^{9}+\frac{382204246019}{269639798040}a^{8}+\frac{438369829097}{14979988780}a^{7}+\frac{107078483377}{17975986536}a^{6}+\frac{5158844586349}{13481989902}a^{5}-\frac{7945656534014}{6740994951}a^{4}+\frac{1186556301839}{1497998878}a^{3}-\frac{185786947827149}{17975986536}a^{2}+\frac{25558708946575}{2995997756}a-\frac{252873945870587}{5991995512}$, $\frac{3487190467}{44939966340}a^{9}-\frac{30181053512}{33704974755}a^{8}+\frac{79536104993}{13481989902}a^{7}-\frac{82096817765}{1497998878}a^{6}+\frac{810653750851}{2995997756}a^{5}-\frac{31909185340097}{26963979804}a^{4}+\frac{132291933824477}{26963979804}a^{3}-\frac{37365876777691}{2995997756}a^{2}+\frac{60190102206265}{2995997756}a-\frac{28678668178367}{1497998878}$, $\frac{11\cdots 01}{269639798040}a^{9}+\frac{36\cdots 29}{269639798040}a^{8}+\frac{14\cdots 21}{53927959608}a^{7}+\frac{26\cdots 99}{17975986536}a^{6}+\frac{24\cdots 89}{6740994951}a^{5}-\frac{11\cdots 33}{13481989902}a^{4}+\frac{51\cdots 47}{53927959608}a^{3}-\frac{14\cdots 63}{17975986536}a^{2}+\frac{47\cdots 35}{5991995512}a-\frac{19\cdots 97}{5991995512}$, $\frac{13\cdots 57}{539279596080}a^{9}+\frac{54502888111863}{59919955120}a^{8}+\frac{11\cdots 51}{107855919216}a^{7}-\frac{11\cdots 59}{35951973072}a^{6}+\frac{41\cdots 03}{53927959608}a^{5}-\frac{97\cdots 49}{17975986536}a^{4}-\frac{35\cdots 81}{107855919216}a^{3}+\frac{63\cdots 63}{11983991024}a^{2}-\frac{16\cdots 85}{11983991024}a+\frac{26\cdots 29}{11983991024}$ 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:  \( 2918061.90553 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 2918061.90553 \cdot 1}{2\cdot\sqrt{40500000000000000000}}\cr\approx \mathstrut & 2.24510333790 \end{aligned}\] (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 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225) 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 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225, 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 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225); 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 + 50*x^8 - 150*x^7 + 725*x^6 - 4440*x^5 + 8125*x^4 - 24750*x^3 + 74250*x^2 - 130500*x + 234225); 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

$\POPlus(4,5)$ (as 10T41):

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 14400
The 25 conjugacy class representatives for $(A_5^2 : C_2):C_2$
Character table for $(A_5^2 : C_2):C_2$

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 sibling: 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 ${\href{/padicField/7.10.0.1}{10} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.1.6.11a1.1$x^{6} + 2$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.1.2.1a1.1$x^{2} + 3$$2$$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)