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Properties

Label 16.0.233...753.2
Degree $16$
Signature $[0, 8]$
Discriminant $2.335\times 10^{27}$
Root discriminant \(51.35\)
Ramified primes $13,17$
Class number $1808$ (GRH)
Class group [2, 2, 2, 226] (GRH)
Galois group $C_{16}$ (as 16T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629)
 
gp: K = bnfinit(y^16 - y^15 + 52*y^14 - 52*y^13 + 1123*y^12 - 1123*y^11 + 13057*y^10 - 13057*y^9 + 88792*y^8 - 88792*y^7 + 361438*y^6 - 361438*y^5 + 881944*y^4 - 881944*y^3 + 1328092*y^2 - 1328092*y + 1439629, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629)
 

\( x^{16} - x^{15} + 52 x^{14} - 52 x^{13} + 1123 x^{12} - 1123 x^{11} + 13057 x^{10} - 13057 x^{9} + \cdots + 1439629 \) 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:   \(2334966419615122175401076753\) \(\medspace = 13^{8}\cdot 17^{15}\) 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:  \(51.35\)
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:  $13^{1/2}17^{15/16}\approx 51.34729148263172$
Ramified primes:   \(13\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(66,·)$, $\chi_{221}(196,·)$, $\chi_{221}(129,·)$, $\chi_{221}(12,·)$, $\chi_{221}(194,·)$, $\chi_{221}(142,·)$, $\chi_{221}(207,·)$, $\chi_{221}(144,·)$, $\chi_{221}(90,·)$, $\chi_{221}(157,·)$, $\chi_{221}(116,·)$, $\chi_{221}(53,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$, $\chi_{221}(181,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{399331}a^{9}+\frac{80968}{399331}a^{8}+\frac{27}{399331}a^{7}-\frac{53423}{399331}a^{6}+\frac{243}{399331}a^{5}+\frac{198324}{399331}a^{4}+\frac{810}{399331}a^{3}-\frac{162952}{399331}a^{2}+\frac{729}{399331}a-\frac{61107}{399331}$, $\frac{1}{399331}a^{10}+\frac{30}{399331}a^{8}+\frac{156427}{399331}a^{7}+\frac{315}{399331}a^{6}+\frac{90319}{399331}a^{5}+\frac{1350}{399331}a^{4}+\frac{142583}{399331}a^{3}+\frac{2025}{399331}a^{2}+\frac{14209}{399331}a+\frac{486}{399331}$, $\frac{1}{399331}a^{11}+\frac{123373}{399331}a^{8}-\frac{495}{399331}a^{7}+\frac{95685}{399331}a^{6}-\frac{5940}{399331}a^{5}+\frac{182828}{399331}a^{4}-\frac{22275}{399331}a^{3}+\frac{110797}{399331}a^{2}-\frac{21384}{399331}a-\frac{163445}{399331}$, $\frac{1}{399331}a^{12}-\frac{594}{399331}a^{8}-\frac{40738}{399331}a^{7}-\frac{8316}{399331}a^{6}+\frac{153014}{399331}a^{5}-\frac{40095}{399331}a^{4}+\frac{11417}{399331}a^{3}-\frac{64152}{399331}a^{2}+\frac{146444}{399331}a-\frac{16038}{399331}$, $\frac{1}{399331}a^{13}+\frac{134534}{399331}a^{8}+\frac{7722}{399331}a^{7}-\frac{33099}{399331}a^{6}+\frac{104247}{399331}a^{5}+\frac{13228}{399331}a^{4}+\frac{17657}{399331}a^{3}-\frac{8942}{399331}a^{2}+\frac{17657}{399331}a+\frac{41563}{399331}$, $\frac{1}{399331}a^{14}+\frac{9828}{399331}a^{8}-\frac{71538}{399331}a^{7}+\frac{154791}{399331}a^{6}+\frac{66608}{399331}a^{5}-\frac{2594}{399331}a^{4}+\frac{35881}{399331}a^{3}+\frac{128787}{399331}a^{2}-\frac{197628}{399331}a-\frac{58159}{399331}$, $\frac{1}{399331}a^{15}+\frac{41641}{399331}a^{8}-\frac{110565}{399331}a^{7}-\frac{12413}{399331}a^{6}+\frac{5188}{399331}a^{5}+\frac{42220}{399331}a^{4}+\frac{154727}{399331}a^{3}-\frac{22682}{399331}a^{2}-\frac{34813}{399331}a-\frac{34228}{399331}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{226}$, which has order $1808$ (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:   $\frac{1}{399331}a^{15}+\frac{45}{399331}a^{13}+\frac{810}{399331}a^{11}+\frac{7425}{399331}a^{9}+\frac{36450}{399331}a^{7}+\frac{91854}{399331}a^{5}+\frac{102060}{399331}a^{3}-\frac{75316}{399331}a^{2}+\frac{32805}{399331}a-\frac{451896}{399331}$, $\frac{40}{399331}a^{11}+\frac{1320}{399331}a^{9}+\frac{15840}{399331}a^{7}-\frac{2683}{399331}a^{6}+\frac{83160}{399331}a^{5}-\frac{48294}{399331}a^{4}+\frac{178200}{399331}a^{3}-\frac{217323}{399331}a^{2}+\frac{106920}{399331}a-\frac{144882}{399331}$, $\frac{217}{399331}a^{9}-\frac{508}{399331}a^{8}+\frac{5859}{399331}a^{7}-\frac{12192}{399331}a^{6}+\frac{52731}{399331}a^{5}-\frac{91440}{399331}a^{4}+\frac{175770}{399331}a^{3}-\frac{219456}{399331}a^{2}+\frac{158193}{399331}a-\frac{481627}{399331}$, $\frac{1}{399331}a^{15}-\frac{4}{399331}a^{14}+\frac{45}{399331}a^{13}-\frac{168}{399331}a^{12}+\frac{850}{399331}a^{11}-\frac{2869}{399331}a^{10}+\frac{8745}{399331}a^{9}-\frac{25590}{399331}a^{8}+\frac{53449}{399331}a^{7}-\frac{128494}{399331}a^{6}+\frac{199353}{399331}a^{5}-\frac{369756}{399331}a^{4}+\frac{458983}{399331}a^{3}-\frac{631948}{399331}a^{2}+\frac{652977}{399331}a-\frac{661416}{399331}$, $\frac{4}{399331}a^{14}+\frac{168}{399331}a^{12}+\frac{2772}{399331}a^{10}+\frac{22680}{399331}a^{8}+\frac{95256}{399331}a^{6}+\frac{190512}{399331}a^{4}-\frac{32689}{399331}a^{3}+\frac{142884}{399331}a^{2}-\frac{294201}{399331}a+\frac{17496}{399331}$, $\frac{19}{399331}a^{12}+\frac{684}{399331}a^{10}+\frac{9234}{399331}a^{8}+\frac{57456}{399331}a^{6}-\frac{6160}{399331}a^{5}+\frac{161595}{399331}a^{4}-\frac{92400}{399331}a^{3}+\frac{166212}{399331}a^{2}-\frac{277200}{399331}a+\frac{27702}{399331}$, $\frac{7}{399331}a^{13}+\frac{273}{399331}a^{11}+\frac{4095}{399331}a^{9}+\frac{29484}{399331}a^{7}+\frac{103194}{399331}a^{5}-\frac{14209}{399331}a^{4}+\frac{154791}{399331}a^{3}-\frac{170508}{399331}a^{2}+\frac{66339}{399331}a-\frac{255762}{399331}$ 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 1808}{2\cdot\sqrt{2334966419615122175401076753}}\cr\approx \mathstrut & 0.165413099163 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629)
 
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 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629, 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 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629);
 
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 + 52*x^14 - 52*x^13 + 1123*x^12 - 1123*x^11 + 13057*x^10 - 13057*x^9 + 88792*x^8 - 88792*x^7 + 361438*x^6 - 361438*x^5 + 881944*x^4 - 881944*x^3 + 1328092*x^2 - 1328092*x + 1439629);
 
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

$C_{16}$ (as 16T1):

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 cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/padicField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(13\) Copy content Toggle raw display 13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(17\) Copy content Toggle raw display 17.16.15.5$x^{16} + 17$$16$$1$$15$$C_{16}$$[\ ]_{16}$