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Properties

Label 16.0.891...000.15
Degree $16$
Signature $[0, 8]$
Discriminant $8.916\times 10^{20}$
Root discriminant \(20.39\)
Ramified primes $2,3,5$
Class number $4$
Class group [4]
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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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^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625)
 
Copy content gp:K = bnfinit(y^16 - 12*y^14 + 115*y^12 - 624*y^10 + 2109*y^8 - 4320*y^6 + 5275*y^4 - 3000*y^2 + 625, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625)
 

\( x^{16} - 12x^{14} + 115x^{12} - 624x^{10} + 2109x^{8} - 4320x^{6} + 5275x^{4} - 3000x^{2} + 625 \) 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:  $16$
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:  $[0, 8]$
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:   \(891610044825600000000\) \(\medspace = 2^{32}\cdot 3^{12}\cdot 5^{8}\) 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:  \(20.39\)
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:  $2^{2}3^{3/4}5^{1/2}\approx 20.38853093816547$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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:  8.0.3317760000.2

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{8}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{9}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}$, $\frac{1}{25}a^{12}-\frac{2}{25}a^{10}-\frac{1}{5}a^{8}+\frac{1}{25}a^{6}-\frac{6}{25}a^{4}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{13}-\frac{2}{25}a^{11}-\frac{1}{5}a^{9}+\frac{1}{25}a^{7}-\frac{6}{25}a^{5}-\frac{1}{5}a^{3}$, $\frac{1}{2550065375}a^{14}+\frac{8181773}{2550065375}a^{12}+\frac{3868598}{39231775}a^{10}-\frac{88740948}{196158875}a^{8}-\frac{42367021}{231824125}a^{6}-\frac{12655066}{510013075}a^{4}+\frac{38654581}{102002615}a^{2}-\frac{4582596}{20400523}$, $\frac{1}{2550065375}a^{15}+\frac{8181773}{2550065375}a^{13}+\frac{3868598}{39231775}a^{11}-\frac{88740948}{196158875}a^{9}-\frac{42367021}{231824125}a^{7}-\frac{12655066}{510013075}a^{5}+\frac{38654581}{102002615}a^{3}-\frac{4582596}{20400523}a$ 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:  $C_{4}$, which has order $4$
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:  $C_{4}$, which has order $4$
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:   \( -\frac{9328001}{2550065375} a^{14} + \frac{95851222}{2550065375} a^{12} - \frac{14298754}{39231775} a^{10} + \frac{342650098}{196158875} a^{8} - \frac{1283169559}{231824125} a^{6} + \frac{5206926742}{510013075} a^{4} - \frac{1150864834}{102002615} a^{2} + \frac{60998362}{20400523} \)  (order $12$) 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:   $\frac{15835669}{2550065375}a^{14}-\frac{178519708}{2550065375}a^{12}+\frac{25677643}{39231775}a^{10}-\frac{652121687}{196158875}a^{8}+\frac{2314658306}{231824125}a^{6}-\frac{346790417}{20400523}a^{4}+\frac{312037210}{20400523}a^{2}-\frac{100210538}{20400523}$, $\frac{17304847}{2550065375}a^{14}-\frac{195791789}{2550065375}a^{12}+\frac{28425092}{39231775}a^{10}-\frac{728220956}{196158875}a^{8}+\frac{2675231018}{231824125}a^{6}-\frac{10523714118}{510013075}a^{4}+\frac{2105090264}{102002615}a^{2}-\frac{128035120}{20400523}$, $\frac{3988318}{2550065375}a^{14}-\frac{42034221}{2550065375}a^{12}+\frac{5732297}{39231775}a^{10}-\frac{133684789}{196158875}a^{8}+\frac{355081512}{231824125}a^{6}-\frac{839999206}{510013075}a^{4}-\frac{31927288}{102002615}a^{2}+\frac{19243218}{20400523}$, $\frac{828667}{231824125}a^{15}-\frac{5627597}{2550065375}a^{14}-\frac{9437984}{231824125}a^{13}+\frac{55216289}{2550065375}a^{12}+\frac{1369566}{3566525}a^{11}-\frac{7474817}{39231775}a^{10}-\frac{35520966}{17832625}a^{9}+\frac{159103706}{196158875}a^{8}+\frac{1452889098}{231824125}a^{7}-\frac{351578518}{231824125}a^{6}-\frac{558744562}{46364825}a^{5}+\frac{242485368}{510013075}a^{4}+\frac{130018388}{9272965}a^{3}+\frac{229124736}{102002615}a^{2}-\frac{12980813}{1854593}a-\frac{52441839}{20400523}$, $\frac{2349249}{2550065375}a^{15}-\frac{7797446}{2550065375}a^{14}-\frac{44724278}{2550065375}a^{13}+\frac{92386917}{2550065375}a^{12}+\frac{6651521}{39231775}a^{11}-\frac{13437598}{39231775}a^{10}-\frac{226467152}{196158875}a^{9}+\frac{356867558}{196158875}a^{8}+\frac{1040482941}{231824125}a^{7}-\frac{1357136959}{231824125}a^{6}-\frac{5074302008}{510013075}a^{5}+\frac{5576156876}{510013075}a^{4}+\frac{1183350166}{102002615}a^{3}-\frac{1214981081}{102002615}a^{2}-\frac{99078074}{20400523}a+\frac{90391258}{20400523}$, $\frac{1138491}{2550065375}a^{15}+\frac{936756}{196158875}a^{14}-\frac{3877257}{2550065375}a^{13}-\frac{10308322}{196158875}a^{12}+\frac{938888}{39231775}a^{11}+\frac{19092163}{39231775}a^{10}-\frac{5159768}{196158875}a^{9}-\frac{471175119}{196158875}a^{8}+\frac{60827639}{231824125}a^{7}+\frac{121340264}{17832625}a^{6}-\frac{829402806}{510013075}a^{5}-\frac{401347934}{39231775}a^{4}+\frac{475976838}{102002615}a^{3}+\frac{49458516}{7846355}a^{2}-\frac{96152708}{20400523}a+\frac{1045886}{1569271}$, $\frac{3808918}{2550065375}a^{15}+\frac{2500326}{2550065375}a^{14}-\frac{34480571}{2550065375}a^{13}-\frac{34693327}{2550065375}a^{12}+\frac{5043557}{39231775}a^{11}+\frac{5109598}{39231775}a^{10}-\frac{104981489}{196158875}a^{9}-\frac{150502073}{196158875}a^{8}+\frac{320157012}{231824125}a^{7}+\frac{623405079}{231824125}a^{6}-\frac{1099368706}{510013075}a^{5}-\frac{2721433431}{510013075}a^{4}+\frac{228828363}{102002615}a^{3}+\frac{596955802}{102002615}a^{2}-\frac{24511805}{20400523}a-\frac{30183346}{20400523}$ Copy content Toggle raw display
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:  \( 14466.6129733 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 14466.6129733 \cdot 4}{12\cdot\sqrt{891610044825600000000}}\cr\approx \mathstrut & 0.392280823887 \end{aligned}\]

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^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625) 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^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625, 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^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625); 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^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625); 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

$D_4:C_2^2$ (as 16T18):

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 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), 8.0.3317760000.2, 8.0.4665600.1, 8.0.1194393600.9

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: deg 32
Degree 16 siblings: 16.0.1426576071720960000.2, 16.8.891610044825600000000.1, 16.0.891610044825600000000.4, 16.0.55725627801600000000.2, 16.0.55725627801600000000.5
Minimal sibling: 16.0.1426576071720960000.2

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.4.4.32b1.1$x^{16} + 4 x^{13} + 4 x^{12} + 6 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} + 16 x^{5} + 13 x^{4} + 4 x^{3} + 8 x^{2} + 12 x + 9$$4$$4$$32$$C_4\times C_2^2$$$[2, 3]^{4}$$
\(3\) Copy content Toggle raw display 3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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