Properties

Label 16.0.225...433.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.259\times 10^{25}$
Root discriminant \(38.43\)
Ramified primes $17,53$
Class number $162$ (GRH)
Class group [9, 18] (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211)
 
gp: K = bnfinit(y^16 - y^15 + 35*y^14 - 52*y^13 + 426*y^12 - 834*y^11 + 2908*y^10 - 5441*y^9 + 12734*y^8 - 20894*y^7 + 36823*y^6 - 49114*y^5 + 65638*y^4 - 66318*y^3 + 63428*y^2 - 39339*y + 19211, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211)
 

\( x^{16} - x^{15} + 35 x^{14} - 52 x^{13} + 426 x^{12} - 834 x^{11} + 2908 x^{10} - 5441 x^{9} + 12734 x^{8} - 20894 x^{7} + 36823 x^{6} - 49114 x^{5} + 65638 x^{4} + \cdots + 19211 \) 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:   \(22585894701900222828166433\) \(\medspace = 17^{15}\cdot 53^{4}\) 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:  \(38.43\)
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:  $17^{15/16}53^{1/2}\approx 103.67732863884214$
Ramified primes:   \(17\), \(53\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{16013}a^{14}-\frac{2829}{16013}a^{13}-\frac{970}{16013}a^{12}+\frac{4752}{16013}a^{11}-\frac{1829}{16013}a^{10}+\frac{6191}{16013}a^{9}+\frac{3715}{16013}a^{8}-\frac{3625}{16013}a^{7}+\frac{3919}{16013}a^{6}-\frac{1729}{16013}a^{5}-\frac{5464}{16013}a^{4}+\frac{1202}{16013}a^{3}+\frac{6129}{16013}a^{2}-\frac{199}{16013}a-\frac{5898}{16013}$, $\frac{1}{69\!\cdots\!91}a^{15}+\frac{14\!\cdots\!10}{69\!\cdots\!91}a^{14}-\frac{20\!\cdots\!73}{10\!\cdots\!73}a^{13}-\frac{36\!\cdots\!47}{69\!\cdots\!91}a^{12}+\frac{11\!\cdots\!72}{69\!\cdots\!91}a^{11}+\frac{10\!\cdots\!86}{69\!\cdots\!91}a^{10}-\frac{41\!\cdots\!19}{69\!\cdots\!91}a^{9}-\frac{25\!\cdots\!74}{69\!\cdots\!91}a^{8}+\frac{23\!\cdots\!96}{69\!\cdots\!91}a^{7}-\frac{88\!\cdots\!48}{28\!\cdots\!69}a^{6}+\frac{11\!\cdots\!45}{69\!\cdots\!91}a^{5}-\frac{57\!\cdots\!22}{69\!\cdots\!91}a^{4}+\frac{15\!\cdots\!61}{69\!\cdots\!91}a^{3}+\frac{42\!\cdots\!95}{69\!\cdots\!91}a^{2}-\frac{32\!\cdots\!04}{69\!\cdots\!91}a-\frac{31\!\cdots\!72}{69\!\cdots\!91}$ 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

$C_{9}\times C_{18}$, which has order $162$ (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:   $\frac{13\!\cdots\!18}{69\!\cdots\!91}a^{15}+\frac{41\!\cdots\!85}{69\!\cdots\!91}a^{14}+\frac{70\!\cdots\!02}{10\!\cdots\!73}a^{13}-\frac{54\!\cdots\!28}{69\!\cdots\!91}a^{12}+\frac{51\!\cdots\!97}{69\!\cdots\!91}a^{11}-\frac{27\!\cdots\!80}{69\!\cdots\!91}a^{10}+\frac{27\!\cdots\!66}{69\!\cdots\!91}a^{9}-\frac{57\!\cdots\!65}{28\!\cdots\!69}a^{8}+\frac{88\!\cdots\!19}{69\!\cdots\!91}a^{7}-\frac{32\!\cdots\!37}{69\!\cdots\!91}a^{6}+\frac{16\!\cdots\!31}{69\!\cdots\!91}a^{5}+\frac{72\!\cdots\!34}{69\!\cdots\!91}a^{4}+\frac{12\!\cdots\!21}{69\!\cdots\!91}a^{3}+\frac{13\!\cdots\!65}{69\!\cdots\!91}a^{2}-\frac{13\!\cdots\!94}{69\!\cdots\!91}a+\frac{22\!\cdots\!84}{69\!\cdots\!91}$, $\frac{28\!\cdots\!35}{69\!\cdots\!91}a^{15}-\frac{42\!\cdots\!07}{69\!\cdots\!91}a^{14}+\frac{89\!\cdots\!30}{69\!\cdots\!91}a^{13}-\frac{19\!\cdots\!84}{69\!\cdots\!91}a^{12}+\frac{92\!\cdots\!60}{69\!\cdots\!91}a^{11}-\frac{27\!\cdots\!03}{69\!\cdots\!91}a^{10}+\frac{55\!\cdots\!65}{69\!\cdots\!91}a^{9}-\frac{14\!\cdots\!59}{69\!\cdots\!91}a^{8}+\frac{22\!\cdots\!54}{69\!\cdots\!91}a^{7}-\frac{47\!\cdots\!80}{69\!\cdots\!91}a^{6}+\frac{57\!\cdots\!09}{69\!\cdots\!91}a^{5}-\frac{91\!\cdots\!10}{69\!\cdots\!91}a^{4}+\frac{79\!\cdots\!66}{69\!\cdots\!91}a^{3}-\frac{94\!\cdots\!14}{69\!\cdots\!91}a^{2}+\frac{52\!\cdots\!27}{69\!\cdots\!91}a-\frac{47\!\cdots\!68}{69\!\cdots\!91}$, $\frac{95\!\cdots\!10}{69\!\cdots\!91}a^{15}-\frac{61\!\cdots\!57}{69\!\cdots\!91}a^{14}+\frac{32\!\cdots\!11}{69\!\cdots\!91}a^{13}-\frac{22\!\cdots\!72}{69\!\cdots\!91}a^{12}+\frac{47\!\cdots\!79}{69\!\cdots\!91}a^{11}-\frac{26\!\cdots\!71}{69\!\cdots\!91}a^{10}+\frac{46\!\cdots\!01}{69\!\cdots\!91}a^{9}-\frac{15\!\cdots\!94}{69\!\cdots\!91}a^{8}+\frac{23\!\cdots\!01}{69\!\cdots\!91}a^{7}-\frac{54\!\cdots\!86}{69\!\cdots\!91}a^{6}+\frac{73\!\cdots\!27}{69\!\cdots\!91}a^{5}-\frac{12\!\cdots\!54}{69\!\cdots\!91}a^{4}+\frac{12\!\cdots\!96}{69\!\cdots\!91}a^{3}-\frac{15\!\cdots\!78}{69\!\cdots\!91}a^{2}+\frac{10\!\cdots\!84}{69\!\cdots\!91}a-\frac{70\!\cdots\!56}{69\!\cdots\!91}$, $\frac{27\!\cdots\!74}{69\!\cdots\!91}a^{15}+\frac{31\!\cdots\!90}{69\!\cdots\!91}a^{14}+\frac{88\!\cdots\!83}{69\!\cdots\!91}a^{13}-\frac{47\!\cdots\!65}{69\!\cdots\!91}a^{12}+\frac{87\!\cdots\!89}{69\!\cdots\!91}a^{11}-\frac{10\!\cdots\!34}{69\!\cdots\!91}a^{10}+\frac{42\!\cdots\!97}{69\!\cdots\!91}a^{9}-\frac{56\!\cdots\!28}{69\!\cdots\!91}a^{8}+\frac{13\!\cdots\!25}{69\!\cdots\!91}a^{7}-\frac{14\!\cdots\!40}{69\!\cdots\!91}a^{6}+\frac{24\!\cdots\!48}{69\!\cdots\!91}a^{5}-\frac{16\!\cdots\!87}{69\!\cdots\!91}a^{4}+\frac{14\!\cdots\!28}{69\!\cdots\!91}a^{3}+\frac{55\!\cdots\!20}{69\!\cdots\!91}a^{2}-\frac{71\!\cdots\!83}{69\!\cdots\!91}a+\frac{16\!\cdots\!05}{69\!\cdots\!91}$, $\frac{63\!\cdots\!61}{69\!\cdots\!91}a^{15}-\frac{42\!\cdots\!97}{69\!\cdots\!91}a^{14}+\frac{79\!\cdots\!47}{69\!\cdots\!91}a^{13}-\frac{14\!\cdots\!19}{69\!\cdots\!91}a^{12}+\frac{48\!\cdots\!71}{69\!\cdots\!91}a^{11}-\frac{16\!\cdots\!69}{69\!\cdots\!91}a^{10}+\frac{20\!\cdots\!04}{10\!\cdots\!73}a^{9}-\frac{93\!\cdots\!31}{69\!\cdots\!91}a^{8}+\frac{89\!\cdots\!29}{69\!\cdots\!91}a^{7}-\frac{33\!\cdots\!40}{69\!\cdots\!91}a^{6}+\frac{33\!\cdots\!61}{69\!\cdots\!91}a^{5}-\frac{74\!\cdots\!23}{69\!\cdots\!91}a^{4}+\frac{64\!\cdots\!38}{69\!\cdots\!91}a^{3}-\frac{10\!\cdots\!34}{69\!\cdots\!91}a^{2}+\frac{59\!\cdots\!10}{69\!\cdots\!91}a-\frac{57\!\cdots\!82}{69\!\cdots\!91}$, $\frac{20\!\cdots\!06}{69\!\cdots\!91}a^{15}-\frac{19\!\cdots\!31}{69\!\cdots\!91}a^{14}+\frac{64\!\cdots\!56}{69\!\cdots\!91}a^{13}-\frac{10\!\cdots\!71}{69\!\cdots\!91}a^{12}+\frac{66\!\cdots\!08}{69\!\cdots\!91}a^{11}-\frac{17\!\cdots\!79}{69\!\cdots\!91}a^{10}+\frac{37\!\cdots\!12}{69\!\cdots\!91}a^{9}-\frac{98\!\cdots\!00}{69\!\cdots\!91}a^{8}+\frac{15\!\cdots\!73}{69\!\cdots\!91}a^{7}-\frac{32\!\cdots\!11}{69\!\cdots\!91}a^{6}+\frac{41\!\cdots\!01}{69\!\cdots\!91}a^{5}-\frac{66\!\cdots\!76}{69\!\cdots\!91}a^{4}+\frac{64\!\cdots\!13}{69\!\cdots\!91}a^{3}-\frac{75\!\cdots\!61}{69\!\cdots\!91}a^{2}+\frac{45\!\cdots\!51}{69\!\cdots\!91}a-\frac{33\!\cdots\!21}{69\!\cdots\!91}$, $\frac{13\!\cdots\!84}{69\!\cdots\!91}a^{15}-\frac{35\!\cdots\!25}{69\!\cdots\!91}a^{14}+\frac{44\!\cdots\!33}{69\!\cdots\!91}a^{13}-\frac{14\!\cdots\!37}{69\!\cdots\!91}a^{12}+\frac{52\!\cdots\!71}{69\!\cdots\!91}a^{11}-\frac{20\!\cdots\!63}{69\!\cdots\!91}a^{10}+\frac{39\!\cdots\!20}{69\!\cdots\!91}a^{9}-\frac{12\!\cdots\!96}{69\!\cdots\!91}a^{8}+\frac{19\!\cdots\!75}{69\!\cdots\!91}a^{7}-\frac{43\!\cdots\!92}{69\!\cdots\!91}a^{6}+\frac{57\!\cdots\!90}{69\!\cdots\!91}a^{5}-\frac{94\!\cdots\!98}{69\!\cdots\!91}a^{4}+\frac{97\!\cdots\!31}{69\!\cdots\!91}a^{3}-\frac{11\!\cdots\!48}{69\!\cdots\!91}a^{2}+\frac{76\!\cdots\!57}{69\!\cdots\!91}a-\frac{56\!\cdots\!25}{69\!\cdots\!91}$ 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:  \( 3640.01221338 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 162}{2\cdot\sqrt{22585894701900222828166433}}\cr\approx \mathstrut & 0.150698231956 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211)
 
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 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211, 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 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211);
 
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 + 35*x^14 - 52*x^13 + 426*x^12 - 834*x^11 + 2908*x^10 - 5441*x^9 + 12734*x^8 - 20894*x^7 + 36823*x^6 - 49114*x^5 + 65638*x^4 - 66318*x^3 + 63428*x^2 - 39339*x + 19211);
 
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

$\OD_{32}$ (as 16T22):

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 $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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
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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ R ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(17\) Copy content Toggle raw display 17.16.15.1$x^{16} + 272$$16$$1$$15$$C_{16}$$[\ ]_{16}$
\(53\) Copy content Toggle raw display 53.8.4.2$x^{8} + 25281 x^{4} - 5657326 x^{2} + 15780962$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
53.8.0.1$x^{8} + 8 x^{4} + 29 x^{3} + 18 x^{2} + x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$