Properties

Label 16.0.7213895789838336.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}$
Root discriminant $9.80$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois Group $Q_8 : C_2$ (as 16T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 36, -108, 244, -436, 636, -780, 831, -780, 636, -436, 244, -108, 36, -8, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*x^2 - 8*x + 1)
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 8 x^{15} \) \(\mathstrut +\mathstrut 36 x^{14} \) \(\mathstrut -\mathstrut 108 x^{13} \) \(\mathstrut +\mathstrut 244 x^{12} \) \(\mathstrut -\mathstrut 436 x^{11} \) \(\mathstrut +\mathstrut 636 x^{10} \) \(\mathstrut -\mathstrut 780 x^{9} \) \(\mathstrut +\mathstrut 831 x^{8} \) \(\mathstrut -\mathstrut 780 x^{7} \) \(\mathstrut +\mathstrut 636 x^{6} \) \(\mathstrut -\mathstrut 436 x^{5} \) \(\mathstrut +\mathstrut 244 x^{4} \) \(\mathstrut -\mathstrut 108 x^{3} \) \(\mathstrut +\mathstrut 36 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $16$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 8]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(7213895789838336=2^{40}\cdot 3^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.80$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{2}{13} a^{13} - \frac{3}{13} a^{12} + \frac{6}{13} a^{11} - \frac{3}{13} a^{10} - \frac{5}{13} a^{9} - \frac{2}{13} a^{8} + \frac{6}{13} a^{7} - \frac{2}{13} a^{6} - \frac{5}{13} a^{5} - \frac{3}{13} a^{4} + \frac{6}{13} a^{3} - \frac{3}{13} a^{2} - \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{949} a^{15} - \frac{9}{949} a^{14} - \frac{28}{949} a^{13} + \frac{66}{949} a^{12} + \frac{397}{949} a^{11} - \frac{322}{949} a^{10} + \frac{228}{949} a^{9} + \frac{306}{949} a^{8} - \frac{278}{949} a^{7} + \frac{9}{949} a^{6} - \frac{176}{949} a^{5} + \frac{105}{949} a^{4} + \frac{358}{949} a^{3} + \frac{45}{949} a^{2} + \frac{210}{949} a - \frac{72}{949}$

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

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $7$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{101}{73} a^{15} - \frac{5320}{949} a^{14} + \frac{11927}{949} a^{13} - \frac{212}{949} a^{12} - \frac{50913}{949} a^{11} + \frac{170777}{949} a^{10} - \frac{336685}{949} a^{9} + \frac{477990}{949} a^{8} - \frac{549047}{949} a^{7} + \frac{561580}{949} a^{6} - \frac{497027}{949} a^{5} + \frac{353726}{949} a^{4} - \frac{193224}{949} a^{3} + \frac{79452}{949} a^{2} - \frac{20066}{949} a + \frac{2116}{949} \) (order $24$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2718}{949} a^{15} - \frac{20593}{949} a^{14} + \frac{86978}{949} a^{13} - \frac{243136}{949} a^{12} + \frac{508186}{949} a^{11} - \frac{841251}{949} a^{10} + \frac{1126105}{949} a^{9} - \frac{1275218}{949} a^{8} + \frac{1270000}{949} a^{7} - \frac{1112586}{949} a^{6} + \frac{817601}{949} a^{5} - \frac{487315}{949} a^{4} + \frac{229466}{949} a^{3} - \frac{81944}{949} a^{2} + \frac{17367}{949} a - \frac{2027}{949} \),  \( \frac{3272}{949} a^{15} - \frac{24776}{949} a^{14} + \frac{105922}{949} a^{13} - \frac{300085}{949} a^{12} + \frac{639940}{949} a^{11} - \frac{1079937}{949} a^{10} + \frac{1482805}{949} a^{9} - \frac{1718456}{949} a^{8} + \frac{1748095}{949} a^{7} - \frac{1561879}{949} a^{6} + \frac{1188684}{949} a^{5} - \frac{743775}{949} a^{4} + \frac{371880}{949} a^{3} - \frac{10849}{73} a^{2} + \frac{37201}{949} a - \frac{5999}{949} \),  \( \frac{2398}{949} a^{15} - \frac{20341}{949} a^{14} + \frac{90755}{949} a^{13} - \frac{267760}{949} a^{12} + \frac{44531}{73} a^{11} - \frac{75816}{73} a^{10} + \frac{1349087}{949} a^{9} - \frac{1540600}{949} a^{8} + \frac{1540584}{949} a^{7} - \frac{1363593}{949} a^{6} + \frac{1015176}{949} a^{5} - \frac{602237}{949} a^{4} + \frac{21423}{73} a^{3} - \frac{97950}{949} a^{2} + \frac{1681}{73} a - \frac{2493}{949} \),  \( \frac{3871}{949} a^{15} - \frac{30021}{949} a^{14} + \frac{129665}{949} a^{13} - \frac{371073}{949} a^{12} + \frac{794158}{949} a^{11} - \frac{1342530}{949} a^{10} + \frac{1838815}{949} a^{9} - \frac{2118140}{949} a^{8} + \frac{2134767}{949} a^{7} - \frac{1893675}{949} a^{6} + \frac{1423221}{949} a^{5} - \frac{867322}{949} a^{4} + \frac{414480}{949} a^{3} - \frac{146786}{949} a^{2} + \frac{32686}{949} a - \frac{3429}{949} \),  \( \frac{40}{949} a^{15} - \frac{4010}{949} a^{14} + \frac{24211}{949} a^{13} - \frac{90800}{949} a^{12} + \frac{220791}{949} a^{11} - \frac{420439}{949} a^{10} + \frac{625240}{949} a^{9} - \frac{764334}{949} a^{8} + \frac{61992}{73} a^{7} - \frac{760081}{949} a^{6} + \frac{609080}{949} a^{5} - \frac{401461}{949} a^{4} + \frac{210690}{949} a^{3} - \frac{87844}{949} a^{2} + \frac{24241}{949} a - \frac{5581}{949} \),  \( \frac{144}{949} a^{15} + \frac{1113}{949} a^{14} - \frac{9799}{949} a^{13} + \frac{44033}{949} a^{12} - \frac{119127}{949} a^{11} + \frac{245340}{949} a^{10} - \frac{30010}{73} a^{9} + \frac{504256}{949} a^{8} - \frac{550375}{949} a^{7} + \frac{534561}{949} a^{6} - \frac{449255}{949} a^{5} + \frac{314420}{949} a^{4} - \frac{172193}{949} a^{3} + \frac{73275}{949} a^{2} - \frac{22028}{949} a + \frac{4378}{949} \),  \( \frac{343}{949} a^{15} - \frac{580}{73} a^{14} + \frac{42007}{949} a^{13} - \frac{11466}{73} a^{12} + \frac{355244}{949} a^{11} - \frac{661742}{949} a^{10} + \frac{970702}{949} a^{9} - \frac{1168235}{949} a^{8} + \frac{1216018}{949} a^{7} - \frac{1127756}{949} a^{6} + \frac{68390}{73} a^{5} - \frac{43725}{73} a^{4} + \frac{285876}{949} a^{3} - \frac{110709}{949} a^{2} + \frac{26843}{949} a - \frac{4475}{949} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 168.461389974 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4:C_2$ (as 16T11):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{24})\), 8.0.9437184.1 x2, 8.0.5308416.2 x2, 8.4.84934656.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$