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Properties

Label 16.0.206...016.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.068\times 10^{21}$
Root discriminant \(21.49\)
Ramified primes $2,7,17$
Class number $4$
Class group [4]
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4)
 
gp: K = bnfinit(y^16 + 28*y^14 + 282*y^12 + 1296*y^10 + 2917*y^8 + 3228*y^6 + 1616*y^4 + 296*y^2 + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4)
 

\( x^{16} + 28x^{14} + 282x^{12} + 1296x^{10} + 2917x^{8} + 3228x^{6} + 1616x^{4} + 296x^{2} + 4 \) 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:   \(2067949204473187926016\) \(\medspace = 2^{32}\cdot 7^{8}\cdot 17^{4}\) 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:  \(21.49\)
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:  $2^{2}7^{1/2}17^{1/2}\approx 43.634848458542855$
Ramified primes:   \(2\), \(7\), \(17\) 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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{3}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{2}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{16}a^{11}+\frac{7}{16}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{160}a^{12}+\frac{1}{20}a^{10}+\frac{1}{32}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{20}a^{4}-\frac{9}{20}a^{2}-\frac{1}{2}a-\frac{17}{40}$, $\frac{1}{160}a^{13}-\frac{1}{80}a^{11}+\frac{1}{32}a^{9}+\frac{5}{16}a^{7}+\frac{1}{5}a^{5}-\frac{13}{40}a^{3}+\frac{13}{40}a-\frac{1}{2}$, $\frac{1}{25120}a^{14}-\frac{3}{25120}a^{12}+\frac{1317}{25120}a^{10}-\frac{119}{5024}a^{8}-\frac{1}{2}a^{7}+\frac{226}{785}a^{6}-\frac{1}{2}a^{5}-\frac{291}{6280}a^{4}-\frac{309}{6280}a^{2}-\frac{1023}{6280}$, $\frac{1}{25120}a^{15}-\frac{3}{25120}a^{13}-\frac{253}{25120}a^{11}-\frac{119}{5024}a^{9}-\frac{1879}{12560}a^{7}+\frac{1279}{6280}a^{5}+\frac{119}{1570}a^{3}-\frac{1}{2}a^{2}-\frac{2593}{6280}a-\frac{1}{2}$ 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

$C_{4}$, which has order $4$

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:   \( \frac{1}{6280} a^{15} + \frac{1}{5024} a^{14} - \frac{4}{157} a^{13} - \frac{3}{5024} a^{12} - \frac{4649}{6280} a^{11} - \frac{567}{5024} a^{10} - \frac{8597}{1256} a^{9} - \frac{6875}{5024} a^{8} - \frac{20291}{785} a^{7} - \frac{7769}{1256} a^{6} - \frac{253463}{6280} a^{5} - \frac{14421}{1256} a^{4} - \frac{78961}{3140} a^{3} - \frac{9415}{1256} a^{2} - \frac{19473}{3140} a - \frac{1023}{1256} \)  (order $8$) 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:   $\frac{539}{5024}a^{15}-\frac{151}{3140}a^{14}+\frac{37327}{12560}a^{13}-\frac{33271}{25120}a^{12}+\frac{738387}{25120}a^{11}-\frac{81379}{6280}a^{10}+\frac{329095}{2512}a^{9}-\frac{283759}{5024}a^{8}+\frac{176239}{628}a^{7}-\frac{723179}{6280}a^{6}+\frac{905561}{3140}a^{5}-\frac{82444}{785}a^{4}+\frac{806953}{6280}a^{3}-\frac{26346}{785}a^{2}+\frac{60491}{3140}a-\frac{8441}{6280}$, $\frac{31}{2512}a^{15}+\frac{1}{5024}a^{14}+\frac{334}{785}a^{13}-\frac{3}{5024}a^{12}+\frac{70057}{12560}a^{11}-\frac{567}{5024}a^{10}+\frac{43765}{1256}a^{9}-\frac{6875}{5024}a^{8}+\frac{133605}{1256}a^{7}-\frac{7769}{1256}a^{6}+\frac{932959}{6280}a^{5}-\frac{14421}{1256}a^{4}+\frac{263593}{3140}a^{3}-\frac{9415}{1256}a^{2}+\frac{48047}{3140}a-\frac{2279}{1256}$, $\frac{3491}{25120}a^{15}-\frac{703}{12560}a^{14}+\frac{95659}{25120}a^{13}-\frac{485}{314}a^{12}+\frac{928243}{25120}a^{11}-\frac{190463}{12560}a^{10}+\frac{799123}{5024}a^{9}-\frac{10442}{157}a^{8}+\frac{2007573}{6280}a^{7}-\frac{430289}{3140}a^{6}+\frac{183257}{628}a^{5}-\frac{198447}{1570}a^{4}+\frac{125009}{1256}a^{3}-\frac{122521}{3140}a^{2}+\frac{36253}{6280}a-\frac{12}{785}$, $\frac{787}{25120}a^{15}+\frac{423}{5024}a^{14}+\frac{1083}{1256}a^{13}+\frac{5881}{2512}a^{12}+\frac{211287}{25120}a^{11}+\frac{116863}{5024}a^{10}+\frac{45549}{1256}a^{9}+\frac{260927}{2512}a^{8}+\frac{111589}{1570}a^{7}+\frac{275101}{1256}a^{6}+\frac{43377}{785}a^{5}+\frac{259673}{1256}a^{4}+\frac{23089}{6280}a^{3}+\frac{82813}{1256}a^{2}-\frac{10167}{1570}a+\frac{187}{314}$, $\frac{473}{25120}a^{15}-\frac{359}{5024}a^{14}+\frac{2903}{6280}a^{13}-\frac{4721}{2512}a^{12}+\frac{89769}{25120}a^{11}-\frac{85327}{5024}a^{10}+\frac{10381}{1256}a^{9}-\frac{161903}{2512}a^{8}-\frac{59341}{6280}a^{7}-\frac{132385}{1256}a^{6}-\frac{8312}{157}a^{5}-\frac{96805}{1256}a^{4}-\frac{69235}{1256}a^{3}-\frac{36021}{1256}a^{2}-\frac{12227}{785}a-\frac{1483}{314}$, $\frac{208}{785}a^{15}-\frac{1571}{25120}a^{14}+\frac{18319}{2512}a^{13}-\frac{8917}{5024}a^{12}+\frac{447689}{6280}a^{11}-\frac{456931}{25120}a^{10}+\frac{780369}{2512}a^{9}-\frac{426293}{5024}a^{8}+\frac{3996433}{6280}a^{7}-\frac{1178523}{6280}a^{6}+\frac{3746033}{6280}a^{5}-\frac{1128853}{6280}a^{4}+\frac{169324}{785}a^{3}-\frac{314947}{6280}a^{2}+\frac{31479}{1570}a+\frac{8559}{6280}$, $\frac{1037}{6280}a^{15}+\frac{687}{2512}a^{14}+\frac{112057}{25120}a^{13}+\frac{181763}{25120}a^{12}+\frac{66329}{1570}a^{11}+\frac{829057}{12560}a^{10}+\frac{876213}{5024}a^{9}+\frac{1275123}{5024}a^{8}+\frac{2060319}{6280}a^{7}+\frac{523419}{1256}a^{6}+\frac{348875}{1256}a^{5}+\frac{872797}{3140}a^{4}+\frac{60635}{628}a^{3}+\frac{215623}{3140}a^{2}+\frac{75509}{6280}a+\frac{25529}{6280}$ 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:  \( 87872.62030226529 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 87872.62030226529 \cdot 4}{8\cdot\sqrt{2067949204473187926016}}\cr\approx \mathstrut & 2.34688924489768 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4)
 
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 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4, 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 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4);
 
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 + 28*x^14 + 282*x^12 + 1296*x^10 + 2917*x^8 + 3228*x^6 + 1616*x^4 + 296*x^2 + 4);
 
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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{14}) \), 4.0.213248.3, 4.4.4352.1, 4.4.53312.1, 4.0.1088.2, \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(i, \sqrt{14})\), 8.0.157351936.1, 8.0.45474709504.20, 8.0.2842169344.2, 8.0.45474709504.3, 8.0.18939904.2, 8.0.45474709504.2, 8.8.45474709504.1

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, 16.0.674676505885957534253056.3, deg 16, deg 16, deg 16, deg 16, deg 16
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
\(7\) 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}$
\(17\) Copy content Toggle raw display 17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$