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Properties

Label 16.0.964...184.1
Degree $16$
Signature $[0, 8]$
Discriminant $9.647\times 10^{46}$
Root discriminant \(864.02\)
Ramified primes $2,257$
Class number $14079590400$ (GRH)
Class group [2, 2, 2, 2, 2, 4, 4, 40, 687480] (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224)
 
gp: K = bnfinit(y^16 + 1028*y^14 + 255458*y^12 + 23931840*y^10 + 862829184*y^8 + 12850493440*y^6 + 71556958208*y^4 + 147171966976*y^2 + 69257396224, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224)
 

\( x^{16} + 1028 x^{14} + 255458 x^{12} + 23931840 x^{10} + 862829184 x^{8} + 12850493440 x^{6} + \cdots + 69257396224 \) 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:   \(96468058006688010602395260507818496203564253184\) \(\medspace = 2^{44}\cdot 257^{14}\) 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:  \(864.02\)
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:  $2^{11/4}257^{7/8}\approx 864.0203491841434$
Ramified primes:   \(2\), \(257\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4112=2^{4}\cdot 257\)
Dirichlet character group:    $\lbrace$$\chi_{4112}(1,·)$, $\chi_{4112}(1795,·)$, $\chi_{4112}(513,·)$, $\chi_{4112}(3851,·)$, $\chi_{4112}(2057,·)$, $\chi_{4112}(2763,·)$, $\chi_{4112}(273,·)$, $\chi_{4112}(1803,·)$, $\chi_{4112}(2329,·)$, $\chi_{4112}(2891,·)$, $\chi_{4112}(241,·)$, $\chi_{4112}(3859,·)$, $\chi_{4112}(707,·)$, $\chi_{4112}(2569,·)$, $\chi_{4112}(2297,·)$, $\chi_{4112}(835,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{7}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16448}a^{8}-\frac{1}{16}a^{6}+\frac{1}{32}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{65792}a^{9}+\frac{1}{64}a^{7}+\frac{1}{128}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{4473856}a^{10}-\frac{23}{1118464}a^{8}-\frac{287}{8704}a^{6}+\frac{9}{136}a^{4}+\frac{133}{272}a^{2}+\frac{2}{17}$, $\frac{1}{17895424}a^{11}-\frac{23}{4473856}a^{9}+\frac{801}{34816}a^{7}+\frac{9}{544}a^{5}-\frac{343}{1088}a^{3}+\frac{19}{68}a$, $\frac{1}{4853525315584}a^{12}-\frac{60899}{1213381328896}a^{10}-\frac{47042495}{2426762657792}a^{8}+\frac{3372913}{590166016}a^{6}+\frac{9283361}{295083008}a^{4}+\frac{1645255}{4610672}a^{2}+\frac{120883}{576334}$, $\frac{1}{9707050631168}a^{13}-\frac{60899}{2426762657792}a^{11}+\frac{104001}{18885312512}a^{9}+\frac{21815601}{1180332032}a^{7}+\frac{13894033}{590166016}a^{5}+\frac{3950591}{9221344}a^{3}-\frac{41821}{288167}a$, $\frac{1}{22\!\cdots\!36}a^{14}-\frac{5}{121443644309504}a^{12}-\frac{981442167}{11\!\cdots\!68}a^{10}+\frac{22964557313}{27\!\cdots\!92}a^{8}+\frac{60354586683}{1358562168832}a^{6}+\frac{64419170253}{339640542208}a^{4}-\frac{1574765255}{5306883472}a^{2}-\frac{257620721}{663360434}$, $\frac{1}{44\!\cdots\!72}a^{15}-\frac{5}{242887288619008}a^{13}+\frac{267236297}{22\!\cdots\!36}a^{11}-\frac{5755047359}{55\!\cdots\!84}a^{9}-\frac{46953718817}{2717124337664}a^{7}+\frac{75657276429}{679281084416}a^{5}+\frac{8732425499}{21227533888}a^{3}-\frac{137569934}{331680217}a$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{40}\times C_{687480}$, which has order $14079590400$ (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:   $\frac{2071}{3789964476416}a^{14}+\frac{530795}{947491119104}a^{12}+\frac{261526311}{1894982238208}a^{10}+\frac{1501547903}{118436389888}a^{8}+\frac{24952713}{57605248}a^{6}+\frac{80596713}{14401312}a^{4}+\frac{138147409}{7200656}a^{2}+\frac{5723515}{450041}$, $\frac{680059}{11\!\cdots\!68}a^{14}+\frac{71883615}{13\!\cdots\!96}a^{12}+\frac{24550174931}{55\!\cdots\!84}a^{10}-\frac{1564956819013}{13\!\cdots\!96}a^{8}-\frac{104291382023}{679281084416}a^{6}-\frac{689270182081}{169820271104}a^{4}-\frac{78055189739}{2653441736}a^{2}-\frac{20633534050}{331680217}$, $\frac{5865687}{11\!\cdots\!68}a^{14}+\frac{737803565}{13\!\cdots\!96}a^{12}+\frac{685140085999}{55\!\cdots\!84}a^{10}+\frac{13841566207899}{13\!\cdots\!96}a^{8}+\frac{9608302521}{39957710848}a^{6}+\frac{27022869309}{15438206464}a^{4}+\frac{2275677777}{482443952}a^{2}+\frac{1275367458}{331680217}$, $\frac{6222961}{11\!\cdots\!68}a^{14}+\frac{796651479}{13\!\cdots\!96}a^{12}+\frac{782605251033}{55\!\cdots\!84}a^{10}+\frac{1624517934799}{126963809959936}a^{8}+\frac{293385744263}{679281084416}a^{6}+\frac{2385629847}{434322944}a^{4}+\frac{97210070359}{5306883472}a^{2}+\frac{3250459746}{331680217}$, $\frac{24413047}{13\!\cdots\!96}a^{14}+\frac{6264782111}{349150477389824}a^{12}+\frac{3098390741799}{698300954779648}a^{10}+\frac{4480350419681}{10910952418432}a^{8}+\frac{302472323205}{21227533888}a^{6}+\frac{993228729723}{5306883472}a^{4}+\frac{1712556425089}{2653441736}a^{2}+\frac{115092465145}{331680217}$, $\frac{868589567}{22\!\cdots\!36}a^{14}+\frac{223081377397}{55\!\cdots\!84}a^{12}+\frac{110643771977103}{11\!\cdots\!68}a^{10}+\frac{12\!\cdots\!23}{13\!\cdots\!96}a^{8}+\frac{44719729730395}{1358562168832}a^{6}+\frac{81067415577985}{169820271104}a^{4}+\frac{13062114379003}{5306883472}a^{2}+\frac{1353871239350}{331680217}$, $\frac{1723874345}{22\!\cdots\!36}a^{14}+\frac{26021641969}{328612214013952}a^{12}+\frac{218821204138313}{11\!\cdots\!68}a^{10}+\frac{79265933403489}{43643809673728}a^{8}+\frac{86705020267257}{1358562168832}a^{6}+\frac{76293588816443}{84910135552}a^{4}+\frac{22818630956397}{5306883472}a^{2}+\frac{2123792203815}{331680217}$ 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:  \( 112709711.05474126 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 112709711.05474126 \cdot 14079590400}{2\cdot\sqrt{96468058006688010602395260507818496203564253184}}\cr\approx \mathstrut & 6.20538226717008 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224)
 
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 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224, 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 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224);
 
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 + 1028*x^14 + 255458*x^12 + 23931840*x^10 + 862829184*x^8 + 12850493440*x^6 + 71556958208*x^4 + 147171966976*x^2 + 69257396224);
 
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

$C_2\times C_8$ (as 16T5):

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)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{257}) \), \(\Q(\sqrt{514}) \), \(\Q(\sqrt{2}, \sqrt{257})\), 4.4.1086373952.2, 4.4.16974593.1, 8.8.1180208363584098304.1, 8.0.310593074627699982467072.6, 8.0.310593074627699982467072.5

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 R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.1.0.1}{1} }^{16}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
\(257\) Copy content Toggle raw display Deg $8$$8$$1$$7$
Deg $8$$8$$1$$7$