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Properties

Label 16.8.43029097861...4969.1
Degree $16$
Signature $[8, 4]$
Discriminant $53^{14}\cdot 89^{10}$
Root discriminant $533.47$
Ramified primes $53, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4334460464259, 6211470910074, -1781065804040, -92952384657, 11289021552, -16828496949, -4410149877, 612567235, -25307029, 1229215, 1442572, -206720, 17170, 1992, -337, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 337*x^14 + 1992*x^13 + 17170*x^12 - 206720*x^11 + 1442572*x^10 + 1229215*x^9 - 25307029*x^8 + 612567235*x^7 - 4410149877*x^6 - 16828496949*x^5 + 11289021552*x^4 - 92952384657*x^3 - 1781065804040*x^2 + 6211470910074*x - 4334460464259)
 
gp: K = bnfinit(x^16 - 4*x^15 - 337*x^14 + 1992*x^13 + 17170*x^12 - 206720*x^11 + 1442572*x^10 + 1229215*x^9 - 25307029*x^8 + 612567235*x^7 - 4410149877*x^6 - 16828496949*x^5 + 11289021552*x^4 - 92952384657*x^3 - 1781065804040*x^2 + 6211470910074*x - 4334460464259, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 337 x^{14} + 1992 x^{13} + 17170 x^{12} - 206720 x^{11} + 1442572 x^{10} + 1229215 x^{9} - 25307029 x^{8} + 612567235 x^{7} - 4410149877 x^{6} - 16828496949 x^{5} + 11289021552 x^{4} - 92952384657 x^{3} - 1781065804040 x^{2} + 6211470910074 x - 4334460464259 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43029097861869577828178499597148046196454969=53^{14}\cdot 89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $533.47$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $53, 89$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{53} a^{8} - \frac{2}{53} a^{7} + \frac{15}{53} a^{6} + \frac{19}{53} a^{5} + \frac{4}{53} a^{4} - \frac{22}{53} a^{3} + \frac{16}{53} a^{2} + \frac{14}{53} a - \frac{17}{53}$, $\frac{1}{53} a^{9} + \frac{11}{53} a^{7} - \frac{4}{53} a^{6} - \frac{11}{53} a^{5} - \frac{14}{53} a^{4} + \frac{25}{53} a^{3} - \frac{7}{53} a^{2} + \frac{11}{53} a + \frac{19}{53}$, $\frac{1}{53} a^{10} + \frac{18}{53} a^{7} - \frac{17}{53} a^{6} - \frac{11}{53} a^{5} - \frac{19}{53} a^{4} + \frac{23}{53} a^{3} - \frac{6}{53} a^{2} + \frac{24}{53} a - \frac{25}{53}$, $\frac{1}{53} a^{11} + \frac{19}{53} a^{7} - \frac{16}{53} a^{6} + \frac{10}{53} a^{5} + \frac{4}{53} a^{4} + \frac{19}{53} a^{3} + \frac{1}{53} a^{2} - \frac{12}{53} a - \frac{12}{53}$, $\frac{1}{53} a^{12} + \frac{22}{53} a^{7} - \frac{10}{53} a^{6} + \frac{14}{53} a^{5} - \frac{4}{53} a^{4} - \frac{5}{53} a^{3} + \frac{2}{53} a^{2} - \frac{13}{53} a + \frac{5}{53}$, $\frac{1}{53} a^{13} - \frac{19}{53} a^{7} + \frac{2}{53} a^{6} + \frac{2}{53} a^{5} + \frac{13}{53} a^{4} + \frac{9}{53} a^{3} + \frac{6}{53} a^{2} + \frac{15}{53} a + \frac{3}{53}$, $\frac{1}{1649667453} a^{14} + \frac{4112449}{1649667453} a^{13} + \frac{11890456}{1649667453} a^{12} - \frac{2165533}{1649667453} a^{11} - \frac{7195942}{1649667453} a^{10} + \frac{1785395}{1649667453} a^{9} + \frac{7090910}{1649667453} a^{8} + \frac{544718435}{1649667453} a^{7} - \frac{11250571}{78555593} a^{6} - \frac{675378581}{1649667453} a^{5} - \frac{604613680}{1649667453} a^{4} + \frac{613686253}{1649667453} a^{3} - \frac{71067925}{1649667453} a^{2} + \frac{28294666}{1649667453} a - \frac{59890049}{549889151}$, $\frac{1}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{15} + \frac{7419342682648709866788915038532740704378421918903875631571783623630}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{14} - \frac{125203800289553003803802115803614276962293781031293439147472114171883516412}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{13} + \frac{137401702021694430958159164447074703716160086030639542072377336538331281642}{15667638682075301411771938977138153402908742435191361208681992922544579453207} a^{12} + \frac{76231119193433532930089162051246933813586913828844212793755058789637964210}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{11} + \frac{1756908497049776546523122765615733245749184807679369077102223971623536373}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{10} + \frac{25150980322955947437452139042565802193281644008514257025018624035069482786}{2764877414483876719724459819494968247572131017974946095649763456919631668213} a^{9} + \frac{187696828001123620365877757187439425704232005922232800236209319031609475882}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{8} - \frac{3232498432625181675226220025691501559709767606516898594111239914774749792181}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{7} + \frac{1648457637535990921907840936752628206930807773965393586233333507296700330179}{47002916046225904235315816931414460208726227305574083626045978767633738359621} a^{6} - \frac{1976158050316299202156031457525611657396157796208908769774655672845404819287}{15667638682075301411771938977138153402908742435191361208681992922544579453207} a^{5} + \frac{6332658656728118283220836992858977891093762278531754160595378810178722173580}{15667638682075301411771938977138153402908742435191361208681992922544579453207} a^{4} + \frac{7066553205393022880962771601952514568581158390222483317009442344600980165331}{15667638682075301411771938977138153402908742435191361208681992922544579453207} a^{3} + \frac{272289990074193531728017483552620034640926719960607475536146124542004406271}{921625804827958906574819939831656082524043672658315365216587818973210556071} a^{2} - \frac{1637552795351366828130569762066928899069008492349417784863852931432403018016}{6714702292317986319330830990202065744103746757939154803720854109661962622803} a + \frac{287183884391323047215250229275495978088936815848277765897611639677950758664}{15667638682075301411771938977138153402908742435191361208681992922544579453207}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14089077231500000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{4717}) \), \(\Q(\sqrt{89}) \), 4.4.13250053.1 x2, 4.4.1179254717.1 x2, \(\Q(\sqrt{53}, \sqrt{89})\), 8.8.1390641687566750089.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$89$89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$