↑

Properties

Label 16.0.549378366500390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.494\times 10^{17}$
Root discriminant \(12.85\)
Ramified primes $3,5,11$
Class number $1$
Class group trivial
Galois group $D_4\times C_2$ (as 16T9)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^16 - 3*y^15 - y^13 + 24*y^12 - 20*y^11 - 4*y^10 - 73*y^9 + 105*y^8 + 49*y^7 - 111*y^6 + 10*y^5 + 59*y^4 - 22*y^3 + 5*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*x + 1)
 

\( x^{16} - 3 x^{15} - x^{13} + 24 x^{12} - 20 x^{11} - 4 x^{10} - 73 x^{9} + 105 x^{8} + 49 x^{7} + \cdots + 1 \) 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:   \(549378366500390625\) \(\medspace = 3^{8}\cdot 5^{8}\cdot 11^{8}\) 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:  \(12.85\)
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:  $3^{1/2}5^{1/2}11^{1/2}\approx 12.84523257866513$
Ramified primes:   \(3\), \(5\), \(11\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{10}+\frac{1}{6}a^{9}-\frac{1}{2}a^{8}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}$, $\frac{1}{30}a^{12}+\frac{1}{3}a^{9}+\frac{1}{6}a^{8}+\frac{1}{6}a^{7}+\frac{11}{30}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a-\frac{1}{30}$, $\frac{1}{90}a^{13}+\frac{1}{18}a^{11}-\frac{1}{6}a^{10}+\frac{1}{9}a^{9}-\frac{1}{3}a^{8}+\frac{1}{90}a^{7}-\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{7}{18}a^{3}-\frac{7}{18}a^{2}-\frac{7}{30}a+\frac{1}{18}$, $\frac{1}{180}a^{14}-\frac{1}{180}a^{13}-\frac{1}{180}a^{12}+\frac{1}{18}a^{11}-\frac{1}{36}a^{10}+\frac{1}{9}a^{9}-\frac{89}{180}a^{8}-\frac{61}{180}a^{7}-\frac{19}{45}a^{6}-\frac{1}{6}a^{5}+\frac{7}{36}a^{4}+\frac{1}{6}a^{3}-\frac{4}{45}a^{2}-\frac{1}{45}a-\frac{29}{180}$, $\frac{1}{1896120}a^{15}+\frac{47}{63204}a^{14}-\frac{913}{189612}a^{13}-\frac{3127}{210680}a^{12}+\frac{31565}{379224}a^{11}+\frac{7151}{126408}a^{10}-\frac{861539}{1896120}a^{9}-\frac{7403}{15801}a^{8}-\frac{151703}{379224}a^{7}+\frac{8477}{41220}a^{6}-\frac{72325}{379224}a^{5}-\frac{183967}{379224}a^{4}-\frac{6134}{79005}a^{3}-\frac{7507}{189612}a^{2}-\frac{16309}{42136}a-\frac{266419}{632040}$ 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

Trivial group, which has order $1$

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:   \( \frac{666413}{632040} a^{15} - \frac{59375}{21068} a^{14} - \frac{94233}{105340} a^{13} - \frac{60465}{42136} a^{12} + \frac{1047315}{42136} a^{11} - \frac{546801}{42136} a^{10} - \frac{4923527}{632040} a^{9} - \frac{845483}{10534} a^{8} + \frac{17779199}{210680} a^{7} + \frac{212393}{2748} a^{6} - \frac{3709043}{42136} a^{5} - \frac{730589}{42136} a^{4} + \frac{4195684}{79005} a^{3} - \frac{132513}{21068} a^{2} + \frac{1127521}{210680} a - \frac{303229}{126408} \)  (order $6$) 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:   $\frac{1662569}{1896120}a^{15}-\frac{750259}{316020}a^{14}-\frac{628399}{948060}a^{13}-\frac{741041}{632040}a^{12}+\frac{7851337}{379224}a^{11}-\frac{484443}{42136}a^{10}-\frac{11491951}{1896120}a^{9}-\frac{1750437}{26335}a^{8}+\frac{137315617}{1896120}a^{7}+\frac{2541787}{41220}a^{6}-\frac{28766765}{379224}a^{5}-\frac{4447439}{379224}a^{4}+\frac{2364151}{52670}a^{3}-\frac{4924723}{948060}a^{2}+\frac{2799301}{632040}a-\frac{609953}{210680}$, $\frac{6631}{189612}a^{15}-\frac{59279}{948060}a^{14}-\frac{68597}{948060}a^{13}-\frac{38314}{237015}a^{12}+\frac{132151}{189612}a^{11}+\frac{22435}{94806}a^{10}+\frac{49645}{189612}a^{9}-\frac{2760809}{948060}a^{8}-\frac{29963}{237015}a^{7}+\frac{2228}{1145}a^{6}+\frac{161327}{189612}a^{5}+\frac{45169}{31602}a^{4}-\frac{43015}{47403}a^{3}+\frac{45437}{474030}a^{2}+\frac{1952057}{948060}a+\frac{2753}{79005}$, $\frac{68881}{632040}a^{15}-\frac{3982}{15801}a^{14}-\frac{13727}{94806}a^{13}-\frac{69771}{210680}a^{12}+\frac{939419}{379224}a^{11}-\frac{58097}{126408}a^{10}+\frac{22163}{1896120}a^{9}-\frac{591359}{63204}a^{8}+\frac{1969177}{379224}a^{7}+\frac{96217}{13740}a^{6}-\frac{532087}{379224}a^{5}-\frac{319843}{379224}a^{4}-\frac{28496}{237015}a^{3}-\frac{559}{189612}a^{2}+\frac{149409}{42136}a-\frac{16361}{1896120}$, $\frac{179947}{316020}a^{15}-\frac{1438387}{948060}a^{14}-\frac{88553}{189612}a^{13}-\frac{188168}{237015}a^{12}+\frac{2515973}{189612}a^{11}-\frac{335030}{47403}a^{10}-\frac{3596389}{948060}a^{9}-\frac{40194427}{948060}a^{8}+\frac{2142046}{47403}a^{7}+\frac{819943}{20610}a^{6}-\frac{9483809}{189612}a^{5}-\frac{33197}{5267}a^{4}+\frac{14723087}{474030}a^{3}-\frac{1741529}{474030}a^{2}+\frac{568853}{189612}a-\frac{203012}{237015}$, $\frac{14652}{26335}a^{15}-\frac{85247}{52670}a^{14}-\frac{14231}{158010}a^{13}-\frac{18874}{26335}a^{12}+\frac{418355}{31602}a^{11}-\frac{52607}{5267}a^{10}-\frac{96081}{52670}a^{9}-\frac{6555301}{158010}a^{8}+\frac{4281272}{79005}a^{7}+\frac{32257}{1145}a^{6}-\frac{576705}{10534}a^{5}+\frac{76988}{15801}a^{4}+\frac{2142839}{79005}a^{3}-\frac{265579}{26335}a^{2}+\frac{919391}{158010}a-\frac{294511}{158010}$, $\frac{903451}{1896120}a^{15}-\frac{294974}{237015}a^{14}-\frac{278119}{474030}a^{13}-\frac{737879}{1896120}a^{12}+\frac{4273499}{379224}a^{11}-\frac{1910699}{379224}a^{10}-\frac{12071789}{1896120}a^{9}-\frac{33278951}{948060}a^{8}+\frac{69447809}{1896120}a^{7}+\frac{207469}{4580}a^{6}-\frac{17578307}{379224}a^{5}-\frac{2212985}{126408}a^{4}+\frac{15208361}{474030}a^{3}-\frac{96359}{948060}a^{2}-\frac{5918039}{1896120}a-\frac{19349}{210680}$, $\frac{39989}{52670}a^{15}-\frac{333323}{158010}a^{14}-\frac{11366}{26335}a^{13}-\frac{52673}{52670}a^{12}+\frac{95023}{5267}a^{11}-\frac{351917}{31602}a^{10}-\frac{696803}{158010}a^{9}-\frac{4558504}{79005}a^{8}+\frac{1749464}{26335}a^{7}+\frac{334337}{6870}a^{6}-\frac{1061179}{15801}a^{5}-\frac{84944}{15801}a^{4}+\frac{3044449}{79005}a^{3}-\frac{1412087}{158010}a^{2}+\frac{834461}{158010}a-\frac{52481}{26335}$ 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:  \( 535.532332979 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 535.532332979 \cdot 1}{6\cdot\sqrt{549378366500390625}}\cr\approx \mathstrut & 0.292507911023 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*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 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*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 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*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 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*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

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{33})\), 4.0.605.1 x2, 4.0.5445.1 x2, 4.2.275.1 x2, 4.2.2475.1 x2, 8.0.741200625.1, 8.0.9150625.1, 8.0.741200625.3, 8.0.29648025.1 x2, 8.0.6125625.1 x2, 8.0.741200625.2 x2, 8.4.741200625.1 x2

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: 8.0.6125625.1, 8.0.29648025.1, 8.0.741200625.2, 8.4.741200625.1
Minimal sibling: 8.0.6125625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ R ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(3\) 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}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(11\) Copy content Toggle raw display 11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$