↑

Properties

Label 16.16.123...000.1
Degree $16$
Signature $[16, 0]$
Discriminant $1.234\times 10^{31}$
Root discriminant \(87.74\)
Ramified primes $2,3,5,7$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $(C_2 \times Q_8):C_2$ (as 16T20)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481)
 
gp: K = bnfinit(y^16 - 84*y^14 + 2583*y^12 - 37044*y^10 + 268128*y^8 - 981666*y^6 + 1694763*y^4 - 1166886*y^2 + 194481, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481)
 

\( x^{16} - 84 x^{14} + 2583 x^{12} - 37044 x^{10} + 268128 x^{8} - 981666 x^{6} + 1694763 x^{4} + \cdots + 194481 \) 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:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(12341030701727613557145600000000\) \(\medspace = 2^{32}\cdot 3^{12}\cdot 5^{8}\cdot 7^{12}\) 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:  \(87.74\)
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:  $2^{5/2}3^{3/4}5^{1/2}7^{3/4}\approx 124.08647925963417$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{21}a^{4}$, $\frac{1}{21}a^{5}$, $\frac{1}{21}a^{6}$, $\frac{1}{21}a^{7}$, $\frac{1}{441}a^{8}$, $\frac{1}{441}a^{9}$, $\frac{1}{1323}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{1323}a^{11}+\frac{1}{3}a^{3}$, $\frac{1}{18522}a^{12}-\frac{1}{2}$, $\frac{1}{18522}a^{13}-\frac{1}{2}a$, $\frac{1}{298333854}a^{14}-\frac{383}{14206374}a^{12}-\frac{2120}{7103187}a^{10}-\frac{6}{5369}a^{8}-\frac{6271}{338247}a^{6}-\frac{277}{16107}a^{4}-\frac{9985}{32214}a^{2}-\frac{69}{1534}$, $\frac{1}{298333854}a^{15}-\frac{383}{14206374}a^{13}-\frac{2120}{7103187}a^{11}-\frac{6}{5369}a^{9}-\frac{6271}{338247}a^{7}-\frac{277}{16107}a^{5}-\frac{9985}{32214}a^{3}-\frac{69}{1534}a$ 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

$C_{2}$, 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:  $15$
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:   $\frac{1}{187278}a^{14}-\frac{55}{120393}a^{12}+\frac{64}{4459}a^{10}-\frac{19}{91}a^{8}+\frac{934}{637}a^{6}-\frac{59}{13}a^{4}+\frac{771}{182}a^{2}+\frac{11}{13}$, $\frac{1268}{149166927}a^{14}-\frac{10237}{14206374}a^{12}+\frac{159119}{7103187}a^{10}-\frac{12171}{37583}a^{8}+\frac{783596}{338247}a^{6}-\frac{127222}{16107}a^{4}+\frac{170930}{16107}a^{2}-\frac{3943}{1534}$, $\frac{397}{16574103}a^{14}-\frac{1991}{1014741}a^{12}+\frac{45538}{789243}a^{10}-\frac{28614}{37583}a^{8}+\frac{530567}{112749}a^{6}-\frac{9670}{767}a^{4}+\frac{59259}{5369}a^{2}-\frac{2631}{767}$, $\frac{353}{33148206}a^{14}-\frac{12601}{14206374}a^{12}+\frac{63590}{2367729}a^{10}-\frac{42113}{112749}a^{8}+\frac{95007}{37583}a^{6}-\frac{41767}{5369}a^{4}+\frac{77981}{10738}a^{2}+\frac{6285}{1534}$, $\frac{6619}{298333854}a^{14}-\frac{26987}{14206374}a^{12}+\frac{426437}{7103187}a^{10}-\frac{301459}{338247}a^{8}+\frac{2238863}{338247}a^{6}-\frac{376930}{16107}a^{4}+\frac{1000309}{32214}a^{2}-\frac{10317}{1534}$, $\frac{541}{21309561}a^{14}-\frac{3329}{1578486}a^{12}+\frac{64685}{1014741}a^{10}-\frac{301192}{338247}a^{8}+\frac{299557}{48321}a^{6}-\frac{48905}{2301}a^{4}+\frac{74495}{2301}a^{2}-\frac{26357}{1534}$, $\frac{997}{298333854}a^{14}-\frac{620}{2367729}a^{12}+\frac{5563}{789243}a^{10}-\frac{25580}{338247}a^{8}+\frac{77864}{338247}a^{6}+\frac{5864}{5369}a^{4}-\frac{71893}{10738}a^{2}+\frac{5104}{767}$, $\frac{1151}{298333854}a^{15}-\frac{2269}{16574103}a^{14}-\frac{2972}{7103187}a^{13}+\frac{162125}{14206374}a^{12}+\frac{126262}{7103187}a^{11}-\frac{2459825}{7103187}a^{10}-\frac{127511}{338247}a^{9}+\frac{234278}{48321}a^{8}+\frac{1431538}{338247}a^{7}-\frac{3784220}{112749}a^{6}-\frac{400129}{16107}a^{5}+\frac{1810871}{16107}a^{4}+\frac{2241167}{32214}a^{3}-\frac{2545103}{16107}a^{2}-\frac{53132}{767}a+\frac{89879}{1534}$, $\frac{2911}{33148206}a^{15}+\frac{4894}{49722309}a^{14}-\frac{51167}{7103187}a^{13}-\frac{114941}{14206374}a^{12}+\frac{503038}{2367729}a^{11}+\frac{1698479}{7103187}a^{10}-\frac{318211}{112749}a^{9}-\frac{358045}{112749}a^{8}+\frac{665475}{37583}a^{7}+\frac{2225995}{112749}a^{6}-\frac{262237}{5369}a^{5}-\frac{852697}{16107}a^{4}+\frac{466027}{10738}a^{3}+\frac{704114}{16107}a^{2}-\frac{2257}{767}a-\frac{8139}{1534}$, $\frac{2395}{298333854}a^{15}+\frac{5168}{149166927}a^{14}-\frac{10691}{14206374}a^{13}-\frac{18892}{7103187}a^{12}+\frac{189589}{7103187}a^{11}+\frac{165962}{2367729}a^{10}-\frac{16987}{37583}a^{9}-\frac{258389}{338247}a^{8}+\frac{1281239}{338247}a^{7}+\frac{1157216}{338247}a^{6}-\frac{237730}{16107}a^{5}-\frac{84998}{16107}a^{4}+\frac{729635}{32214}a^{3}+\frac{3239}{5369}a^{2}-\frac{13389}{1534}a+\frac{1597}{767}$, $\frac{176}{5524701}a^{15}-\frac{211}{2528253}a^{14}-\frac{35899}{14206374}a^{13}+\frac{1609}{240786}a^{12}+\frac{166898}{2367729}a^{11}-\frac{22822}{120393}a^{10}-\frac{96092}{112749}a^{9}+\frac{13565}{5733}a^{8}+\frac{519565}{112749}a^{7}-\frac{77905}{5733}a^{6}-\frac{55312}{5369}a^{5}+\frac{9332}{273}a^{4}+\frac{43786}{5369}a^{3}-\frac{9097}{273}a^{2}-\frac{1525}{1534}a+\frac{167}{26}$, $\frac{10670}{49722309}a^{15}-\frac{671}{3683134}a^{14}-\frac{4691}{263081}a^{13}+\frac{35930}{2367729}a^{12}+\frac{3823279}{7103187}a^{11}-\frac{3264341}{7103187}a^{10}-\frac{2527208}{338247}a^{9}+\frac{2165057}{338247}a^{8}+\frac{1918251}{37583}a^{7}-\frac{4930444}{112749}a^{6}-\frac{899602}{5369}a^{5}+\frac{326492}{2301}a^{4}+\frac{3674485}{16107}a^{3}-\frac{5952637}{32214}a^{2}-\frac{62624}{767}a+\frac{41596}{767}$, $\frac{57599}{298333854}a^{15}+\frac{12440}{49722309}a^{14}-\frac{4161}{263081}a^{13}-\frac{145478}{7103187}a^{12}+\frac{1103320}{2367729}a^{11}+\frac{4277461}{7103187}a^{10}-\frac{2095804}{338247}a^{9}-\frac{299281}{37583}a^{8}+\frac{13268314}{338247}a^{7}+\frac{5600527}{112749}a^{6}-\frac{1802239}{16107}a^{5}-\frac{2194415}{16107}a^{4}+\frac{1238871}{10738}a^{3}+\frac{1987144}{16107}a^{2}-\frac{24796}{767}a-\frac{17135}{767}$, $\frac{12937}{149166927}a^{15}-\frac{16511}{298333854}a^{14}-\frac{51440}{7103187}a^{13}+\frac{66527}{14206374}a^{12}+\frac{521524}{2367729}a^{11}-\frac{1033408}{7103187}a^{10}-\frac{348592}{112749}a^{9}+\frac{238052}{112749}a^{8}+\frac{7238107}{338247}a^{7}-\frac{5181769}{338247}a^{6}-\frac{379237}{5369}a^{5}+\frac{287089}{5369}a^{4}+\frac{492890}{5369}a^{3}-\frac{2306849}{32214}a^{2}-\frac{16739}{767}a+\frac{19439}{1534}$, $\frac{8914}{149166927}a^{15}+\frac{6658}{149166927}a^{14}-\frac{71621}{14206374}a^{13}-\frac{17467}{4735458}a^{12}+\frac{1108414}{7103187}a^{11}+\frac{784156}{7103187}a^{10}-\frac{764054}{338247}a^{9}-\frac{56887}{37583}a^{8}+\frac{5588428}{338247}a^{7}+\frac{3457057}{338247}a^{6}-\frac{979102}{16107}a^{5}-\frac{182300}{5369}a^{4}+\frac{1649275}{16107}a^{3}+\frac{836392}{16107}a^{2}-\frac{92671}{1534}a-\frac{45191}{1534}$ 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:  \( 16027729714.4 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 16027729714.4 \cdot 2}{2\cdot\sqrt{12341030701727613557145600000000}}\cr\approx \mathstrut & 0.299003467150 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481)
 
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 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481, 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 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481);
 
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 - 84*x^14 + 2583*x^12 - 37044*x^10 + 268128*x^8 - 981666*x^6 + 1694763*x^4 - 1166886*x^2 + 194481);
 
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

$C_4.C_2^3$ (as 16T20):

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 17 conjugacy class representatives for $(C_2 \times Q_8):C_2$
Character table for $(C_2 \times Q_8):C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{7}, \sqrt{15})\), 8.8.31116960000.1

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: deg 16, deg 16, deg 16, 16.16.5054886175427630513006837760000.1
Minimal sibling: 16.16.5054886175427630513006837760000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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 2.8.16.22$x^{8} + 8 x^{7} + 20 x^{6} + 12 x^{5} + 40 x^{3} + 40 x^{2} - 24 x + 36$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.16.22$x^{8} + 8 x^{7} + 20 x^{6} + 12 x^{5} + 40 x^{3} + 40 x^{2} - 24 x + 36$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.1$x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(7\) Copy content Toggle raw display 7.8.6.2$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$