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Properties

Label 16.0.897...896.1
Degree $16$
Signature $[0, 8]$
Discriminant $8.979\times 10^{19}$
Root discriminant \(17.66\)
Ramified primes $2,3,13$
Class number $2$
Class group [2]
Galois group $D_4\times C_2$ (as 16T9)

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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 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 28*y^13 + 53*y^12 - 2*y^11 - 24*y^10 + 76*y^9 - 108*y^8 - 10*y^7 + 50*y^6 - 146*y^5 + 261*y^4 - 88*y^3 + 18*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 8*x^14 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*x + 1)
 

\( x^{16} - 4 x^{15} + 8 x^{14} - 28 x^{13} + 53 x^{12} - 2 x^{11} - 24 x^{10} + 76 x^{9} - 108 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(89791815397090000896\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 13^{8}\) 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:  \(17.66\)
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:  $2^{3/2}3^{1/2}13^{1/2}\approx 17.663521732655695$
Ramified primes:   \(2\), \(3\), \(13\) 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 over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{18}a^{13}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{4}{9}a^{8}-\frac{2}{9}a^{6}-\frac{1}{3}a^{5}-\frac{2}{9}a^{4}-\frac{1}{3}a^{3}+\frac{2}{9}a^{2}-\frac{1}{6}a-\frac{2}{9}$, $\frac{1}{413838}a^{14}+\frac{3457}{413838}a^{13}+\frac{27437}{413838}a^{12}+\frac{7478}{206919}a^{11}-\frac{1022}{22991}a^{10}+\frac{78547}{206919}a^{9}+\frac{96209}{206919}a^{8}-\frac{77177}{206919}a^{7}-\frac{20}{747}a^{6}-\frac{62153}{206919}a^{5}+\frac{22204}{206919}a^{4}-\frac{14653}{206919}a^{3}-\frac{111929}{413838}a^{2}-\frac{31345}{413838}a+\frac{183713}{413838}$, $\frac{1}{23767130178}a^{15}+\frac{721}{3961188363}a^{14}-\frac{96846296}{3961188363}a^{13}+\frac{1938590}{516676743}a^{12}-\frac{49995854}{3961188363}a^{11}+\frac{506781079}{3961188363}a^{10}+\frac{5304277910}{11883565089}a^{9}-\frac{1633494509}{3961188363}a^{8}-\frac{705144169}{3961188363}a^{7}+\frac{3446003714}{11883565089}a^{6}+\frac{515716823}{3961188363}a^{5}-\frac{28427000}{172225581}a^{4}+\frac{1094673863}{7922376726}a^{3}+\frac{888150038}{3961188363}a^{2}+\frac{1443280070}{3961188363}a-\frac{3445329254}{11883565089}$ 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:  $2$, $3$

Class group and class number

$C_{2}$, which has order $2$

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:   \( -\frac{4957783102}{11883565089} a^{15} + \frac{12636334541}{7922376726} a^{14} - \frac{36188101511}{11883565089} a^{13} + \frac{5735328002}{516676743} a^{12} - \frac{238439687029}{11883565089} a^{11} - \frac{35704621015}{11883565089} a^{10} + \frac{39857355037}{3961188363} a^{9} - \frac{356044289906}{11883565089} a^{8} + \frac{156446339762}{3961188363} a^{7} + \frac{138249837940}{11883565089} a^{6} - \frac{79132239821}{3961188363} a^{5} + \frac{29600926387}{516676743} a^{4} - \frac{129779108649}{1320396121} a^{3} + \frac{439208418887}{23767130178} a^{2} - \frac{5497296557}{3961188363} a + \frac{4935594205}{3961188363} \)  (order $12$) 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:   $\frac{112757941}{2640792242}a^{15}-\frac{402988255}{3961188363}a^{14}+\frac{1522932403}{23767130178}a^{13}-\frac{221087545}{344451162}a^{12}+\frac{4007776093}{11883565089}a^{11}+\frac{42494421289}{11883565089}a^{10}-\frac{13431657322}{11883565089}a^{9}+\frac{16205112386}{11883565089}a^{8}+\frac{2568137894}{3961188363}a^{7}-\frac{89767477868}{11883565089}a^{6}+\frac{5375951560}{3961188363}a^{5}-\frac{1438737370}{516676743}a^{4}+\frac{8932451929}{7922376726}a^{3}+\frac{169526352878}{11883565089}a^{2}-\frac{42086212747}{7922376726}a+\frac{23451730781}{23767130178}$, $\frac{1889258815}{23767130178}a^{15}-\frac{3062199838}{11883565089}a^{14}+\frac{9672480241}{23767130178}a^{13}-\frac{1852606441}{1033353486}a^{12}+\frac{31127424926}{11883565089}a^{11}+\frac{10695631405}{3961188363}a^{10}-\frac{5440426787}{3961188363}a^{9}+\frac{55086805490}{11883565089}a^{8}-\frac{51382024115}{11883565089}a^{7}-\frac{8827425264}{1320396121}a^{6}+\frac{20223988153}{11883565089}a^{5}-\frac{4601766001}{516676743}a^{4}+\frac{284296829143}{23767130178}a^{3}+\frac{75426798893}{11883565089}a^{2}-\frac{18494027407}{23767130178}a-\frac{3618907709}{23767130178}$, $\frac{248229}{114817054}a^{15}-\frac{832691}{114817054}a^{14}+\frac{309292}{57408527}a^{13}-\frac{2970949}{114817054}a^{12}+\frac{2047378}{57408527}a^{11}+\frac{13041002}{57408527}a^{10}-\frac{20291830}{57408527}a^{9}+\frac{1427895}{57408527}a^{8}-\frac{953219}{57408527}a^{7}-\frac{68349}{691669}a^{6}+\frac{41693827}{57408527}a^{5}-\frac{38893443}{57408527}a^{4}+\frac{23479759}{114817054}a^{3}-\frac{5382555}{114817054}a^{2}-\frac{188806169}{57408527}a-\frac{227955}{114817054}$, $\frac{650656559}{7922376726}a^{15}-\frac{1621004299}{7922376726}a^{14}+\frac{4201985587}{23767130178}a^{13}-\frac{233818504}{172225581}a^{12}+\frac{11499537157}{11883565089}a^{11}+\frac{71675868703}{11883565089}a^{10}-\frac{19934028097}{11883565089}a^{9}+\frac{43044926528}{11883565089}a^{8}+\frac{829490749}{1320396121}a^{7}-\frac{156503895755}{11883565089}a^{6}+\frac{7554379480}{3961188363}a^{5}-\frac{3439937893}{516676743}a^{4}+\frac{21157643095}{7922376726}a^{3}+\frac{536186624803}{23767130178}a^{2}-\frac{53933944211}{7922376726}a+\frac{18534121702}{11883565089}$, $\frac{111539083}{3961188363}a^{15}-\frac{2252380177}{11883565089}a^{14}+\frac{4068688843}{7922376726}a^{13}-\frac{681205447}{516676743}a^{12}+\frac{41238659515}{11883565089}a^{11}-\frac{42316205581}{11883565089}a^{10}-\frac{20250871723}{11883565089}a^{9}+\frac{16479634739}{3961188363}a^{8}-\frac{95425021052}{11883565089}a^{7}+\frac{25325429236}{3961188363}a^{6}+\frac{64179865813}{11883565089}a^{5}-\frac{1371465244}{172225581}a^{4}+\frac{207589707818}{11883565089}a^{3}-\frac{25612297066}{1320396121}a^{2}+\frac{200186215}{286350966}a+\frac{1175562703}{1320396121}$, $\frac{9547063}{1320396121}a^{15}-\frac{331947361}{23767130178}a^{14}+\frac{54909299}{2640792242}a^{13}-\frac{171704695}{1033353486}a^{12}+\frac{1408925900}{11883565089}a^{11}+\frac{2310142366}{11883565089}a^{10}+\frac{9422459803}{11883565089}a^{9}+\frac{667073702}{1320396121}a^{8}-\frac{1735788304}{11883565089}a^{7}+\frac{2747527}{47725161}a^{6}-\frac{19730097628}{11883565089}a^{5}-\frac{257468429}{172225581}a^{4}+\frac{7957459870}{11883565089}a^{3}-\frac{819943237}{7922376726}a^{2}+\frac{26635980925}{23767130178}a-\frac{79831337}{7922376726}$, $\frac{3898817555}{11883565089}a^{15}-\frac{4845145090}{3961188363}a^{14}+\frac{53439334373}{23767130178}a^{13}-\frac{8677375037}{1033353486}a^{12}+\frac{175270975397}{11883565089}a^{11}+\frac{54509648573}{11883565089}a^{10}-\frac{35691325682}{3961188363}a^{9}+\frac{269038985188}{11883565089}a^{8}-\frac{111416880403}{3961188363}a^{7}-\frac{166928745149}{11883565089}a^{6}+\frac{68705871658}{3961188363}a^{5}-\frac{22173712211}{516676743}a^{4}+\frac{94624063869}{1320396121}a^{3}-\frac{35780913920}{11883565089}a^{2}-\frac{54994235063}{7922376726}a+\frac{10302822025}{7922376726}$ Copy content Toggle raw display
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:  \( 14943.0195941 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 14943.0195941 \cdot 2}{12\cdot\sqrt{89791815397090000896}}\cr\approx \mathstrut & 0.638421767674 \end{aligned}\]

# 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 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*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 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*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 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*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 - 28*x^13 + 53*x^12 - 2*x^11 - 24*x^10 + 76*x^9 - 108*x^8 - 10*x^7 + 50*x^6 - 146*x^5 + 261*x^4 - 88*x^3 + 18*x^2 - 6*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

$C_2\times D_4$ (as 16T9):

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 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(i, \sqrt{39})\), \(\Q(i, \sqrt{13})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), \(\Q(\sqrt{3}, \sqrt{13})\), 4.0.8112.1 x2, 4.4.8112.1 x2, 4.4.7488.1 x2, 4.0.7488.1 x2, 8.0.592240896.1, 8.0.1052872704.3 x2, 8.0.56070144.2 x2, 8.0.9475854336.6 x2, 8.0.592240896.6 x2, 8.8.9475854336.1, 8.0.9475854336.3

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 8 siblings: 8.0.1052872704.3, 8.0.56070144.2, 8.0.592240896.6, 8.0.9475854336.6
Minimal sibling: 8.0.56070144.2

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$$4$$2$$12$$D_4$$[2, 2]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(13\) Copy content Toggle raw display 13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} + 13$$2$$1$$1$$C_2$$[\ ]_{2}$