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Properties

Label 16.0.113...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.133\times 10^{18}$
Root discriminant \(13.44\)
Ramified primes $3,5,29$
Class number $1$
Class group trivial
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1)
 
gp: K = bnfinit(y^16 - 3*y^15 + 4*y^14 - 9*y^13 + 12*y^12 + 12*y^11 + 8*y^10 - 54*y^9 - 19*y^8 + 54*y^7 + 8*y^6 - 12*y^5 + 12*y^4 + 9*y^3 + 4*y^2 + 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1)
 

\( x^{16} - 3 x^{15} + 4 x^{14} - 9 x^{13} + 12 x^{12} + 12 x^{11} + 8 x^{10} - 54 x^{9} - 19 x^{8} + \cdots + 1 \) 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:   \(1132927402587890625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 29^{4}\) 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:  \(13.44\)
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:  $3^{1/2}5^{3/4}29^{1/2}\approx 31.187971562966524$
Ramified primes:   \(3\), \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{20}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{10}a^{9}+\frac{1}{4}a^{7}-\frac{1}{20}a^{6}-\frac{1}{4}a^{5}+\frac{1}{10}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{20}$, $\frac{1}{20}a^{13}+\frac{3}{20}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{5}a^{7}+\frac{1}{4}a^{5}+\frac{1}{10}a^{4}+\frac{1}{4}a^{3}-\frac{1}{5}a-\frac{1}{4}$, $\frac{1}{2360}a^{14}-\frac{13}{1180}a^{13}+\frac{13}{2360}a^{12}+\frac{69}{1180}a^{11}+\frac{37}{2360}a^{10}+\frac{479}{2360}a^{9}+\frac{237}{1180}a^{8}-\frac{919}{2360}a^{7}+\frac{44}{295}a^{6}-\frac{583}{2360}a^{5}+\frac{553}{2360}a^{4}-\frac{231}{590}a^{3}+\frac{931}{2360}a^{2}+\frac{141}{590}a+\frac{353}{2360}$, $\frac{1}{342200}a^{15}+\frac{7}{171100}a^{14}-\frac{6573}{342200}a^{13}-\frac{15}{1711}a^{12}-\frac{54033}{342200}a^{11}-\frac{62469}{342200}a^{10}+\frac{10601}{34220}a^{9}+\frac{77631}{342200}a^{8}+\frac{71299}{171100}a^{7}-\frac{11201}{68440}a^{6}+\frac{156593}{342200}a^{5}+\frac{5983}{42775}a^{4}-\frac{483}{13688}a^{3}+\frac{51647}{171100}a^{2}+\frac{116487}{342200}a+\frac{4479}{42775}$ 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:  $2$

Class group and class number

Trivial group, which has order $1$

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:   \( \frac{18133}{342200} a^{15} + \frac{6001}{171100} a^{14} - \frac{162059}{342200} a^{13} + \frac{20307}{34220} a^{12} - \frac{501719}{342200} a^{11} + \frac{1283023}{342200} a^{10} + \frac{63899}{34220} a^{9} - \frac{1073567}{342200} a^{8} - \frac{1003129}{85550} a^{7} + \frac{259233}{68440} a^{6} + \frac{4734049}{342200} a^{5} - \frac{4104}{725} a^{4} - \frac{70737}{68440} a^{3} + \frac{159699}{42775} a^{2} - \frac{30179}{342200} a + \frac{19319}{85550} \)  (order $30$) 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:   $\frac{23169}{85550}a^{15}-\frac{136443}{171100}a^{14}+\frac{177411}{171100}a^{13}-\frac{83121}{34220}a^{12}+\frac{140424}{42775}a^{11}+\frac{269329}{85550}a^{10}+\frac{103953}{34220}a^{9}-\frac{2666397}{171100}a^{8}-\frac{1001361}{171100}a^{7}+\frac{222009}{17110}a^{6}+\frac{268721}{42775}a^{5}-\frac{561873}{171100}a^{4}+\frac{91353}{34220}a^{3}+\frac{112797}{171100}a^{2}+\frac{76833}{85550}a+\frac{30259}{42775}$, $\frac{40487}{342200}a^{15}-\frac{21961}{171100}a^{14}-\frac{144941}{342200}a^{13}+\frac{20763}{34220}a^{12}-\frac{649241}{342200}a^{11}+\frac{2332577}{342200}a^{10}-\frac{15119}{34220}a^{9}-\frac{1675713}{342200}a^{8}-\frac{699373}{42775}a^{7}+\frac{1016487}{68440}a^{6}+\frac{3807911}{342200}a^{5}-\frac{397389}{42775}a^{4}+\frac{83177}{68440}a^{3}+\frac{236347}{85550}a^{2}+\frac{266579}{342200}a-\frac{3992}{42775}$, $\frac{3754}{42775}a^{15}-\frac{171297}{342200}a^{14}+\frac{205927}{171100}a^{13}-\frac{144731}{68440}a^{12}+\frac{307621}{85550}a^{11}-\frac{960763}{342200}a^{10}-\frac{68679}{68440}a^{9}-\frac{708719}{171100}a^{8}+\frac{4256821}{342200}a^{7}+\frac{70947}{34220}a^{6}-\frac{5906839}{342200}a^{5}+\frac{1557103}{342200}a^{4}+\frac{117967}{17110}a^{3}-\frac{1344537}{342200}a^{2}+\frac{23787}{171100}a+\frac{308269}{342200}$, $\frac{79941}{342200}a^{15}-\frac{122493}{171100}a^{14}+\frac{337237}{342200}a^{13}-\frac{75309}{34220}a^{12}+\frac{1035517}{342200}a^{11}+\frac{849011}{342200}a^{10}+\frac{67929}{34220}a^{9}-\frac{4456319}{342200}a^{8}-\frac{310953}{85550}a^{7}+\frac{824757}{68440}a^{6}+\frac{899693}{342200}a^{5}-\frac{200229}{85550}a^{4}+\frac{191119}{68440}a^{3}+\frac{33443}{42775}a^{2}+\frac{311097}{342200}a+\frac{28409}{42775}$, $\frac{39286}{42775}a^{15}-\frac{260237}{85550}a^{14}+\frac{767473}{171100}a^{13}-\frac{316569}{34220}a^{12}+\frac{2279703}{171100}a^{11}+\frac{349361}{42775}a^{10}+\frac{52233}{17110}a^{9}-\frac{8884571}{171100}a^{8}-\frac{435673}{171100}a^{7}+\frac{394027}{6844}a^{6}-\frac{771419}{85550}a^{5}-\frac{636966}{42775}a^{4}+\frac{539427}{34220}a^{3}+\frac{861821}{171100}a^{2}+\frac{159513}{171100}a+\frac{138169}{85550}$, $\frac{111367}{171100}a^{15}-\frac{899529}{342200}a^{14}+\frac{877639}{171100}a^{13}-\frac{704893}{68440}a^{12}+\frac{1435247}{85550}a^{11}-\frac{2161441}{342200}a^{10}+\frac{423529}{68440}a^{9}-\frac{6744283}{171100}a^{8}+\frac{9599447}{342200}a^{7}+\frac{671707}{34220}a^{6}-\frac{9111173}{342200}a^{5}+\frac{3282271}{342200}a^{4}+\frac{137751}{17110}a^{3}-\frac{719459}{342200}a^{2}+\frac{162909}{171100}a+\frac{435673}{342200}$, $\frac{74543}{342200}a^{15}-\frac{35611}{42775}a^{14}+\frac{516881}{342200}a^{13}-\frac{25857}{8555}a^{12}+\frac{1625111}{342200}a^{11}-\frac{205707}{342200}a^{10}+\frac{20027}{17110}a^{9}-\frac{4259227}{342200}a^{8}+\frac{1004897}{171100}a^{7}+\frac{628803}{68440}a^{6}-\frac{2584281}{342200}a^{5}+\frac{406671}{171100}a^{4}+\frac{253443}{68440}a^{3}-\frac{525999}{171100}a^{2}+\frac{679711}{342200}a-\frac{44351}{85550}$ Copy content Toggle raw display
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:  \( 3158.73230855 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3158.73230855 \cdot 1}{30\cdot\sqrt{1132927402587890625}}\cr\approx \mathstrut & 0.240286556676 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1)
 
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 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1, 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 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1);
 
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 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1);
 
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 16T21):

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 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), 4.4.32625.1, \(\Q(\zeta_{15})^+\), 4.0.3625.1, 4.0.6525.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.725.1, 8.0.42575625.1, 8.0.1064390625.3, 8.0.1064390625.1, \(\Q(\zeta_{15})\), 8.0.1064390625.2, 8.0.13140625.1, 8.8.1064390625.1

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 siblings: deg 16, deg 16, 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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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

# 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
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(29\) Copy content Toggle raw display 29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$