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Properties

Label 16.0.162...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.623\times 10^{27}$
Root discriminant \(50.19\)
Ramified primes $3,5,29$
Class number $64$ (GRH)
Class group [2, 2, 4, 4] (GRH)
Galois group $\OD_{16}:C_2$ (as 16T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521)
 
gp: K = bnfinit(y^16 - 3*y^14 + 43*y^12 - 156*y^10 + 2595*y^8 + 20244*y^6 - 2597*y^4 + 170097*y^2 + 923521, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521)
 

\( x^{16} - 3x^{14} + 43x^{12} - 156x^{10} + 2595x^{8} + 20244x^{6} - 2597x^{4} + 170097x^{2} + 923521 \) 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:   \(1622628503115275885009765625\) \(\medspace = 3^{12}\cdot 5^{14}\cdot 29^{8}\) 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:  \(50.19\)
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:  $3^{3/4}5^{7/8}29^{1/2}\approx 50.19248452708817$
Ramified primes:   \(3\), \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{22}a^{12}-\frac{5}{22}a^{10}-\frac{3}{22}a^{8}+\frac{9}{22}a^{6}+\frac{3}{11}a^{4}-\frac{3}{11}a^{2}-\frac{1}{2}a+\frac{5}{22}$, $\frac{1}{682}a^{13}+\frac{14}{341}a^{11}+\frac{37}{341}a^{9}+\frac{185}{682}a^{7}-\frac{1}{2}a^{6}+\frac{135}{341}a^{5}-\frac{61}{682}a^{3}-\frac{1}{2}a^{2}+\frac{19}{341}a$, $\frac{1}{64\!\cdots\!82}a^{14}+\frac{713492889428735}{32\!\cdots\!41}a^{12}+\frac{68\!\cdots\!43}{64\!\cdots\!82}a^{10}+\frac{14\!\cdots\!63}{64\!\cdots\!82}a^{8}-\frac{15\!\cdots\!85}{64\!\cdots\!82}a^{6}-\frac{1}{2}a^{5}-\frac{8807867284017}{71623194432691}a^{4}-\frac{1}{2}a^{3}-\frac{11\!\cdots\!34}{32\!\cdots\!41}a^{2}-\frac{1}{2}a+\frac{2687686199629}{33612966377881}$, $\frac{1}{20\!\cdots\!42}a^{15}+\frac{713492889428735}{10\!\cdots\!71}a^{13}-\frac{20\!\cdots\!45}{10\!\cdots\!71}a^{11}+\frac{14\!\cdots\!63}{20\!\cdots\!42}a^{9}-\frac{91\!\cdots\!33}{20\!\cdots\!42}a^{7}-\frac{1}{2}a^{6}+\frac{22\!\cdots\!87}{44\!\cdots\!42}a^{5}+\frac{35\!\cdots\!17}{10\!\cdots\!71}a^{3}-\frac{1}{2}a^{2}+\frac{307892069800187}{20\!\cdots\!22}a-\frac{1}{2}$ 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

$C_{2}\times C_{2}\times C_{4}\times C_{4}$, which has order $64$ (assuming GRH)

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:   $\frac{6633754}{33612966377881}a^{14}-\frac{2515680}{33612966377881}a^{12}-\frac{1596116823}{33612966377881}a^{10}-\frac{4001859573}{33612966377881}a^{8}+\frac{3250680292}{33612966377881}a^{6}-\frac{195048207}{74529858931}a^{4}-\frac{195812358638}{33612966377881}a^{2}-\frac{20415540535162}{33612966377881}$, $\frac{95497715054075}{20\!\cdots\!42}a^{15}-\frac{3316877}{33612966377881}a^{14}-\frac{327271979667652}{10\!\cdots\!71}a^{13}+\frac{1257840}{33612966377881}a^{12}+\frac{33\!\cdots\!63}{10\!\cdots\!71}a^{11}+\frac{1596116823}{67225932755762}a^{10}-\frac{20\!\cdots\!66}{10\!\cdots\!71}a^{9}+\frac{4001859573}{67225932755762}a^{8}+\frac{40\!\cdots\!69}{20\!\cdots\!42}a^{7}-\frac{1625340146}{33612966377881}a^{6}+\frac{894910850112901}{44\!\cdots\!42}a^{5}+\frac{195048207}{149059717862}a^{4}-\frac{19\!\cdots\!17}{20\!\cdots\!42}a^{3}+\frac{97906179319}{33612966377881}a^{2}+\frac{25\!\cdots\!77}{20\!\cdots\!22}a-\frac{13197425842719}{67225932755762}$, $\frac{77507449547301}{10\!\cdots\!71}a^{15}+\frac{3316877}{33612966377881}a^{14}-\frac{10\!\cdots\!53}{20\!\cdots\!42}a^{13}-\frac{1257840}{33612966377881}a^{12}+\frac{53\!\cdots\!96}{10\!\cdots\!71}a^{11}-\frac{1596116823}{67225932755762}a^{10}-\frac{65\!\cdots\!01}{20\!\cdots\!42}a^{9}-\frac{4001859573}{67225932755762}a^{8}+\frac{64\!\cdots\!07}{20\!\cdots\!42}a^{7}+\frac{1625340146}{33612966377881}a^{6}+\frac{723699824177963}{22\!\cdots\!21}a^{5}-\frac{195048207}{149059717862}a^{4}-\frac{15\!\cdots\!55}{10\!\cdots\!71}a^{3}-\frac{97906179319}{33612966377881}a^{2}+\frac{20\!\cdots\!06}{10\!\cdots\!11}a-\frac{43820736645462}{33612966377881}$, $\frac{6585191}{49955991662}a^{15}-\frac{43840155}{49955991662}a^{13}+\frac{222409875}{24977995831}a^{11}-\frac{2691904205}{49955991662}a^{9}+\frac{13530873480}{24977995831}a^{7}+\frac{32854492587}{49955991662}a^{5}-\frac{129199150535}{49955991662}a^{3}+\frac{1624479645}{51983342}a-\frac{5}{2}$, $\frac{427222788324335}{20\!\cdots\!42}a^{15}+\frac{9950631}{33612966377881}a^{14}-\frac{14\!\cdots\!85}{10\!\cdots\!71}a^{13}-\frac{3773520}{33612966377881}a^{12}+\frac{14\!\cdots\!24}{10\!\cdots\!71}a^{11}-\frac{4788350469}{67225932755762}a^{10}-\frac{87\!\cdots\!73}{10\!\cdots\!71}a^{9}-\frac{12005578719}{67225932755762}a^{8}+\frac{17\!\cdots\!29}{20\!\cdots\!42}a^{7}+\frac{4876020438}{33612966377881}a^{6}+\frac{47\!\cdots\!43}{44\!\cdots\!42}a^{5}-\frac{585144621}{149059717862}a^{4}-\frac{83\!\cdots\!91}{20\!\cdots\!42}a^{3}-\frac{293718537957}{33612966377881}a^{2}+\frac{10\!\cdots\!01}{20\!\cdots\!22}a-\frac{296537386250653}{67225932755762}$, $\frac{17590960538505}{10\!\cdots\!71}a^{15}-\frac{50156004545479}{10\!\cdots\!71}a^{13}+\frac{643716063399085}{10\!\cdots\!71}a^{11}-\frac{33\!\cdots\!42}{10\!\cdots\!71}a^{9}+\frac{43\!\cdots\!15}{10\!\cdots\!71}a^{7}+\frac{761958782051690}{22\!\cdots\!21}a^{5}-\frac{77\!\cdots\!81}{10\!\cdots\!71}a^{3}-\frac{13\!\cdots\!72}{10\!\cdots\!11}a$, $\frac{62156192495953}{10\!\cdots\!71}a^{15}-\frac{417024824442767}{10\!\cdots\!71}a^{13}+\frac{41\!\cdots\!29}{10\!\cdots\!71}a^{11}-\frac{25\!\cdots\!35}{10\!\cdots\!71}a^{9}+\frac{25\!\cdots\!49}{10\!\cdots\!71}a^{7}+\frac{657767319314620}{22\!\cdots\!21}a^{5}-\frac{12\!\cdots\!24}{10\!\cdots\!71}a^{3}+\frac{15\!\cdots\!91}{10\!\cdots\!11}a$ Copy content Toggle raw display (assuming GRH)
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:  \( 175816.836849 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 175816.836849 \cdot 64}{2\cdot\sqrt{1622628503115275885009765625}}\cr\approx \mathstrut & 0.339265608486 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521)
 
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^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521, 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^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521);
 
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^14 + 43*x^12 - 156*x^10 + 2595*x^8 + 20244*x^6 - 2597*x^4 + 170097*x^2 + 923521);
 
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

$\OD_{16}:C_2$ (as 16T36):

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 11 conjugacy class representatives for $\OD_{16}:C_2$
Character table for $\OD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.4.946125.1, \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{5}, \sqrt{29})\), 8.0.40281863203125.1 x2, 8.0.47897578125.1 x2, 8.8.895152515625.1

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 8 siblings: 8.0.40281863203125.1, 8.0.47897578125.1
Degree 16 siblings: 16.0.1929403689792242431640625.1, deg 16
Minimal sibling: 8.0.47897578125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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 3.8.6.3$x^{8} - 6 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 6 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
\(5\) Copy content Toggle raw display 5.8.7.1$x^{8} + 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} + 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
\(29\) Copy content Toggle raw display 29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$