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Properties

Label 16.0.924...625.2
Degree $16$
Signature $[0, 8]$
Discriminant $9.248\times 10^{25}$
Root discriminant \(41.96\)
Ramified primes $5,109$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^3:C_4$ (as 16T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441)
 
gp: K = bnfinit(y^16 + 23*y^14 + 276*y^12 + 2818*y^10 + 20250*y^8 + 86927*y^6 + 243976*y^4 + 501552*y^2 + 531441, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441)
 

\( x^{16} + 23x^{14} + 276x^{12} + 2818x^{10} + 20250x^{8} + 86927x^{6} + 243976x^{4} + 501552x^{2} + 531441 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(92475143538754579453515625\) \(\medspace = 5^{8}\cdot 109^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(41.96\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}109^{3/4}\approx 75.43187023636793$
Ramified primes:   \(5\), \(109\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{30}a^{8}-\frac{1}{6}a^{6}+\frac{7}{30}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}+\frac{1}{5}$, $\frac{1}{30}a^{9}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{11}{30}a-\frac{1}{2}$, $\frac{1}{60}a^{10}-\frac{1}{20}a^{6}-\frac{1}{2}a^{3}-\frac{19}{60}a^{2}+\frac{1}{4}$, $\frac{1}{120}a^{11}-\frac{1}{120}a^{10}-\frac{1}{60}a^{9}-\frac{1}{60}a^{8}+\frac{7}{120}a^{7}+\frac{13}{120}a^{6}-\frac{7}{60}a^{5}+\frac{23}{60}a^{4}-\frac{13}{40}a^{3}+\frac{59}{120}a^{2}+\frac{1}{40}a+\frac{11}{40}$, $\frac{1}{4800}a^{12}-\frac{11}{4800}a^{10}+\frac{61}{4800}a^{8}+\frac{823}{4800}a^{6}-\frac{687}{1600}a^{4}+\frac{309}{800}a^{2}+\frac{423}{1600}$, $\frac{1}{86400}a^{13}-\frac{1}{9600}a^{12}+\frac{149}{86400}a^{11}+\frac{11}{9600}a^{10}-\frac{193}{28800}a^{9}-\frac{61}{9600}a^{8}+\frac{343}{86400}a^{7}-\frac{823}{9600}a^{6}+\frac{1051}{9600}a^{5}-\frac{913}{3200}a^{4}+\frac{9007}{43200}a^{3}-\frac{309}{1600}a^{2}+\frac{39829}{86400}a+\frac{1177}{3200}$, $\frac{1}{19889452800}a^{14}-\frac{128939}{2486181600}a^{12}+\frac{8681441}{1657454400}a^{10}-\frac{62169169}{9944726400}a^{8}+\frac{4153997}{30693600}a^{6}-\frac{1165847237}{3977890560}a^{4}-\frac{1}{2}a^{3}+\frac{480470147}{3977890560}a^{2}-\frac{1}{2}a-\frac{8588761}{27283200}$, $\frac{1}{1074030451200}a^{15}-\frac{1}{39778905600}a^{14}-\frac{128939}{134253806400}a^{13}+\frac{128939}{4972363200}a^{12}+\frac{174426881}{89502537600}a^{11}-\frac{8681441}{3314908800}a^{10}+\frac{2258266991}{537015225600}a^{9}-\frac{269321711}{19889452800}a^{8}+\frac{35870717}{1657454400}a^{7}-\frac{14385197}{61387200}a^{6}+\frac{19253990971}{214806090240}a^{5}+\frac{2226618053}{7955781120}a^{4}-\frac{91276205437}{214806090240}a^{3}+\frac{182511613}{7955781120}a^{2}-\frac{11317081}{1473292800}a+\frac{3132121}{54566400}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{20219}{685843200}a^{14}+\frac{43823}{85730400}a^{12}+\frac{300283}{57153600}a^{10}+\frac{3683633}{68584320}a^{8}+\frac{12487}{42336}a^{6}+\frac{614884093}{685843200}a^{4}+\frac{1455752693}{685843200}a^{2}+\frac{1070653}{940800}$, $\frac{85015957}{1074030451200}a^{15}+\frac{530039}{39778905600}a^{14}+\frac{364677197}{268507612800}a^{13}+\frac{3831217}{9944726400}a^{12}+\frac{626889313}{44751268800}a^{11}+\frac{3725269}{828727200}a^{10}+\frac{76789270073}{537015225600}a^{9}+\frac{866894167}{19889452800}a^{8}+\frac{2596437007}{3314908800}a^{7}+\frac{41007923}{122774400}a^{6}+\frac{2641089662903}{1074030451200}a^{5}+\frac{47740934629}{39778905600}a^{4}+\frac{6440002097443}{1074030451200}a^{3}+\frac{117395592929}{39778905600}a^{2}+\frac{2255901859}{294658560}a+\frac{238213597}{54566400}$, $\frac{85015957}{1074030451200}a^{15}-\frac{530039}{39778905600}a^{14}+\frac{364677197}{268507612800}a^{13}-\frac{3831217}{9944726400}a^{12}+\frac{626889313}{44751268800}a^{11}-\frac{3725269}{828727200}a^{10}+\frac{76789270073}{537015225600}a^{9}-\frac{866894167}{19889452800}a^{8}+\frac{2596437007}{3314908800}a^{7}-\frac{41007923}{122774400}a^{6}+\frac{2641089662903}{1074030451200}a^{5}-\frac{47740934629}{39778905600}a^{4}+\frac{6440002097443}{1074030451200}a^{3}-\frac{117395592929}{39778905600}a^{2}+\frac{2255901859}{294658560}a-\frac{238213597}{54566400}$, $\frac{545563}{76716460800}a^{15}+\frac{4765391}{19179115200}a^{13}+\frac{10417489}{3196519200}a^{11}+\frac{1189713263}{38358230400}a^{9}+\frac{59702917}{236779200}a^{7}+\frac{3189452753}{3068658432}a^{5}+\frac{20912950889}{15343292160}a^{3}-\frac{20259823}{105235200}a$, $\frac{37081}{44396100}a^{14}+\frac{2961659}{177584400}a^{12}+\frac{10672217}{59194800}a^{10}+\frac{64414979}{35516880}a^{8}+\frac{5041109}{438480}a^{6}+\frac{6839908373}{177584400}a^{4}+\frac{8019939869}{88792200}a^{2}+\frac{37118663}{243600}$, $\frac{120742519}{358010150400}a^{15}-\frac{8383631}{7955781120}a^{14}+\frac{16108723}{2796954300}a^{13}-\frac{98672671}{4972363200}a^{12}+\frac{1680421853}{29834179200}a^{11}-\frac{693659051}{3314908800}a^{10}+\frac{20451041113}{35801015040}a^{9}-\frac{41884018469}{19889452800}a^{8}+\frac{166937291}{55248480}a^{7}-\frac{771149903}{61387200}a^{6}+\frac{2595705844313}{358010150400}a^{5}-\frac{1564029821357}{39778905600}a^{4}+\frac{6690995360233}{358010150400}a^{3}-\frac{3388645388197}{39778905600}a^{2}+\frac{16413176873}{491097600}a-\frac{5629479533}{54566400}$, $\frac{402054851}{1074030451200}a^{15}-\frac{5807519}{13259635200}a^{14}+\frac{98819423}{13425380640}a^{13}-\frac{13918307}{1657454400}a^{12}+\frac{1443676529}{17900507520}a^{11}-\frac{95379007}{1104969600}a^{10}+\frac{441138618937}{537015225600}a^{9}-\frac{5678317201}{6629817600}a^{8}+\frac{540344869}{103590900}a^{7}-\frac{104823667}{20462400}a^{6}+\frac{20260158801373}{1074030451200}a^{5}-\frac{184739264761}{13259635200}a^{4}+\frac{53970924449213}{1074030451200}a^{3}-\frac{366020473361}{13259635200}a^{2}+\frac{124466313421}{1473292800}a-\frac{233472293}{3637760}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 6889381.56239 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6889381.56239 \cdot 1}{2\cdot\sqrt{92475143538754579453515625}}\cr\approx \mathstrut & 0.870115054148 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 23*x^14 + 276*x^12 + 2818*x^10 + 20250*x^8 + 86927*x^6 + 243976*x^4 + 501552*x^2 + 531441);
 
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

$C_2^3:C_4$ (as 16T33):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3:C_4$
Character table for $C_2^3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{545}) \), 4.0.2725.1 x2, 4.0.59405.1 x2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.4.9616399718125.1 x2, 8.0.88223850625.1, 8.0.88223850625.2 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.4.1048187569275625.1, 8.0.88223850625.2, 8.0.41927502771025.1, 8.4.9616399718125.1
Degree 16 siblings: 16.8.1098697180383943158487219140625.1, 16.0.1098697180383943158487219140625.5, 16.0.43947887215357726339488765625.3
Minimal sibling: 8.0.88223850625.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.2.0.1}{2} }^{8}$ R ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(109\) Copy content Toggle raw display 109.4.3.1$x^{4} + 436$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.2.1$x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.3.1$x^{4} + 436$$4$$1$$3$$C_4$$[\ ]_{4}$