Properties

Label 16.0.5236602731044893.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 13^{4}\cdot 181^{2}\cdot 853$
Root discriminant $9.60$
Ramified primes $3, 13, 181, 853$
Class number $1$
Class group Trivial
Galois Group 16T1823

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 9, -2, -22, 51, -58, 21, 60, -148, 195, -180, 125, -66, 26, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*x + 1)
gp: K = bnfinit(x^16 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 7 x^{15} \) \(\mathstrut +\mathstrut 26 x^{14} \) \(\mathstrut -\mathstrut 66 x^{13} \) \(\mathstrut +\mathstrut 125 x^{12} \) \(\mathstrut -\mathstrut 180 x^{11} \) \(\mathstrut +\mathstrut 195 x^{10} \) \(\mathstrut -\mathstrut 148 x^{9} \) \(\mathstrut +\mathstrut 60 x^{8} \) \(\mathstrut +\mathstrut 21 x^{7} \) \(\mathstrut -\mathstrut 58 x^{6} \) \(\mathstrut +\mathstrut 51 x^{5} \) \(\mathstrut -\mathstrut 22 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(5236602731044893=3^{8}\cdot 13^{4}\cdot 181^{2}\cdot 853\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.60$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 13, 181, 853$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2041} a^{15} - \frac{924}{2041} a^{14} + \frac{319}{2041} a^{13} - \frac{726}{2041} a^{12} + \frac{501}{2041} a^{11} - \frac{372}{2041} a^{10} + \frac{472}{2041} a^{9} - \frac{280}{2041} a^{8} - \frac{346}{2041} a^{7} + \frac{948}{2041} a^{6} + \frac{92}{2041} a^{5} - \frac{632}{2041} a^{4} - \frac{122}{2041} a^{3} - \frac{383}{2041} a^{2} + \frac{168}{2041} a - \frac{986}{2041}$

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:  \( -\frac{917}{157} a^{15} + \frac{5945}{157} a^{14} - \frac{20599}{157} a^{13} + \frac{48889}{157} a^{12} - \frac{86228}{157} a^{11} + \frac{113631}{157} a^{10} - \frac{108933}{157} a^{9} + \frac{66162}{157} a^{8} - \frac{9749}{157} a^{7} - \frac{29366}{157} a^{6} + \frac{37311}{157} a^{5} - \frac{23964}{157} a^{4} + \frac{4643}{157} a^{3} + \frac{5654}{157} a^{2} - \frac{5063}{157} a + \frac{1412}{157} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{3262}{2041} a^{15} - \frac{24023}{2041} a^{14} + \frac{91513}{2041} a^{13} - \frac{235367}{2041} a^{12} + \frac{448441}{2041} a^{11} - \frac{646066}{2041} a^{10} + \frac{692649}{2041} a^{9} - \frac{509242}{2041} a^{8} + \frac{179629}{2041} a^{7} + \frac{106393}{2041} a^{6} - \frac{218310}{2041} a^{5} + \frac{173311}{2041} a^{4} - \frac{61199}{2041} a^{3} - \frac{20664}{2041} a^{2} + \frac{35725}{2041} a - \frac{11962}{2041} \),  \( \frac{5790}{2041} a^{15} - \frac{35196}{2041} a^{14} + \frac{118283}{2041} a^{13} - \frac{274615}{2041} a^{12} + \frac{476082}{2041} a^{11} - \frac{617007}{2041} a^{10} + \frac{585748}{2041} a^{9} - \frac{353739}{2041} a^{8} + \frac{62152}{2041} a^{7} + \frac{143541}{2041} a^{6} - \frac{181670}{2041} a^{5} + \frac{118611}{2041} a^{4} - \frac{20604}{2041} a^{3} - \frac{25536}{2041} a^{2} + \frac{23655}{2041} a - \frac{6386}{2041} \),  \( \frac{1343}{2041} a^{15} - \frac{6127}{2041} a^{14} + \frac{14094}{2041} a^{13} - \frac{17789}{2041} a^{12} + \frac{3395}{2041} a^{11} + \frac{41269}{2041} a^{10} - \frac{98823}{2041} a^{9} + \frac{130128}{2041} a^{8} - \frac{99339}{2041} a^{7} + \frac{34277}{2041} a^{6} + \frac{21506}{2041} a^{5} - \frac{36458}{2041} a^{4} + \frac{30049}{2041} a^{3} - \frac{4119}{2041} a^{2} - \frac{9091}{2041} a + \frac{6534}{2041} \),  \( \frac{6956}{2041} a^{15} - \frac{43096}{2041} a^{14} + \frac{145308}{2041} a^{13} - \frac{337387}{2041} a^{12} + \frac{582654}{2041} a^{11} - \frac{748691}{2041} a^{10} + \frac{695244}{2041} a^{9} - \frac{394479}{2041} a^{8} + \frac{26096}{2041} a^{7} + \frac{216163}{2041} a^{6} - \frac{247883}{2041} a^{5} + \frac{147074}{2041} a^{4} - \frac{17945}{2041} a^{3} - \frac{41463}{2041} a^{2} + \frac{31771}{2041} a - \frac{6979}{2041} \),  \( \frac{1180}{2041} a^{15} - \frac{6549}{2041} a^{14} + \frac{19245}{2041} a^{13} - \frac{38239}{2041} a^{12} + \frac{54397}{2041} a^{11} - \frac{51170}{2041} a^{10} + \frac{24259}{2041} a^{9} + \frac{10447}{2041} a^{8} - \frac{24572}{2041} a^{7} + \frac{14459}{2041} a^{6} + \frac{2428}{2041} a^{5} - \frac{11000}{2041} a^{4} + \frac{13197}{2041} a^{3} - \frac{4961}{2041} a^{2} - \frac{3819}{2041} a + \frac{3972}{2041} \),  \( \frac{7097}{2041} a^{15} - \frac{40715}{2041} a^{14} + \frac{131098}{2041} a^{13} - \frac{292801}{2041} a^{12} + \frac{487974}{2041} a^{11} - \frac{601125}{2041} a^{10} + \frac{533204}{2041} a^{9} - \frac{276802}{2041} a^{8} - \frac{239}{2041} a^{7} + \frac{174305}{2041} a^{6} - \frac{177763}{2041} a^{5} + \frac{100823}{2041} a^{4} - \frac{450}{2041} a^{3} - \frac{28113}{2041} a^{2} + \frac{20762}{2041} a - \frac{3135}{2041} \),  \( a^{15} - 7 a^{14} + 25 a^{13} - 60 a^{12} + 106 a^{11} - 139 a^{10} + 130 a^{9} - 74 a^{8} + 4 a^{7} + 39 a^{6} - 44 a^{5} + 26 a^{4} - 3 a^{3} - 9 a^{2} + 5 a \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 29.7678343854 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

16T1823:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 32768
The 230 conjugacy class representatives for t16n1823 are not computed
Character table for t16n1823 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 8.0.2477709.1

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 $16$ R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
181Data not computed
853Data not computed