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Properties

Label 16.0.27434520006...0953.7
Degree $16$
Signature $[0, 8]$
Discriminant $71^{12}\cdot 73^{13}$
Root discriminant $798.72$
Ramified primes $71, 73$
Class number $896$ (GRH)
Class group $[2, 2, 4, 56]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56812866336026263, -8522534230317664, 2915744632504734, 110526109065100, 65896886115085, 6160562935376, 2150065913720, 121079233072, 23302576816, 783771762, 160968008, 1854491, 612545, 544, 1243, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 1243*x^14 + 544*x^13 + 612545*x^12 + 1854491*x^11 + 160968008*x^10 + 783771762*x^9 + 23302576816*x^8 + 121079233072*x^7 + 2150065913720*x^6 + 6160562935376*x^5 + 65896886115085*x^4 + 110526109065100*x^3 + 2915744632504734*x^2 - 8522534230317664*x + 56812866336026263)
 
gp: K = bnfinit(x^16 - 3*x^15 + 1243*x^14 + 544*x^13 + 612545*x^12 + 1854491*x^11 + 160968008*x^10 + 783771762*x^9 + 23302576816*x^8 + 121079233072*x^7 + 2150065913720*x^6 + 6160562935376*x^5 + 65896886115085*x^4 + 110526109065100*x^3 + 2915744632504734*x^2 - 8522534230317664*x + 56812866336026263, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 1243 x^{14} + 544 x^{13} + 612545 x^{12} + 1854491 x^{11} + 160968008 x^{10} + 783771762 x^{9} + 23302576816 x^{8} + 121079233072 x^{7} + 2150065913720 x^{6} + 6160562935376 x^{5} + 65896886115085 x^{4} + 110526109065100 x^{3} + 2915744632504734 x^{2} - 8522534230317664 x + 56812866336026263 \)

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:  \(27434520006428699643943668849708793135373210953=71^{12}\cdot 73^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $798.72$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{15} - \frac{446653574272016499082194004849562533924867781161699385736622775508744287877991970153938277344190610976782069329565890545}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{14} + \frac{1294192704213661095264816068012450383766922100363190635833292409343735616805209206374934727349270205976033499939848433695}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{13} + \frac{318269821372143372366543851018924884312764621305382631960782844932765098312646947099223208641513595102288043917803426365}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{12} - \frac{2121075275470205779708652868866232598761429892699544534224215106278430919863316645502997084937651343212060746398081396934}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{11} - \frac{1098482115783692949571378889427700642032001326454048789996393866763547684992933506956172252880025069430756620725761487360}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{10} - \frac{944987027426957134993916351495960453347895498624037099428568688021407334877150122084677372934709411260386215994544563642}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{9} + \frac{728900043186304517914976467015953214534988968913654413037414033758700776159901253351081107528568987304007018508791197402}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{8} + \frac{1526568697506633298028786887115821763546306517770511049028807036872256021410285945736603301981090159565966992668294063661}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{7} - \frac{702545224445470590832568703441492936815949867245784273562505150266883827387846126890393399449554179680554771037812783851}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{6} + \frac{1224911707728013215909503469844518936701618691933044023674885995712404291155139286045518943484199845030009466213450054686}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{5} + \frac{1306570963172703816518104957422964585801514268063218282501917516731118161293858554538197108003158167026755979280042719331}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{4} - \frac{1776784773239810370444916565162251472927423343840484042895586456741498809709855916515665449607003247972807508303770854520}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{3} + \frac{1584378397683532051097007803193053543414224122362638453785847029957414965953054592638106805032734643260175372470707271315}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a^{2} - \frac{1987196667511973033717522672637902330383652395426689571112330424674466230485543717720670637282280526165856218954147342420}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961} a - \frac{496632636250604831595668601904574148032697809872332496365408955934797843444363139728852900983749611958359174423928805178}{4669266758453259434246881769374972849798103026213254148905937889266785412685656338956294874029743329278513988662067066961}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{56}$, which has order $896$ (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:  $7$
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:  \( 31378584997900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-71}) \), 4.0.367993.1, 8.0.52680234011477833.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $16$ $16$ $16$

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
$71$71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
71.4.3.1$x^{4} + 142$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
73Data not computed