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Properties

Label 10.2.665...000.1
Degree $10$
Signature $[2, 4]$
Discriminant $6.655\times 10^{30}$
Root discriminant \(1208.69\)
Ramified primes $2,3,5,11,41$
Class number $125$ (GRH)
Class group [5, 5, 5] (GRH)
Galois group $F_{5}\times C_2$ (as 10T5)

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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 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552)
 
Copy content gp:K = bnfinit(y^10 - 4*y^9 + 353*y^8 - 104*y^7 + 33864*y^6 + 107536*y^5 + 577512*y^4 + 1013344*y^3 + 4995856*y^2 + 7761536*y - 15005552, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552)
 

\( x^{10} - 4 x^{9} + 353 x^{8} - 104 x^{7} + 33864 x^{6} + 107536 x^{5} + 577512 x^{4} + 1013344 x^{3} + \cdots - 15005552 \) 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:   \(6654854908590083522960688000000\) \(\medspace = 2^{10}\cdot 3^{5}\cdot 5^{6}\cdot 11^{8}\cdot 41^{8}\) 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:  \(1208.69\)
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:  $2\cdot 3^{1/2}5^{3/4}11^{4/5}41^{4/5}\approx 1538.7191534850367$
Ramified primes:   \(2\), \(3\), \(5\), \(11\), \(41\) 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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8}a^{5}+\frac{3}{8}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{64}a^{6}+\frac{1}{64}a^{4}+\frac{5}{16}a^{3}-\frac{3}{16}a^{2}+\frac{3}{8}a-\frac{7}{16}$, $\frac{1}{64}a^{7}+\frac{1}{64}a^{5}-\frac{1}{16}a^{4}+\frac{1}{16}a^{3}-\frac{1}{4}a^{2}+\frac{5}{16}a-\frac{1}{4}$, $\frac{1}{512}a^{8}-\frac{1}{256}a^{7}-\frac{1}{512}a^{6}+\frac{9}{256}a^{5}+\frac{5}{256}a^{4}-\frac{23}{64}a^{3}+\frac{3}{128}a^{2}+\frac{1}{64}a-\frac{9}{64}$, $\frac{1}{97\cdots 08}a^{9}-\frac{30\cdots 43}{48\cdots 04}a^{8}-\frac{24\cdots 37}{97\cdots 08}a^{7}+\frac{18\cdots 03}{48\cdots 04}a^{6}+\frac{44\cdots 01}{48\cdots 04}a^{5}+\frac{73\cdots 71}{15\cdots 72}a^{4}-\frac{11\cdots 85}{24\cdots 52}a^{3}-\frac{27\cdots 93}{12\cdots 76}a^{2}+\frac{56\cdots 79}{12\cdots 76}a-\frac{75\cdots 99}{30\cdots 44}$ 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:  $C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  $C_{5}\times C_{5}\times C_{10}$, which has order $250$ (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:   $\frac{469843099027}{30\cdots 44}a^{9}-\frac{2346894468497}{30\cdots 44}a^{8}+\frac{165604343823513}{30\cdots 44}a^{7}-\frac{197330984178545}{30\cdots 44}a^{6}+\frac{76\cdots 31}{15\cdots 72}a^{5}+\frac{46\cdots 99}{37\cdots 18}a^{4}+\frac{41\cdots 33}{75\cdots 36}a^{3}+\frac{42\cdots 65}{75\cdots 36}a^{2}+\frac{30\cdots 73}{37\cdots 18}a+\frac{52\cdots 41}{18\cdots 09}$, $\frac{44\cdots 89}{97\cdots 08}a^{9}+\frac{91\cdots 73}{97\cdots 08}a^{8}-\frac{10\cdots 13}{97\cdots 08}a^{7}+\frac{16\cdots 75}{97\cdots 08}a^{6}+\frac{12\cdots 47}{24\cdots 52}a^{5}+\frac{15\cdots 31}{48\cdots 04}a^{4}+\frac{12\cdots 19}{24\cdots 52}a^{3}+\frac{72\cdots 55}{24\cdots 52}a^{2}+\frac{15\cdots 99}{30\cdots 44}a-\frac{11\cdots 79}{12\cdots 76}$, $\frac{16\cdots 37}{97\cdots 08}a^{9}-\frac{21\cdots 31}{97\cdots 08}a^{8}-\frac{33\cdots 77}{97\cdots 08}a^{7}-\frac{64\cdots 49}{97\cdots 08}a^{6}-\frac{60\cdots 19}{24\cdots 52}a^{5}-\frac{50\cdots 97}{48\cdots 04}a^{4}-\frac{56\cdots 97}{24\cdots 52}a^{3}-\frac{22\cdots 45}{24\cdots 52}a^{2}-\frac{80\cdots 77}{75\cdots 36}a+\frac{29\cdots 77}{12\cdots 76}$, $\frac{30\cdots 43}{48\cdots 04}a^{9}-\frac{91\cdots 17}{48\cdots 04}a^{8}+\frac{10\cdots 99}{48\cdots 04}a^{7}+\frac{77\cdots 57}{48\cdots 04}a^{6}+\frac{13\cdots 11}{60\cdots 88}a^{5}+\frac{21\cdots 45}{24\cdots 52}a^{4}+\frac{55\cdots 55}{12\cdots 76}a^{3}+\frac{13\cdots 89}{12\cdots 76}a^{2}+\frac{12\cdots 53}{30\cdots 44}a+\frac{56\cdots 83}{60\cdots 88}$, $\frac{16\cdots 23}{24\cdots 52}a^{9}-\frac{40\cdots 29}{15\cdots 72}a^{8}+\frac{56\cdots 05}{24\cdots 52}a^{7}-\frac{12\cdots 29}{30\cdots 44}a^{6}+\frac{24\cdots 43}{12\cdots 76}a^{5}+\frac{42\cdots 19}{60\cdots 88}a^{4}+\frac{90\cdots 69}{60\cdots 88}a^{3}-\frac{25\cdots 79}{37\cdots 18}a^{2}-\frac{45\cdots 07}{30\cdots 44}a+\frac{30\cdots 51}{15\cdots 72}$ 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:  \( 6327756421.905004 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 6327756421.905004 \cdot 125}{2\cdot\sqrt{6654854908590083522960688000000}}\cr\approx \mathstrut & 0.955740389675778 \end{aligned}\] (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 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552) 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 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552, 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 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552); 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 - 4*x^9 + 353*x^8 - 104*x^7 + 33864*x^6 + 107536*x^5 + 577512*x^4 + 1013344*x^3 + 4995856*x^2 + 7761536*x - 15005552); 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

\(\Q(\sqrt{3}) \), 5.1.5171495850125.3

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 R R R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ R ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.2.0.1}{2} }^{5}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{5}$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ R ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.2.2a1.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$$[2]$$
2.4.2.8a1.1$x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$$2$$4$$8$$C_4\times C_2$$$[2]^{4}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$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$$$[\ ]_{4}^{2}$$
\(11\) Copy content Toggle raw display 11.1.5.4a1.3$x^{5} + 44$$5$$1$$4$$C_5$$$[\ ]_{5}$$
11.1.5.4a1.3$x^{5} + 44$$5$$1$$4$$C_5$$$[\ ]_{5}$$
\(41\) 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}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

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