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Properties

Label 16.0.13647448497...6881.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{8}\cdot 89^{14}$
Root discriminant $209.38$
Ramified primes $17, 89$
Class number $51171146$ (GRH)
Class group $[51171146]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8388608, -12058624, -1187840, 8123904, 7502144, -1040544, -2777412, -67436, 486655, -61350, -566, 266, -704, 174, -6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608)
 
gp: K = bnfinit(x^16 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 6 x^{14} + 174 x^{13} - 704 x^{12} + 266 x^{11} - 566 x^{10} - 61350 x^{9} + 486655 x^{8} - 67436 x^{7} - 2777412 x^{6} - 1040544 x^{5} + 7502144 x^{4} + 8123904 x^{3} - 1187840 x^{2} - 12058624 x + 8388608 \)

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:  \(13647448497711210780660288977472356881=17^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.38$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1513=17\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{1513}(1,·)$, $\chi_{1513}(322,·)$, $\chi_{1513}(1412,·)$, $\chi_{1513}(902,·)$, $\chi_{1513}(713,·)$, $\chi_{1513}(611,·)$, $\chi_{1513}(101,·)$, $\chi_{1513}(800,·)$, $\chi_{1513}(1123,·)$, $\chi_{1513}(390,·)$, $\chi_{1513}(1191,·)$, $\chi_{1513}(1512,·)$, $\chi_{1513}(749,·)$, $\chi_{1513}(52,·)$, $\chi_{1513}(1461,·)$, $\chi_{1513}(764,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{4} + \frac{7}{32} a^{3} + \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} - \frac{7}{128} a^{4} + \frac{3}{16} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{16} a^{4} + \frac{1}{16} a^{2}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} - \frac{1}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} + \frac{5}{128} a^{4} - \frac{11}{64} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{512} a^{11} + \frac{1}{512} a^{9} + \frac{7}{512} a^{7} - \frac{1}{32} a^{6} + \frac{11}{512} a^{5} - \frac{5}{128} a^{3} + \frac{1}{32} a^{2}$, $\frac{1}{8192} a^{12} - \frac{1}{1024} a^{11} + \frac{13}{8192} a^{10} + \frac{7}{2048} a^{9} - \frac{17}{8192} a^{8} - \frac{3}{256} a^{7} + \frac{151}{8192} a^{6} - \frac{101}{2048} a^{5} - \frac{117}{2048} a^{4} - \frac{17}{256} a^{3} - \frac{3}{128} a^{2} - \frac{7}{16} a$, $\frac{1}{8192} a^{13} - \frac{3}{8192} a^{11} + \frac{1}{2048} a^{10} - \frac{1}{8192} a^{9} + \frac{3}{1024} a^{8} - \frac{25}{8192} a^{7} - \frac{23}{2048} a^{6} - \frac{89}{2048} a^{5} + \frac{5}{128} a^{4} + \frac{15}{64} a^{3} - \frac{3}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{2195456} a^{14} - \frac{129}{2195456} a^{13} - \frac{111}{2195456} a^{12} - \frac{105}{2195456} a^{11} + \frac{2367}{2195456} a^{10} + \frac{2233}{2195456} a^{9} - \frac{2437}{2195456} a^{8} - \frac{13555}{2195456} a^{7} + \frac{5825}{548864} a^{6} - \frac{26247}{548864} a^{5} - \frac{3637}{137216} a^{4} - \frac{2595}{34304} a^{3} - \frac{1359}{8576} a^{2} + \frac{107}{1072} a + \frac{15}{67}$, $\frac{1}{140047363626455236161014595584} a^{15} - \frac{9219013757224664414859}{70023681813227618080507297792} a^{14} - \frac{4115853235449699489907371}{70023681813227618080507297792} a^{13} + \frac{651802404849431947364543}{70023681813227618080507297792} a^{12} - \frac{3495296338903101535468177}{8752960226653452260063412224} a^{11} + \frac{31994321898779791587165437}{70023681813227618080507297792} a^{10} + \frac{106630738663856887242595085}{70023681813227618080507297792} a^{9} - \frac{262392688629674452585768475}{70023681813227618080507297792} a^{8} + \frac{781992993730381686397242511}{140047363626455236161014595584} a^{7} + \frac{358628037587549400934331709}{35011840906613809040253648896} a^{6} - \frac{270505398246654153655840097}{35011840906613809040253648896} a^{5} - \frac{103430694382596542375645033}{4376480113326726130031706112} a^{4} + \frac{131364841910469185699344261}{2188240056663363065015853056} a^{3} - \frac{56250145971867726779458833}{273530007082920383126981632} a^{2} + \frac{7429376931888759947665221}{17095625442682523945436352} a - \frac{234464884140984139598391}{534238295083828873294886}$

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

Class group and class number

$C_{51171146}$, which has order $51171146$ (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:  \( 651512602.568 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{1513}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.4.704969.1, 4.4.203736041.1, 8.8.41508374402353681.1, 8.0.44231334895529.1, 8.0.3694245321809477609.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$