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Properties

Label 16.0.560...656.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.606\times 10^{19}$
Root discriminant \(17.15\)
Ramified primes $2,13$
Class number $3$
Class group [3]
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 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 8*y^15 + 20*y^14 - 56*y^12 - 28*y^11 + 318*y^10 - 380*y^9 - 21*y^8 + 340*y^7 - 174*y^6 - 136*y^5 + 226*y^4 - 144*y^3 + 50*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*x + 1)
 

\( x^{16} - 8 x^{15} + 20 x^{14} - 56 x^{12} - 28 x^{11} + 318 x^{10} - 380 x^{9} - 21 x^{8} + 340 x^{7} + \cdots + 1 \) 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:   \(56056588304600006656\) \(\medspace = 2^{36}\cdot 13^{8}\) 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:  \(17.15\)
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:  $2^{9/4}13^{1/2}\approx 17.150988921155648$
Ramified primes:   \(2\), \(13\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{81}a^{12}-\frac{2}{27}a^{11}-\frac{5}{81}a^{10}-\frac{1}{81}a^{9}-\frac{10}{81}a^{8}+\frac{7}{81}a^{7}+\frac{40}{81}a^{6}-\frac{26}{81}a^{5}-\frac{11}{27}a^{4}+\frac{19}{81}a^{3}-\frac{2}{27}a^{2}-\frac{34}{81}a+\frac{26}{81}$, $\frac{1}{405}a^{13}+\frac{1}{405}a^{12}+\frac{34}{405}a^{11}-\frac{7}{45}a^{10}-\frac{17}{405}a^{9}+\frac{2}{45}a^{8}+\frac{62}{405}a^{7}+\frac{146}{405}a^{6}+\frac{1}{405}a^{5}+\frac{166}{405}a^{4}+\frac{181}{405}a^{3}-\frac{76}{405}a^{2}-\frac{77}{405}a+\frac{74}{405}$, $\frac{1}{405}a^{14}-\frac{2}{405}a^{12}-\frac{22}{405}a^{11}-\frac{49}{405}a^{10}-\frac{13}{81}a^{9}-\frac{11}{405}a^{8}-\frac{26}{405}a^{7}-\frac{13}{27}a^{6}-\frac{1}{81}a^{5}+\frac{2}{9}a^{4}+\frac{158}{405}a^{3}-\frac{196}{405}a^{2}-\frac{1}{45}a-\frac{13}{135}$, $\frac{1}{6075}a^{15}+\frac{1}{1215}a^{13}+\frac{2}{1215}a^{12}+\frac{283}{2025}a^{11}-\frac{901}{6075}a^{10}-\frac{58}{1215}a^{9}-\frac{1}{405}a^{8}-\frac{74}{675}a^{7}-\frac{2708}{6075}a^{6}+\frac{1607}{6075}a^{5}+\frac{32}{135}a^{4}+\frac{466}{6075}a^{3}-\frac{691}{6075}a^{2}+\frac{964}{2025}a-\frac{452}{6075}$ 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:  $3$

Class group and class number

$C_{3}$, which has order $3$

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:   \( \frac{1046}{6075} a^{15} - \frac{523}{405} a^{14} + \frac{3491}{1215} a^{13} + \frac{221}{243} a^{12} - \frac{16627}{2025} a^{11} - \frac{48191}{6075} a^{10} + \frac{57757}{1215} a^{9} - \frac{6356}{135} a^{8} - \frac{5749}{675} a^{7} + \frac{271682}{6075} a^{6} - \frac{118928}{6075} a^{5} - \frac{8149}{405} a^{4} + \frac{182426}{6075} a^{3} - \frac{105491}{6075} a^{2} + \frac{1531}{225} a - \frac{8962}{6075} \)  (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:   $\frac{1669}{6075}a^{15}-\frac{782}{405}a^{14}+\frac{4243}{1215}a^{13}+\frac{5339}{1215}a^{12}-\frac{27493}{2025}a^{11}-\frac{127909}{6075}a^{10}+\frac{18019}{243}a^{9}-\frac{3881}{135}a^{8}-\frac{150028}{2025}a^{7}+\frac{414193}{6075}a^{6}+\frac{161603}{6075}a^{5}-\frac{2737}{45}a^{4}+\frac{140134}{6075}a^{3}+\frac{40106}{6075}a^{2}-\frac{16649}{2025}a+\frac{13537}{6075}$, $\frac{382}{2025}a^{15}-\frac{139}{81}a^{14}+\frac{2054}{405}a^{13}-\frac{704}{405}a^{12}-\frac{1096}{75}a^{11}+\frac{646}{675}a^{10}+\frac{6413}{81}a^{9}-\frac{8893}{81}a^{8}-\frac{2714}{675}a^{7}+\frac{203429}{2025}a^{6}-\frac{11474}{225}a^{5}-\frac{17432}{405}a^{4}+\frac{129247}{2025}a^{3}-\frac{73547}{2025}a^{2}+\frac{19049}{2025}a-\frac{649}{2025}$, $\frac{1414}{2025}a^{15}-\frac{1961}{405}a^{14}+\frac{3548}{405}a^{13}+\frac{3869}{405}a^{12}-\frac{19483}{675}a^{11}-\frac{105154}{2025}a^{10}+\frac{4510}{27}a^{9}-\frac{10873}{135}a^{8}-\frac{209554}{2025}a^{7}+\frac{226633}{2025}a^{6}+\frac{20693}{2025}a^{5}-\frac{30494}{405}a^{4}+\frac{14656}{225}a^{3}-\frac{64739}{2025}a^{2}+\frac{4681}{675}a-\frac{776}{675}$, $\frac{181}{405}a^{15}-\frac{29}{9}a^{14}+\frac{2578}{405}a^{13}+\frac{716}{135}a^{12}-\frac{8824}{405}a^{11}-\frac{2348}{81}a^{10}+\frac{5476}{45}a^{9}-\frac{10237}{135}a^{8}-\frac{6691}{81}a^{7}+\frac{9145}{81}a^{6}+\frac{569}{81}a^{5}-\frac{32782}{405}a^{4}+\frac{7168}{135}a^{3}-\frac{1453}{135}a^{2}-\frac{979}{405}a+\frac{61}{81}$, $\frac{1591}{6075}a^{15}-\frac{268}{135}a^{14}+\frac{5221}{1215}a^{13}+\frac{3056}{1215}a^{12}-\frac{9919}{675}a^{11}-\frac{94801}{6075}a^{10}+\frac{98447}{1215}a^{9}-\frac{7474}{135}a^{8}-\frac{108007}{2025}a^{7}+\frac{408847}{6075}a^{6}+\frac{61037}{6075}a^{5}-\frac{16559}{405}a^{4}+\frac{153751}{6075}a^{3}-\frac{102646}{6075}a^{2}+\frac{12779}{2025}a+\frac{5233}{6075}$, $\frac{13}{2025}a^{15}-\frac{59}{405}a^{14}+\frac{113}{135}a^{13}-\frac{119}{81}a^{12}-\frac{3103}{2025}a^{11}+\frac{1153}{225}a^{10}+\frac{3599}{405}a^{9}-\frac{12842}{405}a^{8}+\frac{29047}{2025}a^{7}+\frac{17342}{675}a^{6}-\frac{39379}{2025}a^{5}-\frac{983}{81}a^{4}+\frac{12401}{675}a^{3}-\frac{14743}{2025}a^{2}+\frac{4666}{2025}a-\frac{467}{675}$, $\frac{637}{2025}a^{15}-\frac{914}{405}a^{14}+\frac{616}{135}a^{13}+\frac{371}{135}a^{12}-\frac{26752}{2025}a^{11}-\frac{36547}{2025}a^{10}+\frac{31267}{405}a^{9}-\frac{8783}{135}a^{8}-\frac{1306}{75}a^{7}+\frac{141884}{2025}a^{6}-\frac{67736}{2025}a^{5}-\frac{15268}{405}a^{4}+\frac{106187}{2025}a^{3}-\frac{53077}{2025}a^{2}+\frac{14399}{2025}a+\frac{1}{2025}$ 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:  \( 4427.02461923 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4427.02461923 \cdot 3}{8\cdot\sqrt{56056588304600006656}}\cr\approx \mathstrut & 0.538603065081 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*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 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*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 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*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 - 8*x^15 + 20*x^14 - 56*x^12 - 28*x^11 + 318*x^10 - 380*x^9 - 21*x^8 + 340*x^7 - 174*x^6 - 136*x^5 + 226*x^4 - 144*x^3 + 50*x^2 - 8*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

\(\Q(\sqrt{-13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(i, \sqrt{26})\), \(\Q(\sqrt{-2}, \sqrt{13})\), 4.0.832.1 x2, 4.0.3328.1 x2, 4.2.2704.1 x2, 4.2.10816.1 x2, 8.0.1871773696.1, 8.0.116985856.1, 8.0.1871773696.3, 8.0.44302336.1 x2, 8.4.1871773696.1 x2, 8.0.7487094784.1 x2, 8.0.1871773696.4 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: 8.0.44302336.1, 8.0.7487094784.1, 8.4.1871773696.1, 8.0.1871773696.4
Minimal sibling: 8.0.44302336.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{8}$ ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 Deg $16$$8$$2$$36$
\(13\) Copy content Toggle raw display 13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$