Properties

Label 16.0.101...216.4
Degree $16$
Signature $[0, 8]$
Discriminant $1.014\times 10^{20}$
Root discriminant \(17.80\)
Ramified primes $2,7$
Class number $2$
Class group [2]
Galois group $C_2^2\wr C_2$ (as 16T39)

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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 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14)
 
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 84*y^13 + 166*y^12 - 268*y^11 + 428*y^10 - 732*y^9 + 1279*y^8 - 1956*y^7 + 2456*y^6 - 2432*y^5 + 1866*y^4 - 1072*y^3 + 436*y^2 - 112*y + 14, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14)
 

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 166 x^{12} - 268 x^{11} + 428 x^{10} - 732 x^{9} + 1279 x^{8} - 1956 x^{7} + 2456 x^{6} - 2432 x^{5} + 1866 x^{4} - 1072 x^{3} + \cdots + 14 \) 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:   \(101415451701035401216\) \(\medspace = 2^{44}\cdot 7^{8}\) 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:  \(17.80\)
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:  $2^{11/4}7^{1/2}\approx 17.798422345016238$
Ramified primes:   \(2\), \(7\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2381}a^{14}-\frac{7}{2381}a^{13}-\frac{1045}{2381}a^{12}-\frac{782}{2381}a^{11}+\frac{845}{2381}a^{10}-\frac{795}{2381}a^{9}+\frac{263}{2381}a^{8}+\frac{164}{2381}a^{7}+\frac{991}{2381}a^{6}-\frac{251}{2381}a^{5}-\frac{1001}{2381}a^{4}+\frac{846}{2381}a^{3}-\frac{49}{2381}a^{2}+\frac{820}{2381}a-\frac{1077}{2381}$, $\frac{1}{473819}a^{15}+\frac{92}{473819}a^{14}-\frac{77930}{473819}a^{13}-\frac{75665}{473819}a^{12}+\frac{125812}{473819}a^{11}+\frac{192386}{473819}a^{10}+\frac{76323}{473819}a^{9}+\frac{166680}{473819}a^{8}-\frac{94680}{473819}a^{7}-\frac{73574}{473819}a^{6}+\frac{83676}{473819}a^{5}-\frac{81586}{473819}a^{4}-\frac{35345}{473819}a^{3}-\frac{42127}{473819}a^{2}+\frac{58674}{473819}a-\frac{218530}{473819}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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:   \( -1 \)  (order $2$) 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:   $\frac{70160}{473819}a^{15}-\frac{526200}{473819}a^{14}+\frac{1787000}{473819}a^{13}-\frac{3634800}{473819}a^{12}+\frac{5144700}{473819}a^{11}-\frac{5870590}{473819}a^{10}+\frac{9658994}{473819}a^{9}-\frac{21786948}{473819}a^{8}+\frac{37027286}{473819}a^{7}-\frac{41724227}{473819}a^{6}+\frac{19545587}{473819}a^{5}+\frac{13892638}{473819}a^{4}-\frac{38394197}{473819}a^{3}+\frac{34609151}{473819}a^{2}-\frac{17649996}{473819}a+\frac{3925721}{473819}$, $\frac{296745}{473819}a^{15}-\frac{2059124}{473819}a^{14}+\frac{7333570}{473819}a^{13}-\frac{17285964}{473819}a^{12}+\frac{31143492}{473819}a^{11}-\frac{46107604}{473819}a^{10}+\frac{75866596}{473819}a^{9}-\frac{131993881}{473819}a^{8}+\frac{233311373}{473819}a^{7}-\frac{326643549}{473819}a^{6}+\frac{366654588}{473819}a^{5}-\frac{297140089}{473819}a^{4}+\frac{174160806}{473819}a^{3}-\frac{60275899}{473819}a^{2}+\frac{6715532}{473819}a+\frac{2073567}{473819}$, $\frac{296745}{473819}a^{15}-\frac{2392051}{473819}a^{14}+\frac{9664059}{473819}a^{13}-\frac{25689137}{473819}a^{12}+\frac{51266173}{473819}a^{11}-\frac{82940315}{473819}a^{10}+\frac{131115563}{473819}a^{9}-\frac{222396596}{473819}a^{8}+\frac{389560800}{473819}a^{7}-\frac{603032659}{473819}a^{6}+\frac{759623072}{473819}a^{5}-\frac{747576788}{473819}a^{4}+\frac{552060612}{473819}a^{3}-\frac{294138908}{473819}a^{2}+\frac{98556022}{473819}a-\frac{16050159}{473819}$, $\frac{719612}{473819}a^{15}-\frac{5143564}{473819}a^{14}+\frac{18467195}{473819}a^{13}-\frac{43389628}{473819}a^{12}+\frac{77568447}{473819}a^{11}-\frac{114887200}{473819}a^{10}+\frac{188532185}{473819}a^{9}-\frac{333860417}{473819}a^{8}+\frac{584067625}{473819}a^{7}-\frac{818106735}{473819}a^{6}+\frac{908550521}{473819}a^{5}-\frac{744665450}{473819}a^{4}+\frac{447182817}{473819}a^{3}-\frac{172890089}{473819}a^{2}+\frac{35115269}{473819}a-\frac{964939}{473819}$, $\frac{106218}{473819}a^{15}-\frac{796635}{473819}a^{14}+\frac{2957177}{473819}a^{13}-\frac{7139353}{473819}a^{12}+\frac{13066417}{473819}a^{11}-\frac{19913465}{473819}a^{10}+\frac{32403853}{473819}a^{9}-\frac{57306273}{473819}a^{8}+\frac{99554603}{473819}a^{7}-\frac{143083489}{473819}a^{6}+\frac{165192257}{473819}a^{5}-\frac{146807069}{473819}a^{4}+\frac{99146661}{473819}a^{3}-\frac{47773077}{473819}a^{2}+\frac{16026361}{473819}a-\frac{2817093}{473819}$, $\frac{4600}{2381}a^{14}-\frac{32200}{2381}a^{13}+\frac{114527}{2381}a^{12}-\frac{268562}{2381}a^{11}+\frac{482170}{2381}a^{10}-\frac{716465}{2381}a^{9}+\frac{1183609}{2381}a^{8}-\frac{2074228}{2381}a^{7}+\frac{3634772}{2381}a^{6}-\frac{5080869}{2381}a^{5}+\frac{5702749}{2381}a^{4}-\frac{4703810}{2381}a^{3}+\frac{2879424}{2381}a^{2}-\frac{1125717}{2381}a+\frac{233999}{2381}$, $\frac{393376}{473819}a^{15}-\frac{2950320}{473819}a^{14}+\frac{10840640}{473819}a^{13}-\frac{25717640}{473819}a^{12}+\frac{45999170}{473819}a^{11}-\frac{68543739}{473819}a^{10}+\frac{111352831}{473819}a^{9}-\frac{200402877}{473819}a^{8}+\frac{348150024}{473819}a^{7}-\frac{492656783}{473819}a^{6}+\frac{546443319}{473819}a^{5}-\frac{459962818}{473819}a^{4}+\frac{287107412}{473819}a^{3}-\frac{125607886}{473819}a^{2}+\frac{34864665}{473819}a-\frac{4654687}{473819}$ 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:  \( 4749.52173347 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4749.52173347 \cdot 2}{2\cdot\sqrt{101415451701035401216}}\cr\approx \mathstrut & 1.14560990923 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14)
 
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 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14, 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 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14);
 
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 + 32*x^14 - 84*x^13 + 166*x^12 - 268*x^11 + 428*x^10 - 732*x^9 + 1279*x^8 - 1956*x^7 + 2456*x^6 - 2432*x^5 + 1866*x^4 - 1072*x^3 + 436*x^2 - 112*x + 14);
 
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

$C_2^2\wr C_2$ (as 16T39):

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 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), 4.4.7168.1, 4.2.1024.1, 4.2.50176.1, 4.0.7168.1, 4.0.1568.1 x2, 4.2.1792.1 x2, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.0.157351936.3, 8.4.205520896.1, 8.0.10070523904.8, 8.4.10070523904.2, 8.0.205520896.2, 8.0.2517630976.5, 8.0.2517630976.6

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 8 siblings: 8.4.51380224.1, 8.0.51380224.1, 8.4.205520896.1, 8.4.10070523904.2, 8.0.205520896.2, 8.0.10070523904.8, 8.4.2517630976.1, 8.0.2517630976.1
Degree 16 siblings: 16.0.6338465731314712576.3, 16.0.42238838692642816.1, 16.0.101415451701035401216.12, 16.8.101415451701035401216.1, 16.0.101415451701035401216.10, 16.0.6338465731314712576.7
Minimal sibling: 8.4.51380224.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.8.22.99$x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 12 x^{2} + 14$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
2.8.22.99$x^{8} + 4 x^{7} + 4 x^{6} + 6 x^{4} + 12 x^{2} + 14$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
\(7\) Copy content Toggle raw display 7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{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}$