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Properties

Label 16.0.55346356857...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $83.45$
Ramified primes $5, 41, 661$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![126912289471, 48410059339, 23208296551, 6828187756, 1920959023, 280561309, 18719919, -3217617, -3044487, -559901, -90799, 9208, 7373, 103, -146, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 146*x^14 + 103*x^13 + 7373*x^12 + 9208*x^11 - 90799*x^10 - 559901*x^9 - 3044487*x^8 - 3217617*x^7 + 18719919*x^6 + 280561309*x^5 + 1920959023*x^4 + 6828187756*x^3 + 23208296551*x^2 + 48410059339*x + 126912289471)
 
gp: K = bnfinit(x^16 - 2*x^15 - 146*x^14 + 103*x^13 + 7373*x^12 + 9208*x^11 - 90799*x^10 - 559901*x^9 - 3044487*x^8 - 3217617*x^7 + 18719919*x^6 + 280561309*x^5 + 1920959023*x^4 + 6828187756*x^3 + 23208296551*x^2 + 48410059339*x + 126912289471, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 146 x^{14} + 103 x^{13} + 7373 x^{12} + 9208 x^{11} - 90799 x^{10} - 559901 x^{9} - 3044487 x^{8} - 3217617 x^{7} + 18719919 x^{6} + 280561309 x^{5} + 1920959023 x^{4} + 6828187756 x^{3} + 23208296551 x^{2} + 48410059339 x + 126912289471 \)

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:  \(5534635685714755139654541015625=5^{14}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.45$
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:  $5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{15} - \frac{518759720266569958444512129289734468961847018005234242870528993205128430}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{14} + \frac{759647792722403093870706292673466998139785844547099049274127061541145954}{14712137779996805180636156688462046667917954548779870085572757112369233031} a^{13} + \frac{260784439335702800240280084569176085981993504845890261032574662829446176}{14712137779996805180636156688462046667917954548779870085572757112369233031} a^{12} + \frac{293687376828846406851562248530505047077688842574573065086735491743206361}{4012401212726401412900770005944194545795805786030873659701661030646154463} a^{11} + \frac{3582874474518186316384727260524240329018052355585127984221105556616674986}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{10} - \frac{113674276214047985334003053197380441655651964809685105755448280334219345}{723547759671974025277188033858789180389407600759665741913414284214880313} a^{9} - \frac{9383047745682239964467048949239858708694420988532778285791277019680425431}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{8} - \frac{10403228716032621105252202809296130162083021445807171165214569370354762142}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{7} + \frac{16985040864591608222228524006535935193643574069263602768476261793150577535}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{6} - \frac{123353625890802900783286223930468966178736881746896277855246343837725486}{723547759671974025277188033858789180389407600759665741913414284214880313} a^{5} + \frac{11967642847997925026290456475205063298975755930943821347448259550370149222}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{4} + \frac{18529929370280499528452496705220100484953052945375964730924046512045927059}{44136413339990415541908470065386140003753863646339610256718271337107699093} a^{3} + \frac{2898712731678141300212989551190077151527035696332280839411300587164206468}{14712137779996805180636156688462046667917954548779870085572757112369233031} a^{2} + \frac{5255923997098984669567544715554645696031986042425121308357367075741426653}{44136413339990415541908470065386140003753863646339610256718271337107699093} a + \frac{441928697574473539522554211787142370134357229457676468436817162769868622}{1337467070908800470966923335314731515265268595343624553233887010215384821}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( \frac{73468701494017742829763085319759508068180843034600720}{692832600740477450057474327845777011561556574479475534586208953} a^{15} - \frac{599596093720470126460383114884541621854286845686440049}{692832600740477450057474327845777011561556574479475534586208953} a^{14} - \frac{7541397227511430216898668872908261608142281366426990415}{692832600740477450057474327845777011561556574479475534586208953} a^{13} + \frac{54140007262579885613595091318650263079450110653839762026}{692832600740477450057474327845777011561556574479475534586208953} a^{12} + \frac{299153163418015691650356211156060146407298742821530804218}{692832600740477450057474327845777011561556574479475534586208953} a^{11} - \frac{1109951751745244595602217502933638112534906443962681461898}{692832600740477450057474327845777011561556574479475534586208953} a^{10} - \frac{93015391111582965743203266744801846171443688591742323985}{11357911487548810656679907013865196910845189745565172698134573} a^{9} - \frac{5725864895635064205305621242463430270141898942910742143992}{230944200246825816685824775948592337187185524826491844862069651} a^{8} + \frac{14060409782249469836604801270778565618200608974729218236505}{692832600740477450057474327845777011561556574479475534586208953} a^{7} + \frac{213933633749246059299491037842011405589172375929825008430359}{692832600740477450057474327845777011561556574479475534586208953} a^{6} + \frac{16343813888815142754600692654916041580021159160525176481834}{11357911487548810656679907013865196910845189745565172698134573} a^{5} + \frac{11587455454889265771225478408168438798249152524926892290590700}{692832600740477450057474327845777011561556574479475534586208953} a^{4} + \frac{29284954496193561753982574801830542712442445533326438363477069}{692832600740477450057474327845777011561556574479475534586208953} a^{3} + \frac{128398385846698811607021805114542474706948214192211968763174301}{692832600740477450057474327845777011561556574479475534586208953} a^{2} + \frac{9770364782135526030863760246039501944046552124769442360328055}{692832600740477450057474327845777011561556574479475534586208953} a + \frac{1080315442464927906654848867288466192935822264614496759468912755}{692832600740477450057474327845777011561556574479475534586208953} \) (order $10$)
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:  \( 303355098.517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.26265625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
661Data not computed