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Properties

Label 16.8.671...641.1
Degree $16$
Signature $[8, 4]$
Discriminant $6.713\times 10^{31}$
Root discriminant \(97.54\)
Ramified primes $7,29,37,883$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2\wr C_2^3:C_3$ (as 16T1656)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 1)
 
gp: K = bnfinit(y^16 - 22*y^14 + 135*y^12 - 213*y^10 + 551*y^8 - 797*y^6 + 443*y^4 - 86*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 1)
 

\( x^{16} - 22x^{14} + 135x^{12} - 213x^{10} + 551x^{8} - 797x^{6} + 443x^{4} - 86x^{2} + 1 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(67129786029648929079511980455641\) \(\medspace = 7^{2}\cdot 29^{8}\cdot 37^{8}\cdot 883^{2}\) 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:  \(97.54\)
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:  $7^{1/2}29^{1/2}37^{2/3}883^{1/2}\approx 4701.06425007895$
Ramified primes:   \(7\), \(29\), \(37\), \(883\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{980306}a^{14}+\frac{94723}{490153}a^{12}+\frac{50673}{980306}a^{10}-\frac{205213}{980306}a^{8}+\frac{45296}{490153}a^{6}-\frac{191824}{490153}a^{4}-\frac{1}{2}a^{3}+\frac{90463}{490153}a^{2}-\frac{1}{2}a-\frac{143079}{980306}$, $\frac{1}{980306}a^{15}+\frac{94723}{490153}a^{13}+\frac{50673}{980306}a^{11}-\frac{205213}{980306}a^{9}+\frac{45296}{490153}a^{7}-\frac{191824}{490153}a^{5}-\frac{1}{2}a^{4}+\frac{90463}{490153}a^{3}-\frac{1}{2}a^{2}-\frac{143079}{980306}a$ 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

$C_{2}$, which has order $2$ (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:  $11$
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:   $\frac{1745}{490153}a^{14}-\frac{49857}{980306}a^{12}-\frac{96463}{980306}a^{10}+\frac{1185464}{490153}a^{8}-\frac{962911}{980306}a^{6}+\frac{9479383}{980306}a^{4}-\frac{9198231}{980306}a^{2}-\frac{1}{2}a+\frac{2080809}{980306}$, $\frac{1745}{490153}a^{14}-\frac{49857}{980306}a^{12}-\frac{96463}{980306}a^{10}+\frac{1185464}{490153}a^{8}-\frac{962911}{980306}a^{6}+\frac{9479383}{980306}a^{4}-\frac{9198231}{980306}a^{2}+\frac{1}{2}a+\frac{2080809}{980306}$, $a$, $\frac{90491}{980306}a^{15}-\frac{90491}{980306}a^{14}-\frac{1903801}{980306}a^{13}+\frac{1903801}{980306}a^{12}+\frac{10362341}{980306}a^{11}-\frac{10362341}{980306}a^{10}-\frac{8815779}{980306}a^{9}+\frac{8815779}{980306}a^{8}+\frac{19336917}{490153}a^{7}-\frac{19336917}{490153}a^{6}-\frac{16242291}{490153}a^{5}+\frac{16242291}{490153}a^{4}-\frac{938226}{490153}a^{3}+\frac{938226}{490153}a^{2}-\frac{1440753}{980306}a-\frac{14853}{490153}$, $\frac{93981}{980306}a^{15}+\frac{87001}{980306}a^{14}-\frac{976829}{490153}a^{13}-\frac{926972}{490153}a^{12}+\frac{5132939}{490153}a^{11}+\frac{5229402}{490153}a^{10}-\frac{6444851}{980306}a^{9}-\frac{11186707}{980306}a^{8}+\frac{37710923}{980306}a^{7}+\frac{39636745}{980306}a^{6}-\frac{23005199}{980306}a^{5}-\frac{41963965}{980306}a^{4}-\frac{11074683}{980306}a^{3}+\frac{3415813}{490153}a^{2}+\frac{3090821}{980306}a-\frac{90491}{980306}$, $\frac{734037}{980306}a^{15}+\frac{71815}{490153}a^{14}-\frac{16128871}{980306}a^{13}-\frac{1582790}{490153}a^{12}+\frac{49313548}{490153}a^{11}+\frac{19487213}{980306}a^{10}-\frac{153042657}{980306}a^{9}-\frac{30762327}{980306}a^{8}+\frac{198413315}{490153}a^{7}+\frac{77081443}{980306}a^{6}-\frac{285420377}{490153}a^{5}-\frac{57528891}{490153}a^{4}+\frac{295965477}{980306}a^{3}+\frac{57797833}{980306}a^{2}-\frac{22400268}{490153}a-\frac{2591811}{490153}$, $\frac{384809}{490153}a^{15}+\frac{13711}{980306}a^{14}-\frac{17015107}{980306}a^{13}-\frac{158397}{490153}a^{12}+\frac{105702517}{980306}a^{11}+\frac{1095657}{490153}a^{10}-\frac{174483901}{980306}a^{9}-\frac{5098753}{980306}a^{8}+\frac{437617153}{980306}a^{7}+\frac{11332729}{980306}a^{6}-\frac{326992601}{490153}a^{5}-\frac{18991699}{980306}a^{4}+\frac{196192061}{490153}a^{3}+\frac{21578785}{980306}a^{2}-\frac{83197311}{980306}a-\frac{4738385}{490153}$, $\frac{47241}{21311}a^{15}-\frac{13960}{490153}a^{14}-\frac{1044136}{21311}a^{13}+\frac{199428}{490153}a^{12}+\frac{6478418}{21311}a^{11}+\frac{385852}{490153}a^{10}-\frac{10599756}{21311}a^{9}-\frac{9483712}{490153}a^{8}+\frac{26428603}{21311}a^{7}+\frac{3851644}{490153}a^{6}-\frac{39741544}{21311}a^{5}-\frac{37917532}{490153}a^{4}+\frac{22341568}{21311}a^{3}+\frac{35812618}{490153}a^{2}-\frac{4119503}{21311}a-\frac{10283848}{490153}$, $\frac{545915}{490153}a^{15}-\frac{1451397}{980306}a^{14}-\frac{11653276}{490153}a^{13}+\frac{15571070}{490153}a^{12}+\frac{66066436}{490153}a^{11}-\frac{89534341}{490153}a^{10}-\frac{145526177}{980306}a^{9}+\frac{212867289}{980306}a^{8}+\frac{251032622}{490153}a^{7}-\frac{689099433}{980306}a^{6}-\frac{533300493}{980306}a^{5}+\frac{784719537}{980306}a^{4}+\frac{119060005}{980306}a^{3}-\frac{118569574}{490153}a^{2}+\frac{8671937}{490153}a+\frac{1390156}{490153}$, $\frac{206086}{490153}a^{15}-\frac{370006}{490153}a^{14}-\frac{4890083}{490153}a^{13}+\frac{7485996}{490153}a^{12}+\frac{35087076}{490153}a^{11}-\frac{36252804}{490153}a^{10}-\frac{80040111}{490153}a^{9}+\frac{10193003}{980306}a^{8}+\frac{125294310}{490153}a^{7}-\frac{288661105}{980306}a^{6}-\frac{618787059}{980306}a^{5}+\frac{12897859}{980306}a^{4}+\frac{246320505}{980306}a^{3}+\frac{91500061}{980306}a^{2}+\frac{3476451}{490153}a-\frac{2183959}{980306}$, $\frac{11397755}{490153}a^{15}+\frac{2139364347}{490153}a^{13}-\frac{22890749557}{490153}a^{11}+\frac{32150091450}{490153}a^{9}-\frac{92110637540}{490153}a^{7}+\frac{144113389546}{490153}a^{5}-\frac{81634895888}{490153}a^{3}+\frac{15449458461}{490153}a$ 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:  \( 7030073415.12 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 7030073415.12 \cdot 2}{2\cdot\sqrt{67129786029648929079511980455641}}\cr\approx \mathstrut & 0.342343169906 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 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 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 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 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 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 - 22*x^14 + 135*x^12 - 213*x^10 + 551*x^8 - 797*x^6 + 443*x^4 - 86*x^2 + 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

$C_2\wr C_2^3:C_3$ (as 16T1656):

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 6144
The 69 conjugacy class representatives for $C_2\wr C_2^3:C_3$
Character table for $C_2\wr C_2^3:C_3$

Intermediate fields

4.4.1151329.2, 8.4.8193276879835621.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
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 ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{4}$ R ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.6.0.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.0.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
\(29\) Copy content Toggle raw display 29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.12.8.1$x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(883\) Copy content Toggle raw display Deg $4$$2$$2$$2$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$