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Properties

Label 16.0.17197312483...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{26}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}$
Root discriminant $283.28$
Ramified primes $2, 5, 7, 241$
Class number $240$ (GRH)
Class group $[4, 60]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4615706356, -5322863812, 1769942058, -623815062, 661014999, -261805816, 19937397, 1110888, 2943728, -735808, -8277, 364, 3898, -316, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 - 316*x^13 + 3898*x^12 + 364*x^11 - 8277*x^10 - 735808*x^9 + 2943728*x^8 + 1110888*x^7 + 19937397*x^6 - 261805816*x^5 + 661014999*x^4 - 623815062*x^3 + 1769942058*x^2 - 5322863812*x + 4615706356)
 
gp: K = bnfinit(x^16 - 2*x^15 - 15*x^14 - 316*x^13 + 3898*x^12 + 364*x^11 - 8277*x^10 - 735808*x^9 + 2943728*x^8 + 1110888*x^7 + 19937397*x^6 - 261805816*x^5 + 661014999*x^4 - 623815062*x^3 + 1769942058*x^2 - 5322863812*x + 4615706356, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 15 x^{14} - 316 x^{13} + 3898 x^{12} + 364 x^{11} - 8277 x^{10} - 735808 x^{9} + 2943728 x^{8} + 1110888 x^{7} + 19937397 x^{6} - 261805816 x^{5} + 661014999 x^{4} - 623815062 x^{3} + 1769942058 x^{2} - 5322863812 x + 4615706356 \)

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:  \(1719731248304838654458056435302400000000=2^{26}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $283.28$
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:  $2, 5, 7, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{885633562013242337019728317874681660413943621942127647311636359522} a^{15} - \frac{49704547190891802406593136901864420451497053055207039887204259093}{885633562013242337019728317874681660413943621942127647311636359522} a^{14} - \frac{47505915283366960770205696018102664763708451562673663922434385631}{885633562013242337019728317874681660413943621942127647311636359522} a^{13} - \frac{17823534011724606541460545066113822198582536180553773572617920689}{885633562013242337019728317874681660413943621942127647311636359522} a^{12} + \frac{11443646512645763745165656563799635894736363579475734010360910192}{442816781006621168509864158937340830206971810971063823655818179761} a^{11} - \frac{155620640714545441436132029490263538464246355930442052295225067037}{885633562013242337019728317874681660413943621942127647311636359522} a^{10} - \frac{42441164706680418673916197313836999255854540075582980626497022697}{885633562013242337019728317874681660413943621942127647311636359522} a^{9} - \frac{64132972186139247499380882306016230887735537379804673482350595359}{295211187337747445673242772624893886804647873980709215770545453174} a^{8} - \frac{206078654956303145959384154915285686364863680270779301265268501252}{442816781006621168509864158937340830206971810971063823655818179761} a^{7} + \frac{27670506904381798331276124055079127859012370573107344914533549401}{295211187337747445673242772624893886804647873980709215770545453174} a^{6} - \frac{213371683592555834257446344228009463352999503042685521123371346685}{885633562013242337019728317874681660413943621942127647311636359522} a^{5} + \frac{88667159249262814406052577179421947867006752898994735758892631599}{295211187337747445673242772624893886804647873980709215770545453174} a^{4} + \frac{106765038552495517771242884163329439601703994627603494491914352195}{885633562013242337019728317874681660413943621942127647311636359522} a^{3} - \frac{3876395924191168867416208277072047429284254774720857310072328106}{40256071000601924409987650812485530018815619179187620332347107251} a^{2} - \frac{186797654876651215638656941661212495194778192926814674774885195812}{442816781006621168509864158937340830206971810971063823655818179761} a - \frac{213182391696786156372124280722761766291572704358134678624995019903}{442816781006621168509864158937340830206971810971063823655818179761}$

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

Class group and class number

$C_{4}\times C_{60}$, which has order $240$ (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:  \( 91987461555.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{8435}) \), \(\Q(\sqrt{1687}) \), 4.0.6025.1, 4.0.4723600.3, \(\Q(\sqrt{5}, \sqrt{1687})\), 8.0.1295926327833760000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.18.1$x^{8} + 14 x^{6} + 10 x^{4} + 12 x^{2} + 16 x + 4$$4$$2$$18$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7Data not computed
241Data not computed