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Properties

Label 16.8.162...856.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.624\times 10^{20}$
Root discriminant \(18.33\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
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 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*x^2 + 16)
 
gp: K = bnfinit(y^16 - 10*y^14 + 19*y^12 + 10*y^10 - 51*y^8 + 20*y^6 + 76*y^4 - 80*y^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*x^2 + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*x^2 + 16)
 

\( x^{16} - 10x^{14} + 19x^{12} + 10x^{10} - 51x^{8} + 20x^{6} + 76x^{4} - 80x^{2} + 16 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(162447943996702457856\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 7^{8}\) 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:  \(18.33\)
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:  $2^{2}3^{1/2}7^{1/2}\approx 18.33030277982336$
Ramified primes:   \(2\), \(3\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{176}a^{12}+\frac{5}{44}a^{10}-\frac{1}{4}a^{9}+\frac{23}{176}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{75}{176}a^{4}-\frac{1}{2}a^{3}-\frac{37}{88}a^{2}-\frac{1}{2}a-\frac{9}{44}$, $\frac{1}{176}a^{13}+\frac{5}{44}a^{11}-\frac{21}{176}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{31}{176}a^{5}+\frac{1}{4}a^{4}-\frac{37}{88}a^{3}-\frac{9}{44}a$, $\frac{1}{5984}a^{14}-\frac{1}{374}a^{12}+\frac{183}{5984}a^{10}-\frac{1}{4}a^{9}-\frac{9}{44}a^{8}+\frac{81}{352}a^{6}-\frac{1}{2}a^{5}-\frac{381}{2992}a^{4}-\frac{1}{4}a^{3}-\frac{113}{1496}a^{2}-\frac{75}{187}$, $\frac{1}{5984}a^{15}-\frac{1}{374}a^{13}+\frac{183}{5984}a^{11}+\frac{1}{22}a^{9}-\frac{7}{352}a^{7}+\frac{367}{2992}a^{5}+\frac{635}{1496}a^{3}-\frac{75}{187}a$ 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 $1$ (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:  $11$
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:   $\frac{21}{748}a^{14}-\frac{417}{1496}a^{12}+\frac{409}{748}a^{10}+\frac{5}{88}a^{8}-\frac{59}{44}a^{6}+\frac{1231}{1496}a^{4}+\frac{1643}{748}a^{2}-\frac{567}{374}$, $\frac{135}{2992}a^{14}-\frac{587}{1496}a^{12}+\frac{1041}{2992}a^{10}+\frac{75}{88}a^{8}-\frac{153}{176}a^{6}-\frac{447}{748}a^{4}+\frac{1017}{374}a^{2}-\frac{431}{374}$, $\frac{5}{44}a^{15}-\frac{3}{352}a^{14}-\frac{185}{176}a^{13}+\frac{5}{88}a^{12}+\frac{61}{44}a^{11}+\frac{35}{352}a^{10}+\frac{361}{176}a^{9}-\frac{35}{88}a^{8}-\frac{47}{11}a^{7}-\frac{83}{352}a^{6}-\frac{21}{176}a^{5}+\frac{125}{176}a^{4}+\frac{61}{8}a^{3}-\frac{23}{88}a^{2}-\frac{191}{44}a-\frac{15}{22}$, $\frac{249}{5984}a^{15}+\frac{43}{1496}a^{14}-\frac{1125}{2992}a^{13}-\frac{87}{374}a^{12}+\frac{2455}{5984}a^{11}+\frac{83}{1496}a^{10}+\frac{173}{176}a^{9}+\frac{35}{44}a^{8}-\frac{511}{352}a^{7}-\frac{15}{88}a^{6}-\frac{1033}{1496}a^{5}-\frac{1083}{748}a^{4}+\frac{514}{187}a^{3}+\frac{445}{374}a^{2}-\frac{971}{748}a+\frac{156}{187}$, $\frac{141}{5984}a^{15}-\frac{185}{2992}a^{14}-\frac{805}{2992}a^{13}+\frac{1651}{2992}a^{12}+\frac{4315}{5984}a^{11}-\frac{1691}{2992}a^{10}+\frac{25}{176}a^{9}-\frac{243}{176}a^{8}-\frac{723}{352}a^{7}+\frac{327}{176}a^{6}+\frac{95}{136}a^{5}+\frac{4273}{2992}a^{4}+\frac{883}{374}a^{3}-\frac{5501}{1496}a^{2}-\frac{2197}{748}a+\frac{109}{748}$, $\frac{141}{5984}a^{15}+\frac{185}{2992}a^{14}-\frac{805}{2992}a^{13}-\frac{1651}{2992}a^{12}+\frac{4315}{5984}a^{11}+\frac{1691}{2992}a^{10}+\frac{25}{176}a^{9}+\frac{243}{176}a^{8}-\frac{723}{352}a^{7}-\frac{327}{176}a^{6}+\frac{95}{136}a^{5}-\frac{4273}{2992}a^{4}+\frac{883}{374}a^{3}+\frac{5501}{1496}a^{2}-\frac{2197}{748}a-\frac{109}{748}$, $\frac{69}{1496}a^{15}+\frac{117}{1496}a^{14}-\frac{577}{1496}a^{13}-\frac{2129}{2992}a^{12}+\frac{353}{1496}a^{11}+\frac{1283}{1496}a^{10}+\frac{79}{88}a^{9}+\frac{23}{16}a^{8}-\frac{65}{88}a^{7}-\frac{225}{88}a^{6}-\frac{77}{136}a^{5}-\frac{1729}{2992}a^{4}+\frac{1933}{748}a^{3}+\frac{7789}{1496}a^{2}+\frac{23}{187}a-\frac{145}{68}$, $\frac{369}{5984}a^{15}-\frac{185}{2992}a^{14}-\frac{449}{748}a^{13}+\frac{1651}{2992}a^{12}+\frac{6055}{5984}a^{11}-\frac{1691}{2992}a^{10}+\frac{10}{11}a^{9}-\frac{243}{176}a^{8}-\frac{1087}{352}a^{7}+\frac{327}{176}a^{6}+\frac{1599}{2992}a^{5}+\frac{4273}{2992}a^{4}+\frac{753}{136}a^{3}-\frac{5501}{1496}a^{2}-\frac{1273}{374}a+\frac{109}{748}$, $\frac{23}{352}a^{15}-\frac{25}{44}a^{13}+\frac{177}{352}a^{11}+\frac{5}{4}a^{9}-\frac{449}{352}a^{7}-\frac{235}{176}a^{5}+\frac{321}{88}a^{3}$, $\frac{53}{1496}a^{15}+\frac{7}{352}a^{14}-\frac{127}{374}a^{13}-\frac{9}{44}a^{12}+\frac{791}{1496}a^{11}+\frac{145}{352}a^{10}+\frac{27}{44}a^{9}+\frac{3}{11}a^{8}-\frac{151}{88}a^{7}-\frac{393}{352}a^{6}+\frac{343}{748}a^{5}+\frac{57}{176}a^{4}+\frac{1809}{748}a^{3}+\frac{97}{88}a^{2}-\frac{413}{187}a-\frac{9}{11}$, $\frac{129}{5984}a^{15}-\frac{249}{5984}a^{14}-\frac{607}{2992}a^{13}+\frac{1125}{2992}a^{12}+\frac{1711}{5984}a^{11}-\frac{2455}{5984}a^{10}+\frac{67}{176}a^{9}-\frac{173}{176}a^{8}-\frac{199}{352}a^{7}+\frac{511}{352}a^{6}-\frac{247}{748}a^{5}+\frac{1033}{1496}a^{4}+\frac{186}{187}a^{3}-\frac{514}{187}a^{2}-\frac{263}{748}a+\frac{971}{748}$ 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:  \( 17122.1735406 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 17122.1735406 \cdot 1}{2\cdot\sqrt{162447943996702457856}}\cr\approx \mathstrut & 0.267997799450 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*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 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*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 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*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 - 10*x^14 + 19*x^12 + 10*x^10 - 51*x^8 + 20*x^6 + 76*x^4 - 80*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

$C_2^2\times D_4$ (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{21}) \), 4.2.16128.1, 4.2.4032.1, 4.2.448.1, 4.2.1792.1, \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{6}, \sqrt{14})\), 8.8.12745506816.1, 8.4.12745506816.3, 8.4.157351936.1, 8.4.260112384.2, 8.4.260112384.1, 8.4.12745506816.5, 8.4.796594176.2

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: 16.0.67658452310163456.1, 16.0.634562281237118976.2, 16.0.10152996499793903616.4, 16.0.162447943996702457856.11, 16.0.162447943996702457856.12, 16.0.162447943996702457856.13, 16.0.162447943996702457856.15
Minimal sibling: 16.0.67658452310163456.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(7\) 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}$