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Properties

Label 16.0.149...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.496\times 10^{24}$
Root discriminant \(32.43\)
Ramified primes $2,3,5$
Class number $12$ (GRH)
Class group [2, 6] (GRH)
Galois group $C_2^2:Q_8$ (as 16T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401)
 
gp: K = bnfinit(y^16 + 4*y^12 - 48*y^10 + 246*y^8 + 768*y^6 + 1924*y^4 + 2352*y^2 + 2401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401)
 

\( x^{16} + 4x^{12} - 48x^{10} + 246x^{8} + 768x^{6} + 1924x^{4} + 2352x^{2} + 2401 \) 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:   \(1495873430980877352960000\) \(\medspace = 2^{52}\cdot 3^{12}\cdot 5^{4}\) 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:  \(32.43\)
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:  $2^{13/4}3^{3/4}5^{1/2}\approx 48.49237211223896$
Ramified primes:   \(2\), \(3\), \(5\) 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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{8}a^{2}+\frac{3}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}+\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}+\frac{1}{8}$, $\frac{1}{112}a^{13}-\frac{1}{16}a^{12}-\frac{3}{112}a^{9}-\frac{1}{16}a^{8}+\frac{1}{14}a^{7}-\frac{13}{112}a^{5}-\frac{3}{16}a^{4}+\frac{5}{14}a^{3}-\frac{1}{112}a+\frac{5}{16}$, $\frac{1}{1408641808}a^{14}-\frac{137835}{28747792}a^{12}+\frac{24479669}{1408641808}a^{10}+\frac{93992585}{1408641808}a^{8}-\frac{116847653}{1408641808}a^{6}+\frac{221156535}{1408641808}a^{4}-\frac{81305393}{1408641808}a^{2}+\frac{5836635}{28747792}$, $\frac{1}{9860492656}a^{15}-\frac{137835}{201234544}a^{13}-\frac{503761009}{9860492656}a^{11}+\frac{974393715}{9860492656}a^{9}+\frac{587473251}{9860492656}a^{7}-\frac{1}{4}a^{6}-\frac{483164369}{9860492656}a^{5}-\frac{1}{4}a^{4}+\frac{1151256189}{9860492656}a^{3}+\frac{1}{4}a^{2}+\frac{2243161}{201234544}a+\frac{1}{4}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:   $\frac{921161}{704320904}a^{14}-\frac{9593}{14373896}a^{12}+\frac{2233019}{704320904}a^{10}-\frac{41217761}{704320904}a^{8}+\frac{228305955}{704320904}a^{6}+\frac{707621237}{704320904}a^{4}+\frac{895248505}{704320904}a^{2}+\frac{28945181}{14373896}$, $\frac{1227011}{704320904}a^{14}-\frac{514}{1796737}a^{12}+\frac{1706923}{704320904}a^{10}-\frac{7394161}{88040113}a^{8}+\frac{320625769}{704320904}a^{6}+\frac{125924263}{88040113}a^{4}+\frac{1275037209}{704320904}a^{2}+\frac{841237}{1796737}$, $\frac{11526495}{9860492656}a^{15}-\frac{2184971}{704320904}a^{14}-\frac{526497}{201234544}a^{13}+\frac{15319}{14373896}a^{12}+\frac{93930127}{9860492656}a^{11}-\frac{4843257}{704320904}a^{10}-\frac{722318085}{9860492656}a^{9}+\frac{99155065}{704320904}a^{8}+\frac{4325421325}{9860492656}a^{7}-\frac{551267897}{704320904}a^{6}+\frac{130073613}{9860492656}a^{5}-\frac{1714341199}{704320904}a^{4}+\frac{16218911589}{9860492656}a^{3}-\frac{2168328899}{704320904}a^{2}-\frac{120219591}{201234544}a-\frac{70203465}{14373896}$, $\frac{172805}{2465123164}a^{15}+\frac{921161}{704320904}a^{14}-\frac{266269}{100617272}a^{13}-\frac{9593}{14373896}a^{12}+\frac{27664129}{4930246328}a^{11}+\frac{2233019}{704320904}a^{10}-\frac{10939083}{616280791}a^{9}-\frac{41217761}{704320904}a^{8}+\frac{101340056}{616280791}a^{7}+\frac{228305955}{704320904}a^{6}-\frac{4188807397}{4930246328}a^{5}+\frac{707621237}{704320904}a^{4}-\frac{2571091571}{4930246328}a^{3}+\frac{895248505}{704320904}a^{2}-\frac{77660203}{50308636}a+\frac{43319077}{14373896}$, $\frac{4273219}{9860492656}a^{15}+\frac{442181}{352160452}a^{14}-\frac{639}{28747792}a^{13}-\frac{7979}{14373896}a^{12}+\frac{13460849}{9860492656}a^{11}+\frac{166213}{88040113}a^{10}-\frac{244960479}{9860492656}a^{9}-\frac{42433745}{704320904}a^{8}+\frac{1072690877}{9860492656}a^{7}+\frac{112984891}{352160452}a^{6}+\frac{3435322977}{9860492656}a^{5}+\frac{708295379}{704320904}a^{4}+\frac{8745136975}{9860492656}a^{3}+\frac{112150665}{88040113}a^{2}+\frac{77236167}{201234544}a+\frac{23164585}{14373896}$, $\frac{10835275}{9860492656}a^{15}+\frac{1545}{861028}a^{14}+\frac{863}{28747792}a^{13}-\frac{7}{17572}a^{12}+\frac{38601869}{9860492656}a^{11}+\frac{3191}{861028}a^{10}-\frac{547292757}{9860492656}a^{9}-\frac{17707}{215257}a^{8}+\frac{2703980429}{9860492656}a^{7}+\frac{394819}{861028}a^{6}+\frac{8507688407}{9860492656}a^{5}+\frac{1230709}{861028}a^{4}+\frac{21361094731}{9860492656}a^{3}+\frac{1556333}{861028}a^{2}+\frac{190421221}{201234544}a+\frac{7647}{8786}$, $\frac{751}{14373896}a^{14}-\frac{807}{7186948}a^{12}+\frac{18435}{14373896}a^{10}+\frac{3102}{1796737}a^{8}+\frac{47677}{14373896}a^{6}-\frac{6879}{7186948}a^{4}-\frac{39935}{14373896}a^{2}+\frac{8632097}{3593474}$ 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:  \( 45687.5845647 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 45687.5845647 \cdot 12}{2\cdot\sqrt{1495873430980877352960000}}\cr\approx \mathstrut & 0.544428649640 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401)
 
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 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401, 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 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401);
 
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 + 4*x^12 - 48*x^10 + 246*x^8 + 768*x^6 + 1924*x^4 + 2352*x^2 + 2401);
 
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

$C_2^2:Q_8$ (as 16T31):

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 14 conjugacy class representatives for $C_2^2:Q_8$
Character table for $C_2^2:Q_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.0.92160.3, \(\Q(\sqrt{2}, \sqrt{3})\), 4.0.92160.6, 8.0.33973862400.12, 8.0.19110297600.3, 8.8.12230590464.1

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 16 sibling: deg 16
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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display 2.16.52.15$x^{16} + 8 x^{15} + 12 x^{14} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 30$$16$$1$$52$16T31$[2, 3, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.1$x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
\(5\) Copy content Toggle raw display 5.4.0.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$