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Properties

Label 16.14.236...528.1
Degree $16$
Signature $[14, 1]$
Discriminant $-2.369\times 10^{21}$
Root discriminant \(21.67\)
Ramified primes $2,223$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{2440}.D_6$ (as 16T1759)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 1)
 
gp: K = bnfinit(y^16 - 4*y^15 - 8*y^14 + 48*y^13 + 4*y^12 - 208*y^11 + 106*y^10 + 400*y^9 - 290*y^8 - 348*y^7 + 202*y^6 + 192*y^5 + 33*y^4 - 152*y^3 - 2*y^2 + 28*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 1)
 

\( x^{16} - 4 x^{15} - 8 x^{14} + 48 x^{13} + 4 x^{12} - 208 x^{11} + 106 x^{10} + 400 x^{9} - 290 x^{8} + \cdots - 1 \) 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:  $[14, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-2368558831812663574528\) \(\medspace = -\,2^{32}\cdot 223^{5}\) 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:  \(21.67\)
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:  not computed
Ramified primes:   \(2\), \(223\) 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:  \(\Q(\sqrt{-223}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{70319}a^{15}-\frac{17367}{70319}a^{14}+\frac{15341}{70319}a^{13}+\frac{2637}{70319}a^{12}-\frac{8558}{70319}a^{11}+\frac{8299}{70319}a^{10}-\frac{11800}{70319}a^{9}-\frac{25766}{70319}a^{8}+\frac{5290}{70319}a^{7}-\frac{14004}{70319}a^{6}-\frac{11448}{70319}a^{5}-\frac{19997}{70319}a^{4}-\frac{27278}{70319}a^{3}+\frac{29297}{70319}a^{2}+\frac{3833}{70319}a-\frac{30577}{70319}$ 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

Trivial group, which has order $1$ (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:  $14$
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:   $\frac{61130}{70319}a^{15}-\frac{179405}{70319}a^{14}-\frac{682044}{70319}a^{13}+\frac{2208551}{70319}a^{12}+\frac{2624623}{70319}a^{11}-\frac{9948694}{70319}a^{10}-\frac{4220838}{70319}a^{9}+\frac{20181254}{70319}a^{8}+\frac{3707526}{70319}a^{7}-\frac{17932359}{70319}a^{6}-\frac{5908348}{70319}a^{5}+\frac{6056320}{70319}a^{4}+\frac{7494440}{70319}a^{3}-\frac{1646338}{70319}a^{2}-\frac{1890231}{70319}a-\frac{22671}{70319}$, $\frac{147749}{70319}a^{15}-\frac{368168}{70319}a^{14}-\frac{1732893}{70319}a^{13}+\frac{4476950}{70319}a^{12}+\frac{7283173}{70319}a^{11}-\frac{19743091}{70319}a^{10}-\frac{13801757}{70319}a^{9}+\frac{38353143}{70319}a^{8}+\frac{14200963}{70319}a^{7}-\frac{30388548}{70319}a^{6}-\frac{14955273}{70319}a^{5}+\frac{6455699}{70319}a^{4}+\frac{14029744}{70319}a^{3}-\frac{1641567}{70319}a^{2}-\frac{2769750}{70319}a+\frac{133939}{70319}$, $\frac{17536}{70319}a^{15}-\frac{66442}{70319}a^{14}-\frac{161356}{70319}a^{13}+\frac{816358}{70319}a^{12}+\frac{339253}{70319}a^{11}-\frac{3685654}{70319}a^{10}+\frac{586569}{70319}a^{9}+\frac{7631770}{70319}a^{8}-\frac{2305848}{70319}a^{7}-\frac{7474010}{70319}a^{6}+\frac{641488}{70319}a^{5}+\frac{3810687}{70319}a^{4}+\frac{2142719}{70319}a^{3}-\frac{1686078}{70319}a^{2}-\frac{712666}{70319}a+\frac{265379}{70319}$, $\frac{136919}{70319}a^{15}-\frac{316564}{70319}a^{14}-\frac{1641807}{70319}a^{13}+\frac{3834883}{70319}a^{12}+\frac{7145233}{70319}a^{11}-\frac{16800181}{70319}a^{10}-\frac{14269613}{70319}a^{9}+\frac{32185059}{70319}a^{8}+\frac{15204714}{70319}a^{7}-\frac{24496515}{70319}a^{6}-\frac{14172321}{70319}a^{5}+\frac{4049843}{70319}a^{4}+\frac{12141752}{70319}a^{3}-\frac{875240}{70319}a^{2}-\frac{2511654}{70319}a+\frac{10040}{70319}$, $\frac{312669}{70319}a^{15}-\frac{792533}{70319}a^{14}-\frac{3671406}{70319}a^{13}+\frac{9651581}{70319}a^{12}+\frac{15497785}{70319}a^{11}-\frac{42614702}{70319}a^{10}-\frac{29741845}{70319}a^{9}+\frac{82772882}{70319}a^{8}+\frac{31408085}{70319}a^{7}-\frac{65178897}{70319}a^{6}-\frac{33036585}{70319}a^{5}+\frac{13051226}{70319}a^{4}+\frac{30173379}{70319}a^{3}-\frac{3075516}{70319}a^{2}-\frac{5893236}{70319}a+\frac{302184}{70319}$, $\frac{334012}{70319}a^{15}-\frac{875284}{70319}a^{14}-\frac{3864664}{70319}a^{13}+\frac{10662338}{70319}a^{12}+\frac{15884748}{70319}a^{11}-\frac{47123122}{70319}a^{10}-\frac{29003397}{70319}a^{9}+\frac{91885194}{70319}a^{8}+\frac{28848757}{70319}a^{7}-\frac{73719118}{70319}a^{6}-\frac{32872086}{70319}a^{5}+\frac{17240406}{70319}a^{4}+\frac{33103424}{70319}a^{3}-\frac{4894487}{70319}a^{2}-\frac{6640023}{70319}a+\frac{327912}{70319}$, $\frac{95281}{70319}a^{15}-\frac{279695}{70319}a^{14}-\frac{1070017}{70319}a^{13}+\frac{3451841}{70319}a^{12}+\frac{4153147}{70319}a^{11}-\frac{15540635}{70319}a^{10}-\frac{6735933}{70319}a^{9}+\frac{31188318}{70319}a^{8}+\frac{5826375}{70319}a^{7}-\frac{26522362}{70319}a^{6}-\frac{8919073}{70319}a^{5}+\frac{7623919}{70319}a^{4}+\frac{11095843}{70319}a^{3}-\frac{2396732}{70319}a^{2}-\frac{2345340}{70319}a+\frac{190309}{70319}$, $\frac{283866}{70319}a^{15}-\frac{679560}{70319}a^{14}-\frac{3372357}{70319}a^{13}+\frac{8236210}{70319}a^{12}+\frac{14470979}{70319}a^{11}-\frac{36096851}{70319}a^{10}-\frac{28311792}{70319}a^{9}+\frac{69192687}{70319}a^{8}+\frac{29733832}{70319}a^{7}-\frac{52795325}{70319}a^{6}-\frac{28947130}{70319}a^{5}+\frac{9033705}{70319}a^{4}+\frac{25476053}{70319}a^{3}-\frac{2174860}{70319}a^{2}-\frac{4698882}{70319}a+\frac{125402}{70319}$, $\frac{8680}{3701}a^{15}-\frac{22335}{3701}a^{14}-\frac{102027}{3701}a^{13}+\frac{272349}{3701}a^{12}+\frac{432348}{3701}a^{11}-\frac{1203769}{3701}a^{10}-\frac{838952}{3701}a^{9}+\frac{2337881}{3701}a^{8}+\frac{905638}{3701}a^{7}-\frac{1831071}{3701}a^{6}-\frac{951648}{3701}a^{5}+\frac{347133}{3701}a^{4}+\frac{849665}{3701}a^{3}-\frac{75471}{3701}a^{2}-\frac{164394}{3701}a+\frac{8855}{3701}$, $\frac{93325}{70319}a^{15}-\frac{203601}{70319}a^{14}-\frac{1121119}{70319}a^{13}+\frac{2442690}{70319}a^{12}+\frac{4859863}{70319}a^{11}-\frac{10537141}{70319}a^{10}-\frac{9462206}{70319}a^{9}+\frac{19635575}{70319}a^{8}+\frac{9191340}{70319}a^{7}-\frac{14038166}{70319}a^{6}-\frac{7622485}{70319}a^{5}+\frac{1804210}{70319}a^{4}+\frac{6719712}{70319}a^{3}-\frac{844661}{70319}a^{2}-\frac{1123132}{70319}a+\frac{87133}{70319}$, $\frac{120619}{70319}a^{15}-\frac{338758}{70319}a^{14}-\frac{1364467}{70319}a^{13}+\frac{4168287}{70319}a^{12}+\frac{5369762}{70319}a^{11}-\frac{18749057}{70319}a^{10}-\frac{8978153}{70319}a^{9}+\frac{37851311}{70319}a^{8}+\frac{8156908}{70319}a^{7}-\frac{33065707}{70319}a^{6}-\frac{11946339}{70319}a^{5}+\frac{10330769}{70319}a^{4}+\frac{14607280}{70319}a^{3}-\frac{2567667}{70319}a^{2}-\frac{3460429}{70319}a-\frac{5932}{70319}$, $\frac{58791}{70319}a^{15}-\frac{132055}{70319}a^{14}-\frac{701953}{70319}a^{13}+\frac{1595809}{70319}a^{12}+\frac{3022784}{70319}a^{11}-\frac{6998613}{70319}a^{10}-\frac{5873342}{70319}a^{9}+\frac{13574559}{70319}a^{8}+\frac{5890249}{70319}a^{7}-\frac{10913757}{70319}a^{6}-\frac{5501101}{70319}a^{5}+\frac{2621537}{70319}a^{4}+\frac{5338460}{70319}a^{3}-\frac{696849}{70319}a^{2}-\frac{1292234}{70319}a+\frac{52828}{70319}$, $\frac{24572}{70319}a^{15}-\frac{46232}{70319}a^{14}-\frac{302383}{70319}a^{13}+\frac{524798}{70319}a^{12}+\frac{1373014}{70319}a^{11}-\frac{2041323}{70319}a^{10}-\frac{2907442}{70319}a^{9}+\frac{2983522}{70319}a^{8}+\frac{3060085}{70319}a^{7}-\frac{527654}{70319}a^{6}-\frac{1782231}{70319}a^{5}-\frac{1453811}{70319}a^{4}+\frac{568244}{70319}a^{3}+\frac{100600}{70319}a^{2}+\frac{238292}{70319}a-\frac{49848}{70319}$, $\frac{312669}{70319}a^{15}-\frac{792533}{70319}a^{14}-\frac{3671406}{70319}a^{13}+\frac{9651581}{70319}a^{12}+\frac{15497785}{70319}a^{11}-\frac{42614702}{70319}a^{10}-\frac{29741845}{70319}a^{9}+\frac{82772882}{70319}a^{8}+\frac{31408085}{70319}a^{7}-\frac{65178897}{70319}a^{6}-\frac{33036585}{70319}a^{5}+\frac{13051226}{70319}a^{4}+\frac{30173379}{70319}a^{3}-\frac{3075516}{70319}a^{2}-\frac{5893236}{70319}a+\frac{231865}{70319}$ 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:  \( 143953.377239 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 143953.377239 \cdot 1}{2\cdot\sqrt{2368558831812663574528}}\cr\approx \mathstrut & 0.152247293846 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 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 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 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 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 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 - 4*x^15 - 8*x^14 + 48*x^13 + 4*x^12 - 208*x^11 + 106*x^10 + 400*x^9 - 290*x^8 - 348*x^7 + 202*x^6 + 192*x^5 + 33*x^4 - 152*x^3 - 2*x^2 + 28*x - 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

$C_{2440}.D_6$ (as 16T1759):

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 12288
The 93 conjugacy class representatives for $C_{2440}.D_6$
Character table for $C_{2440}.D_6$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14272.1, 8.8.3259039744.1

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: 16.12.33011788718388998569984.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }$

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 Deg $16$$4$$4$$32$
\(223\) Copy content Toggle raw display $\Q_{223}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{223}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{223}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{223}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$4$$1$$3$