Properties

Label 16.0.5575888916015625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 59^{2}$
Root discriminant $9.64$
Ramified primes $3, 5, 59$
Class number $1$
Class group Trivial
Galois Group $C_2\times C_2\wr C_4$ (as 16T261)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 15, -30, 49, -70, 90, -105, 111, -105, 90, -70, 49, -30, 15, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1)
gp: K = bnfinit(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 5 x^{15} \) \(\mathstrut +\mathstrut 15 x^{14} \) \(\mathstrut -\mathstrut 30 x^{13} \) \(\mathstrut +\mathstrut 49 x^{12} \) \(\mathstrut -\mathstrut 70 x^{11} \) \(\mathstrut +\mathstrut 90 x^{10} \) \(\mathstrut -\mathstrut 105 x^{9} \) \(\mathstrut +\mathstrut 111 x^{8} \) \(\mathstrut -\mathstrut 105 x^{7} \) \(\mathstrut +\mathstrut 90 x^{6} \) \(\mathstrut -\mathstrut 70 x^{5} \) \(\mathstrut +\mathstrut 49 x^{4} \) \(\mathstrut -\mathstrut 30 x^{3} \) \(\mathstrut +\mathstrut 15 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(5575888916015625=3^{8}\cdot 5^{12}\cdot 59^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.64$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29} a^{15} - \frac{6}{29} a^{14} - \frac{8}{29} a^{13} + \frac{7}{29} a^{12} + \frac{13}{29} a^{11} + \frac{4}{29} a^{10} - \frac{1}{29} a^{9} + \frac{12}{29} a^{8} + \frac{12}{29} a^{7} - \frac{1}{29} a^{6} + \frac{4}{29} a^{5} + \frac{13}{29} a^{4} + \frac{7}{29} a^{3} - \frac{8}{29} a^{2} - \frac{6}{29} a + \frac{1}{29}$

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:  \( -\frac{15}{29} a^{15} + \frac{32}{29} a^{14} - \frac{54}{29} a^{13} + \frac{11}{29} a^{12} - \frac{50}{29} a^{11} + \frac{114}{29} a^{10} - \frac{217}{29} a^{9} + \frac{284}{29} a^{8} - \frac{296}{29} a^{7} + \frac{276}{29} a^{6} - \frac{263}{29} a^{5} + \frac{182}{29} a^{4} - \frac{105}{29} a^{3} + \frac{4}{29} a^{2} + \frac{61}{29} a - \frac{44}{29} \) (order $30$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{4}{29} a^{15} + \frac{5}{29} a^{14} - \frac{32}{29} a^{13} + \frac{115}{29} a^{12} - \frac{180}{29} a^{11} + \frac{306}{29} a^{10} - \frac{439}{29} a^{9} + \frac{541}{29} a^{8} - \frac{590}{29} a^{7} + \frac{576}{29} a^{6} - \frac{506}{29} a^{5} + \frac{400}{29} a^{4} - \frac{291}{29} a^{3} + \frac{171}{29} a^{2} - \frac{82}{29} a + \frac{33}{29} \),  \( \frac{28}{29} a^{15} - \frac{110}{29} a^{14} + \frac{269}{29} a^{13} - \frac{413}{29} a^{12} + \frac{567}{29} a^{11} - \frac{758}{29} a^{10} + \frac{900}{29} a^{9} - \frac{1027}{29} a^{8} + \frac{1061}{29} a^{7} - \frac{927}{29} a^{6} + \frac{808}{29} a^{5} - \frac{622}{29} a^{4} + \frac{428}{29} a^{3} - \frac{282}{29} a^{2} + \frac{93}{29} a - \frac{1}{29} \),  \( \frac{7}{29} a^{15} - \frac{42}{29} a^{14} + \frac{147}{29} a^{13} - \frac{357}{29} a^{12} + \frac{700}{29} a^{11} - \frac{1161}{29} a^{10} + \frac{1646}{29} a^{9} - \frac{2004}{29} a^{8} + \frac{2143}{29} a^{7} - \frac{2095}{29} a^{6} + \frac{1797}{29} a^{5} - \frac{1388}{29} a^{4} + \frac{890}{29} a^{3} - \frac{462}{29} a^{2} + \frac{190}{29} a - \frac{51}{29} \),  \( \frac{12}{29} a^{15} - \frac{72}{29} a^{14} + \frac{252}{29} a^{13} - \frac{583}{29} a^{12} + \frac{1055}{29} a^{11} - \frac{1576}{29} a^{10} + \frac{2076}{29} a^{9} - \frac{2408}{29} a^{8} + \frac{2493}{29} a^{7} - \frac{2361}{29} a^{6} + \frac{1962}{29} a^{5} - \frac{1439}{29} a^{4} + \frac{925}{29} a^{3} - \frac{502}{29} a^{2} + \frac{218}{29} a - \frac{46}{29} \),  \( \frac{15}{29} a^{15} - \frac{90}{29} a^{14} + \frac{315}{29} a^{13} - \frac{736}{29} a^{12} + \frac{1355}{29} a^{11} - \frac{2057}{29} a^{10} + \frac{2740}{29} a^{9} - \frac{3184}{29} a^{8} + \frac{3341}{29} a^{7} - \frac{3147}{29} a^{6} + \frac{2612}{29} a^{5} - \frac{1951}{29} a^{4} + \frac{1236}{29} a^{3} - \frac{642}{29} a^{2} + \frac{258}{29} a - \frac{72}{29} \),  \( \frac{55}{29} a^{15} - \frac{243}{29} a^{14} + \frac{633}{29} a^{13} - \frac{1036}{29} a^{12} + \frac{1382}{29} a^{11} - \frac{1723}{29} a^{10} + \frac{2004}{29} a^{9} - \frac{2211}{29} a^{8} + \frac{2226}{29} a^{7} - \frac{1969}{29} a^{6} + \frac{1612}{29} a^{5} - \frac{1286}{29} a^{4} + \frac{965}{29} a^{3} - \frac{643}{29} a^{2} + \frac{308}{29} a - \frac{61}{29} \),  \( \frac{39}{29} a^{15} - \frac{176}{29} a^{14} + \frac{471}{29} a^{13} - \frac{800}{29} a^{12} + \frac{1087}{29} a^{11} - \frac{1294}{29} a^{10} + \frac{1353}{29} a^{9} - \frac{1301}{29} a^{8} + \frac{1164}{29} a^{7} - \frac{851}{29} a^{6} + \frac{533}{29} a^{5} - \frac{334}{29} a^{4} + \frac{186}{29} a^{3} - \frac{138}{29} a^{2} + \frac{85}{29} a - \frac{19}{29} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 155.628338249 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times C_2\wr C_4$ (as 16T261):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 26 conjugacy class representatives for $C_2\times C_2\wr C_4$
Character table for $C_2\times C_2\wr C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.6.74671875.1, 8.2.74671875.1, \(\Q(\zeta_{15})\)

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$