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Properties

Label 16.4.232292068597265625.1
Degree $16$
Signature $[4, 6]$
Discriminant $2.323\times 10^{17}$
Root discriminant \(12.17\)
Ramified primes $3,5,13$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T44)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*x - 1)
 
gp: K = bnfinit(y^16 - y^15 + 5*y^14 - y^13 + y^11 - 19*y^10 - 13*y^9 - 20*y^8 - 11*y^7 + 17*y^6 + 35*y^5 + 46*y^4 + 36*y^3 + 22*y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*x - 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*x - 1)
 

\( x^{16} - x^{15} + 5 x^{14} - x^{13} + x^{11} - 19 x^{10} - 13 x^{9} - 20 x^{8} - 11 x^{7} + 17 x^{6} + \cdots - 1 \) 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:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(232292068597265625\) \(\medspace = 3^{6}\cdot 5^{8}\cdot 13^{8}\) 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:  \(12.17\)
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:  $3^{1/2}5^{1/2}13^{1/2}\approx 13.96424004376894$
Ramified primes:   \(3\), \(5\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{334930591}a^{15}+\frac{138555887}{334930591}a^{14}+\frac{143733157}{334930591}a^{13}+\frac{108488160}{334930591}a^{12}+\frac{144776765}{334930591}a^{11}-\frac{102678000}{334930591}a^{10}-\frac{66609302}{334930591}a^{9}-\frac{83071435}{334930591}a^{8}+\frac{15454558}{334930591}a^{7}-\frac{113367582}{334930591}a^{6}+\frac{11827202}{334930591}a^{5}-\frac{159074520}{334930591}a^{4}+\frac{43836859}{334930591}a^{3}-\frac{144952960}{334930591}a^{2}+\frac{3839262}{334930591}a-\frac{150689317}{334930591}$ 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

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:  $9$
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:   $\frac{10720645}{334930591}a^{15}+\frac{14297481}{334930591}a^{14}-\frac{4712116}{334930591}a^{13}+\frac{161216741}{334930591}a^{12}-\frac{210217311}{334930591}a^{11}+\frac{119805825}{334930591}a^{10}-\frac{220078193}{334930591}a^{9}-\frac{597459530}{334930591}a^{8}+\frac{35055212}{334930591}a^{7}-\frac{247789778}{334930591}a^{6}+\frac{225219829}{334930591}a^{5}+\frac{1044328259}{334930591}a^{4}+\frac{539700632}{334930591}a^{3}+\frac{294704776}{334930591}a^{2}+\frac{79566591}{334930591}a-\frac{211541388}{334930591}$, $\frac{19202939}{334930591}a^{15}+\frac{4512668}{334930591}a^{14}+\frac{17460304}{334930591}a^{13}+\frac{132471461}{334930591}a^{12}-\frac{213688269}{334930591}a^{11}-\frac{62885141}{334930591}a^{10}-\frac{129929625}{334930591}a^{9}-\frac{576729072}{334930591}a^{8}-\frac{19013181}{334930591}a^{7}+\frac{471714850}{334930591}a^{6}+\frac{604789578}{334930591}a^{5}+\frac{983489165}{334930591}a^{4}+\frac{403091706}{334930591}a^{3}-\frac{449179736}{334930591}a^{2}-\frac{477561084}{334930591}a-\frac{375589560}{334930591}$, $\frac{55836336}{334930591}a^{15}-\frac{142687484}{334930591}a^{14}+\frac{381641954}{334930591}a^{13}-\frac{444422111}{334930591}a^{12}+\frac{69429471}{334930591}a^{11}+\frac{365855588}{334930591}a^{10}-\frac{1375551068}{334930591}a^{9}+\frac{968091736}{334930591}a^{8}-\frac{334142415}{334930591}a^{7}+\frac{17532087}{334930591}a^{6}+\frac{1294300671}{334930591}a^{5}-\frac{128628780}{334930591}a^{4}+\frac{82751074}{334930591}a^{3}-\frac{236401548}{334930591}a^{2}+\frac{342768619}{334930591}a-\frac{254720881}{334930591}$, $\frac{15341678}{334930591}a^{15}-\frac{63538762}{334930591}a^{14}+\frac{160856421}{334930591}a^{13}-\frac{264159280}{334930591}a^{12}+\frac{171709072}{334930591}a^{11}+\frac{110380202}{334930591}a^{10}-\frac{445084022}{334930591}a^{9}+\frac{572686264}{334930591}a^{8}-\frac{187553531}{334930591}a^{7}-\frac{265110563}{334930591}a^{6}+\frac{142120115}{334930591}a^{5}-\frac{322106606}{334930591}a^{4}-\frac{273362050}{334930591}a^{3}+\frac{595996861}{334930591}a^{2}+\frac{497489558}{334930591}a+\frac{276378611}{334930591}$, $\frac{84789373}{334930591}a^{15}-\frac{157120433}{334930591}a^{14}+\frac{540937127}{334930591}a^{13}-\frac{544134168}{334930591}a^{12}+\frac{421425666}{334930591}a^{11}-\frac{330542686}{334930591}a^{10}-\frac{1230283148}{334930591}a^{9}+\frac{34062559}{334930591}a^{8}-\frac{1501138630}{334930591}a^{7}+\frac{759615103}{334930591}a^{6}+\frac{1005313563}{334930591}a^{5}+\frac{1927054315}{334930591}a^{4}+\frac{1861456860}{334930591}a^{3}+\frac{698801927}{334930591}a^{2}+\frac{533243869}{334930591}a-\frac{234240899}{334930591}$, $\frac{9768380}{334930591}a^{15}-\frac{20279216}{334930591}a^{14}+\frac{5994157}{334930591}a^{13}+\frac{24328291}{334930591}a^{12}-\frac{251819661}{334930591}a^{11}+\frac{163114604}{334930591}a^{10}-\frac{3363334}{334930591}a^{9}-\frac{159044590}{334930591}a^{8}+\frac{1055341655}{334930591}a^{7}+\frac{234731150}{334930591}a^{6}+\frac{305690856}{334930591}a^{5}+\frac{69349170}{334930591}a^{4}-\frac{1367278673}{334930591}a^{3}-\frac{1354240245}{334930591}a^{2}-\frac{1152652847}{334930591}a-\frac{36843468}{334930591}$, $\frac{12840730}{334930591}a^{15}+\frac{67012508}{334930591}a^{14}-\frac{44857391}{334930591}a^{13}+\frac{411528230}{334930591}a^{12}-\frac{264180052}{334930591}a^{11}+\frac{160212729}{334930591}a^{10}-\frac{482094942}{334930591}a^{9}-\frac{1208343611}{334930591}a^{8}-\frac{913134297}{334930591}a^{7}-\frac{1291119693}{334930591}a^{6}+\frac{655007375}{334930591}a^{5}+\frac{1535520280}{334930591}a^{4}+\frac{2871314740}{334930591}a^{3}+\frac{2257483453}{334930591}a^{2}+\frac{1162913152}{334930591}a+\frac{176843790}{334930591}$, $\frac{86695208}{334930591}a^{15}-\frac{135053644}{334930591}a^{14}+\frac{530163506}{334930591}a^{13}-\frac{368948461}{334930591}a^{12}+\frac{202524776}{334930591}a^{11}+\frac{230147338}{334930591}a^{10}-\frac{2176759498}{334930591}a^{9}+\frac{271313261}{334930591}a^{8}-\frac{2129626422}{334930591}a^{7}-\frac{780661223}{334930591}a^{6}+\frac{2211785297}{334930591}a^{5}+\frac{1597216176}{334930591}a^{4}+\frac{3599801858}{334930591}a^{3}+\frac{2394210129}{334930591}a^{2}+\frac{1309307835}{334930591}a+\frac{40922675}{334930591}$, $\frac{48105152}{334930591}a^{15}-\frac{58645394}{334930591}a^{14}+\frac{253904410}{334930591}a^{13}-\frac{142095394}{334930591}a^{12}+\frac{94157245}{334930591}a^{11}-\frac{136288604}{334930591}a^{10}-\frac{781771867}{334930591}a^{9}-\frac{283453591}{334930591}a^{8}-\frac{901298702}{334930591}a^{7}+\frac{192036688}{334930591}a^{6}+\frac{1080360049}{334930591}a^{5}+\frac{1278387636}{334930591}a^{4}+\frac{1505705826}{334930591}a^{3}+\frac{441983143}{334930591}a^{2}-\frac{149203169}{334930591}a-\frac{244156176}{334930591}$ 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:  \( 167.322044256 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 167.322044256 \cdot 1}{2\cdot\sqrt{232292068597265625}}\cr\approx \mathstrut & 0.170885482243 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*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 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*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 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*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 - x^15 + 5*x^14 - x^13 + x^11 - 19*x^10 - 13*x^9 - 20*x^8 - 11*x^7 + 17*x^6 + 35*x^5 + 46*x^4 + 36*x^3 + 22*x^2 + 4*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

$D_8:C_2$ (as 16T44):

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 $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), 4.2.507.1, 4.2.12675.1, \(\Q(\sqrt{5}, \sqrt{13})\), 8.4.160655625.1

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: 16.0.3345005787800625.1, 16.0.12370583534765625.1
Minimal sibling: 16.0.12370583534765625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
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 5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(13\) Copy content Toggle raw display 13.8.4.1$x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$