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Properties

Label 16.0.10107540821...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{8}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}$
Root discriminant $365.42$
Ramified primes $2, 3, 5, 13, 1249, 1511$
Class number $992277120$ (GRH)
Class group $[2, 2, 2, 2, 2, 31008660]$ (GRH)
Galois group 16T1547

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3136651030534129, -119034342218376, 490041878843476, -25388246588088, 30925819337576, -1458109683324, 903868150860, -29544777456, 13586028503, -267407364, 112405248, -1113408, 511010, -1716, 1168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1168*x^14 - 1716*x^13 + 511010*x^12 - 1113408*x^11 + 112405248*x^10 - 267407364*x^9 + 13586028503*x^8 - 29544777456*x^7 + 903868150860*x^6 - 1458109683324*x^5 + 30925819337576*x^4 - 25388246588088*x^3 + 490041878843476*x^2 - 119034342218376*x + 3136651030534129)
 
gp: K = bnfinit(x^16 + 1168*x^14 - 1716*x^13 + 511010*x^12 - 1113408*x^11 + 112405248*x^10 - 267407364*x^9 + 13586028503*x^8 - 29544777456*x^7 + 903868150860*x^6 - 1458109683324*x^5 + 30925819337576*x^4 - 25388246588088*x^3 + 490041878843476*x^2 - 119034342218376*x + 3136651030534129, 1)
 

Normalized defining polynomial

\( x^{16} + 1168 x^{14} - 1716 x^{13} + 511010 x^{12} - 1113408 x^{11} + 112405248 x^{10} - 267407364 x^{9} + 13586028503 x^{8} - 29544777456 x^{7} + 903868150860 x^{6} - 1458109683324 x^{5} + 30925819337576 x^{4} - 25388246588088 x^{3} + 490041878843476 x^{2} - 119034342218376 x + 3136651030534129 \)

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:  \(101075408216203634510751650650521600000000=2^{40}\cdot 3^{4}\cdot 5^{8}\cdot 13^{8}\cdot 1249^{2}\cdot 1511^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $365.42$
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:  $2, 3, 5, 13, 1249, 1511$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{2}{9} a^{7} + \frac{1}{3} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{24960987} a^{14} + \frac{264658}{8320329} a^{13} + \frac{132103}{2773443} a^{12} - \frac{1159}{2773443} a^{11} + \frac{2001941}{24960987} a^{10} - \frac{1583251}{8320329} a^{9} - \frac{7848677}{24960987} a^{8} + \frac{2513026}{8320329} a^{7} + \frac{9435106}{24960987} a^{6} + \frac{2266534}{8320329} a^{5} - \frac{6833140}{24960987} a^{4} - \frac{3440027}{8320329} a^{3} - \frac{1302850}{2773443} a^{2} + \frac{1326634}{8320329} a - \frac{5055083}{24960987}$, $\frac{1}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{15} - \frac{328522758127639977281949423104533195681890285170252087750073582380873344130281546680284147}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{14} + \frac{264356100235845600874755743046371581252945310378274976402810447492192999351919813906681262065105}{5782598496100599008719582413014599620071173201132937006755951981594996535858419009397669278257213} a^{13} + \frac{26612952409118701795995933352122655302701490738565030664296259209247353336710948643012768972448}{826085499442942715531368916144942802867310457304705286679421711656428076551202715628238468322459} a^{12} + \frac{2316325069804015776761458183630956628812535457352207418948497937870581069967174678949532935995547}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{11} - \frac{2033131137652400046592652279419173082089887825801295309210551687897965492607407061441593144926373}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{10} + \frac{2563621849962522416852474368256211081271376195088437586772298657825955140609027768073599253012844}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{9} + \frac{3679383119795529308328143711362660983118595475732060815959478657975101874986147207783320292114618}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{8} + \frac{303678797472937077668992300835646716958403970152385347169799316767402136790562627830691178450780}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{7} + \frac{1232416828192233055094355363493178103858086910559359244517864967379459303942555326366075917884480}{2478256498328828146594106748434828408601931371914115860038265134969284229653608146884715404967377} a^{6} + \frac{4352389532604661507061739523185542668133073540832261691296944470252697564690625191367222312908110}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a^{5} + \frac{423137927079468590481498920457969562543889523741648693982103071998342177869685357787781910845854}{1020458558135399825068161602296694050600795270788165354133403290869705271033838648717235754986567} a^{4} - \frac{33982135964418628612034165847308124488122104646409154166711818358860419274828072370180932196835}{91787277715882523947929879571660311429701161922745031853269079072936452950133635069804274258051} a^{3} + \frac{1561182056127067380165976083881738833288049579973758400181323049941463133911813971008335176138335}{5782598496100599008719582413014599620071173201132937006755951981594996535858419009397669278257213} a^{2} - \frac{8526409725962156989737839783871100766956858823496019727911390646146332450344170435901949095607848}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639} a - \frac{6713912611631954243434987271824510264194396612766563658134561534960346582940515364456811406910744}{17347795488301797026158747239043798860213519603398811020267855944784989607575257028193007834771639}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{31008660}$, which has order $992277120$ (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:  \( 113414.110194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1547:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 133 conjugacy class representatives for t16n1547 are not computed
Character table for t16n1547 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.421149081600.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1249Data not computed
1511Data not computed