Properties

Label 16.0.7058653305387264.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{14}\cdot 7^{8}$
Root discriminant $9.78$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois Group $D_8:C_2$ (as 16T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 26, -60, 92, -89, 40, 21, -44, 21, 10, -19, 11, -3, 2, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1)
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 2 x^{15} \) \(\mathstrut +\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut 3 x^{13} \) \(\mathstrut +\mathstrut 11 x^{12} \) \(\mathstrut -\mathstrut 19 x^{11} \) \(\mathstrut +\mathstrut 10 x^{10} \) \(\mathstrut +\mathstrut 21 x^{9} \) \(\mathstrut -\mathstrut 44 x^{8} \) \(\mathstrut +\mathstrut 21 x^{7} \) \(\mathstrut +\mathstrut 40 x^{6} \) \(\mathstrut -\mathstrut 89 x^{5} \) \(\mathstrut +\mathstrut 92 x^{4} \) \(\mathstrut -\mathstrut 60 x^{3} \) \(\mathstrut +\mathstrut 26 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(7058653305387264=2^{8}\cdot 3^{14}\cdot 7^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.78$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{40937} a^{15} - \frac{197}{3149} a^{14} + \frac{3681}{40937} a^{13} - \frac{4172}{40937} a^{12} - \frac{646}{3149} a^{11} - \frac{1462}{40937} a^{10} + \frac{16001}{40937} a^{9} - \frac{9538}{40937} a^{8} + \frac{138}{611} a^{7} + \frac{1093}{40937} a^{6} - \frac{13231}{40937} a^{5} + \frac{3141}{40937} a^{4} - \frac{14075}{40937} a^{3} - \frac{515}{3149} a^{2} - \frac{1544}{3149} a - \frac{11694}{40937}$

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:  \( -\frac{324}{3149} a^{15} - \frac{4721}{3149} a^{14} + \frac{3976}{3149} a^{13} + \frac{807}{3149} a^{12} + \frac{6514}{3149} a^{11} - \frac{39599}{3149} a^{10} + \frac{36718}{3149} a^{9} + \frac{32633}{3149} a^{8} - \frac{1566}{47} a^{7} + \frac{67834}{3149} a^{6} + \frac{86078}{3149} a^{5} - \frac{180050}{3149} a^{4} + \frac{139104}{3149} a^{3} - \frac{54014}{3149} a^{2} + \frac{3792}{3149} a + \frac{3758}{3149} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{82388}{40937} a^{15} - \frac{3639}{3149} a^{14} + \frac{8932}{40937} a^{13} - \frac{138495}{40937} a^{12} + \frac{48985}{3149} a^{11} - \frac{464909}{40937} a^{10} - \frac{617878}{40937} a^{9} + \frac{1770199}{40937} a^{8} - \frac{15219}{611} a^{7} - \frac{1607859}{40937} a^{6} + \frac{3187894}{40937} a^{5} - \frac{2438289}{40937} a^{4} + \frac{584417}{40937} a^{3} + \frac{43892}{3149} a^{2} - \frac{34707}{3149} a + \frac{129834}{40937} \),  \( \frac{101181}{40937} a^{15} + \frac{6811}{3149} a^{14} - \frac{202250}{40937} a^{13} - \frac{66662}{40937} a^{12} + \frac{38655}{3149} a^{11} + \frac{838436}{40937} a^{10} - \frac{2640200}{40937} a^{9} + \frac{1908499}{40937} a^{8} + \frac{38899}{611} a^{7} - \frac{6161491}{40937} a^{6} + \frac{2779679}{40937} a^{5} + \frac{4273038}{40937} a^{4} - \frac{7743312}{40937} a^{3} + \frac{461191}{3149} a^{2} - \frac{184216}{3149} a + \frac{482741}{40937} \),  \( \frac{191458}{40937} a^{15} - \frac{20547}{3149} a^{14} + \frac{108317}{40937} a^{13} - \frac{368465}{40937} a^{12} + \frac{139961}{3149} a^{11} - \frac{2317799}{40937} a^{10} - \frac{533118}{40937} a^{9} + \frac{4944669}{40937} a^{8} - \frac{76033}{611} a^{7} - \frac{1807578}{40937} a^{6} + \frac{9293461}{40937} a^{5} - \frac{10352013}{40937} a^{4} + \frac{6292847}{40937} a^{3} - \frac{153683}{3149} a^{2} + \frac{16968}{3149} a + \frac{57489}{40937} \),  \( \frac{5369}{3149} a^{15} - \frac{4624}{3149} a^{14} + \frac{165}{3149} a^{13} - \frac{10078}{3149} a^{12} + \frac{45755}{3149} a^{11} - \frac{39958}{3149} a^{10} - \frac{39437}{3149} a^{9} + \frac{119178}{3149} a^{8} - \frac{1114}{47} a^{7} - \frac{99038}{3149} a^{6} + \frac{208886}{3149} a^{5} - \frac{168912}{3149} a^{4} + \frac{73454}{3149} a^{3} - \frac{12216}{3149} a^{2} - \frac{1490}{3149} a - \frac{324}{3149} \),  \( \frac{77852}{40937} a^{15} - \frac{7512}{3149} a^{14} + \frac{96086}{40937} a^{13} - \frac{209071}{40937} a^{12} + \frac{56969}{3149} a^{11} - \frac{956315}{40937} a^{10} + \frac{242564}{40937} a^{9} + \frac{1395725}{40937} a^{8} - \frac{32020}{611} a^{7} + \frac{557331}{40937} a^{6} + \frac{2453202}{40937} a^{5} - \frac{4691324}{40937} a^{4} + \frac{4824345}{40937} a^{3} - \frac{236887}{3149} a^{2} + \frac{94610}{3149} a - \frac{248967}{40937} \),  \( \frac{122432}{40937} a^{15} - \frac{913}{3149} a^{14} - \frac{126052}{40937} a^{13} - \frac{138166}{40937} a^{12} + \frac{65341}{3149} a^{11} - \frac{19020}{40937} a^{10} - \frac{2093490}{40937} a^{9} + \frac{2755225}{40937} a^{8} + \frac{8187}{611} a^{7} - \frac{5040128}{40937} a^{6} + \frac{4605179}{40937} a^{5} + \frac{447041}{40937} a^{4} - \frac{4122022}{40937} a^{3} + \frac{302251}{3149} a^{2} - \frac{132796}{3149} a + \frac{379263}{40937} \),  \( \frac{367187}{40937} a^{15} - \frac{41097}{3149} a^{14} + \frac{366851}{40937} a^{13} - \frac{819427}{40937} a^{12} + \frac{272735}{3149} a^{11} - \frac{4892016}{40937} a^{10} + \frac{326569}{40937} a^{9} + \frac{8650325}{40937} a^{8} - \frac{167671}{611} a^{7} - \frac{133768}{40937} a^{6} + \frac{16322078}{40937} a^{5} - \frac{23154076}{40937} a^{4} + \frac{17987757}{40937} a^{3} - \frac{658847}{3149} a^{2} + \frac{191874}{3149} a - \frac{330344}{40937} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 37.0149239843 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_8:C_2$ (as 16T47):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 32
The 14 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 4.2.1323.1 x2, 4.0.189.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.1750329.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
3Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$