↑

Properties

Label 16.0.809...016.1
Degree $16$
Signature $[0, 8]$
Discriminant $8.099\times 10^{19}$
Root discriminant \(17.55\)
Ramified primes $2,7,11$
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 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16)
 
gp: K = bnfinit(y^16 + 11*y^14 + 43*y^12 + 86*y^10 + 231*y^8 + 67*y^6 - 3*y^4 - 52*y^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16)
 

\( x^{16} + 11x^{14} + 43x^{12} + 86x^{10} + 231x^{8} + 67x^{6} - 3x^{4} - 52x^{2} + 16 \) 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:   \(80985213602868822016\) \(\medspace = 2^{16}\cdot 7^{8}\cdot 11^{8}\) 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:  \(17.55\)
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:  $2\cdot 7^{1/2}11^{1/2}\approx 17.549928774784245$
Ramified primes:   \(2\), \(7\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{120}a^{12}+\frac{3}{40}a^{10}-\frac{1}{4}a^{9}+\frac{1}{15}a^{8}-\frac{23}{120}a^{6}-\frac{1}{2}a^{5}-\frac{13}{40}a^{4}+\frac{1}{4}a^{3}+\frac{7}{15}a^{2}-\frac{1}{2}a+\frac{1}{15}$, $\frac{1}{120}a^{13}-\frac{1}{20}a^{11}-\frac{7}{120}a^{9}-\frac{23}{120}a^{7}-\frac{9}{20}a^{5}+\frac{41}{120}a^{3}+\frac{1}{15}a$, $\frac{1}{2576400}a^{14}-\frac{801}{214700}a^{12}+\frac{304139}{2576400}a^{10}-\frac{67931}{2576400}a^{8}+\frac{5193}{11300}a^{6}-\frac{1}{2}a^{5}-\frac{157337}{515280}a^{4}-\frac{1}{2}a^{3}-\frac{88207}{644100}a^{2}-\frac{1}{2}a-\frac{1831}{53675}$, $\frac{1}{2576400}a^{15}-\frac{801}{214700}a^{13}-\frac{17911}{2576400}a^{11}-\frac{389981}{2576400}a^{9}-\frac{1}{4}a^{8}-\frac{457}{11300}a^{7}-\frac{221747}{515280}a^{5}-\frac{337439}{1288200}a^{3}-\frac{1}{4}a^{2}+\frac{50013}{107350}a$ 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

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:   \( \frac{1259}{33900} a^{15} + \frac{29269}{67800} a^{13} + \frac{31523}{16950} a^{11} + \frac{96649}{22600} a^{9} + \frac{739717}{67800} a^{7} + \frac{28939}{3390} a^{5} + \frac{216181}{67800} a^{3} - \frac{15038}{8475} a \)  (order $4$) 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:   $\frac{7021}{429400}a^{15}+\frac{40119}{214700}a^{13}+\frac{340179}{429400}a^{11}+\frac{785419}{429400}a^{9}+\frac{53903}{11300}a^{7}+\frac{277591}{85880}a^{5}+\frac{506151}{214700}a^{3}-\frac{185957}{107350}a$, $\frac{1231}{53675}a^{15}-\frac{36811}{2576400}a^{14}+\frac{27401}{107350}a^{13}-\frac{35639}{214700}a^{12}+\frac{54924}{53675}a^{11}-\frac{1846829}{2576400}a^{10}+\frac{230613}{107350}a^{9}-\frac{4296859}{2576400}a^{8}+\frac{16201}{2825}a^{7}-\frac{48273}{11300}a^{6}+\frac{11203}{4294}a^{5}-\frac{1689553}{515280}a^{4}+\frac{71242}{53675}a^{3}-\frac{181337}{161025}a^{2}-\frac{56473}{53675}a+\frac{38816}{53675}$, $\frac{26063}{1288200}a^{15}-\frac{549}{858800}a^{14}+\frac{284249}{1288200}a^{13}-\frac{17683}{1288200}a^{12}+\frac{273013}{322050}a^{11}-\frac{86001}{858800}a^{10}+\frac{706829}{429400}a^{9}-\frac{878083}{2576400}a^{8}+\frac{304223}{67800}a^{7}-\frac{50821}{67800}a^{6}+\frac{12625}{12882}a^{5}-\frac{287789}{171760}a^{4}-\frac{35498}{161025}a^{3}-\frac{203791}{644100}a^{2}-\frac{362651}{322050}a-\frac{13369}{161025}$, $\frac{3481}{1288200}a^{15}+\frac{271}{12882}a^{14}+\frac{44563}{1288200}a^{13}+\frac{60031}{257640}a^{12}+\frac{56531}{322050}a^{11}+\frac{239449}{257640}a^{10}+\frac{215623}{429400}a^{9}+\frac{63176}{32205}a^{8}+\frac{84601}{67800}a^{7}+\frac{72673}{13560}a^{6}+\frac{10492}{6441}a^{5}+\frac{631021}{257640}a^{4}+\frac{249224}{161025}a^{3}+\frac{13454}{10735}a^{2}+\frac{23813}{322050}a-\frac{12104}{32205}$, $\frac{3907}{429400}a^{15}-\frac{8791}{644100}a^{14}+\frac{17641}{161025}a^{13}-\frac{196151}{1288200}a^{12}+\frac{27236}{53675}a^{11}-\frac{794413}{1288200}a^{10}+\frac{425641}{322050}a^{9}-\frac{142449}{107350}a^{8}+\frac{57869}{16950}a^{7}-\frac{238817}{67800}a^{6}+\frac{79027}{21470}a^{5}-\frac{413093}{257640}a^{4}+\frac{3629167}{1288200}a^{3}-\frac{521557}{644100}a^{2}-\frac{22051}{161025}a-\frac{31208}{161025}$, $\frac{26063}{1288200}a^{15}-\frac{549}{858800}a^{14}+\frac{284249}{1288200}a^{13}-\frac{17683}{1288200}a^{12}+\frac{273013}{322050}a^{11}-\frac{86001}{858800}a^{10}+\frac{706829}{429400}a^{9}-\frac{878083}{2576400}a^{8}+\frac{304223}{67800}a^{7}-\frac{50821}{67800}a^{6}+\frac{12625}{12882}a^{5}-\frac{287789}{171760}a^{4}-\frac{35498}{161025}a^{3}-\frac{203791}{644100}a^{2}-\frac{362651}{322050}a+\frac{147656}{161025}$, $\frac{103}{7600}a^{15}+\frac{11291}{644100}a^{14}+\frac{173}{950}a^{13}+\frac{119843}{644100}a^{12}+\frac{7097}{7600}a^{11}+\frac{108241}{161025}a^{10}+\frac{18667}{7600}a^{9}+\frac{245603}{214700}a^{8}+\frac{138}{25}a^{7}+\frac{109811}{33900}a^{6}+\frac{11853}{1520}a^{5}-\frac{8359}{12882}a^{4}-\frac{91}{1900}a^{3}-\frac{284722}{161025}a^{2}-\frac{13}{950}a+\frac{128818}{161025}$ 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:  \( 8995.10035275 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8995.10035275 \cdot 1}{4\cdot\sqrt{80985213602868822016}}\cr\approx \mathstrut & 0.606990791278 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16)
 
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 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16, 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 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16);
 
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 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(i, \sqrt{77})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{7}, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{11})\), \(\Q(\sqrt{7}, \sqrt{-11})\), 4.0.2156.1 x2, 4.0.2156.2 x2, 4.2.13552.2 x2, 4.2.13552.1 x2, 8.0.8999178496.1, 8.0.74373376.3 x2, 8.0.183656704.1 x2, 8.0.562448656.1 x2, 8.4.8999178496.1 x2, 8.0.8999178496.2, 8.0.8999178496.3

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: 8.4.8999178496.1, 8.0.562448656.1, 8.0.74373376.3, 8.0.183656704.1
Minimal sibling: 8.0.74373376.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.2.0.1}{2} }^{8}$ R R ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$$2$$2$$4$$C_2^2$$[2]^{2}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$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}$