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Properties

Label 10.6.1296000000000000.1
Degree $10$
Signature $[6, 2]$
Discriminant $1.296\times 10^{15}$
Root discriminant \(32.45\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_5^2 : C_4$ (as 10T42)

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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 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64)
 
Copy content gp:K = bnfinit(y^10 - 10*y^8 - 5*y^6 - 64*y^5 - 100*y^3 - 105*y^2 + 120*y + 64, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 10*x^8 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 10*x^8 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64)
 

\( x^{10} - 10x^{8} - 5x^{6} - 64x^{5} - 100x^{3} - 105x^{2} + 120x + 64 \) 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:  $[6, 2]$
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:   \(1296000000000000\) \(\medspace = 2^{16}\cdot 3^{4}\cdot 5^{12}\) 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:  \(32.45\)
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:  $2^{19/8}3^{1/2}5^{271/200}\approx 79.54469536351158$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{225370104}a^{9}-\frac{13217074}{28171263}a^{8}+\frac{9831945}{37561684}a^{7}-\frac{3452757}{9390421}a^{6}-\frac{34614389}{225370104}a^{5}-\frac{1338601}{9390421}a^{4}-\frac{3503960}{9390421}a^{3}+\frac{4987187}{56342526}a^{2}+\frac{96543191}{225370104}a+\frac{12963664}{28171263}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $7$
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:   $\frac{6095}{831624}a^{9}-\frac{539}{103953}a^{8}-\frac{11833}{138604}a^{7}+\frac{3264}{34651}a^{6}-\frac{8395}{831624}a^{5}-\frac{21890}{34651}a^{4}+\frac{22536}{34651}a^{3}-\frac{199871}{207906}a^{2}+\frac{218713}{831624}a+\frac{208169}{103953}$, $\frac{1844125}{75123368}a^{9}-\frac{205493}{9390421}a^{8}-\frac{6923125}{37561684}a^{7}+\frac{938220}{9390421}a^{6}-\frac{32844609}{75123368}a^{5}-\frac{9261198}{9390421}a^{4}-\frac{1819019}{9390421}a^{3}-\frac{57224393}{18780842}a^{2}-\frac{87531677}{75123368}a+\frac{4180729}{9390421}$, $\frac{2249125}{225370104}a^{9}+\frac{411347}{28171263}a^{8}-\frac{3744671}{37561684}a^{7}-\frac{1119466}{9390421}a^{6}-\frac{13563761}{225370104}a^{5}-\frac{8706894}{9390421}a^{4}-\frac{8334118}{9390421}a^{3}-\frac{88484809}{56342526}a^{2}-\frac{981863629}{225370104}a-\frac{18355844}{28171263}$, $\frac{3609559}{225370104}a^{9}+\frac{110030}{28171263}a^{8}-\frac{7152941}{37561684}a^{7}-\frac{914647}{9390421}a^{6}+\frac{27242005}{225370104}a^{5}-\frac{2675198}{9390421}a^{4}+\frac{10492419}{9390421}a^{3}-\frac{4021519}{56342526}a^{2}-\frac{179347231}{225370104}a-\frac{5717873}{28171263}$, $\frac{701057}{225370104}a^{9}+\frac{551164}{28171263}a^{8}-\frac{2956555}{37561684}a^{7}-\frac{1252558}{9390421}a^{6}+\frac{66239027}{225370104}a^{5}-\frac{3878622}{9390421}a^{4}-\frac{7285067}{9390421}a^{3}+\frac{23248255}{56342526}a^{2}+\frac{31700023}{225370104}a-\frac{7421056}{28171263}$, $\frac{2228885}{225370104}a^{9}-\frac{324341}{28171263}a^{8}-\frac{509639}{37561684}a^{7}-\frac{830868}{9390421}a^{6}-\frac{93983737}{225370104}a^{5}+\frac{2603182}{9390421}a^{4}-\frac{14033531}{9390421}a^{3}-\frac{63343097}{56342526}a^{2}-\frac{381137477}{225370104}a-\frac{15771622}{28171263}$, $\frac{31350653}{225370104}a^{9}-\frac{5025650}{28171263}a^{8}-\frac{43997035}{37561684}a^{7}+\frac{13392979}{9390421}a^{6}-\frac{504653953}{225370104}a^{5}-\frac{54979875}{9390421}a^{4}+\frac{67767791}{9390421}a^{3}-\frac{1137925403}{56342526}a^{2}+\frac{2234167867}{225370104}a+\frac{267468917}{28171263}$ 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:  \( 15975.978276 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 15975.978276 \cdot 1}{2\cdot\sqrt{1296000000000000}}\cr\approx \mathstrut & 0.56062785957 \end{aligned}\] (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 - 10*x^8 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64) 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 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64, 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 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64); 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 - 10*x^8 - 5*x^6 - 64*x^5 - 100*x^3 - 105*x^2 + 120*x + 64); 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

$A_5^2:C_4$ (as 10T42):

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 22 conjugacy class representatives for $A_5^2 : C_4$
Character table for $A_5^2 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \)

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 ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.4.16b5.8$x^{8} + 8 x^{7} + 24 x^{6} + 48 x^{5} + 63 x^{4} + 64 x^{3} + 46 x^{2} + 24 x + 13$$4$$2$$16$$(((C_4 \times C_2): C_2):C_2):C_2$$$[2, 2, 2, 3]^{4}$$
\(3\) Copy content Toggle raw display 3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.4.2.4a1.1$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 3 x + 4$$2$$4$$4$$C_8$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.1.10.12a1.1$x^{10} + 5 x^{3} + 5$$10$$1$$12$$(C_5^2 : C_8):C_2$$$[\frac{11}{8}, \frac{11}{8}]_{8}^{2}$$

Spectrum of ring of integers

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