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Properties

Label 16.0.93324816857...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $48.49$
Ramified primes $3, 5, 17$
Class number $100$ (GRH)
Class group $[10, 10]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1079671, -102242, -212699, -125819, 59108, 10441, 18978, 919, 1811, 179, 21, 206, 5, 2, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 10*x^14 + 2*x^13 + 5*x^12 + 206*x^11 + 21*x^10 + 179*x^9 + 1811*x^8 + 919*x^7 + 18978*x^6 + 10441*x^5 + 59108*x^4 - 125819*x^3 - 212699*x^2 - 102242*x + 1079671)
 
gp: K = bnfinit(x^16 - x^15 + 10*x^14 + 2*x^13 + 5*x^12 + 206*x^11 + 21*x^10 + 179*x^9 + 1811*x^8 + 919*x^7 + 18978*x^6 + 10441*x^5 + 59108*x^4 - 125819*x^3 - 212699*x^2 - 102242*x + 1079671, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 10 x^{14} + 2 x^{13} + 5 x^{12} + 206 x^{11} + 21 x^{10} + 179 x^{9} + 1811 x^{8} + 919 x^{7} + 18978 x^{6} + 10441 x^{5} + 59108 x^{4} - 125819 x^{3} - 212699 x^{2} - 102242 x + 1079671 \)

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:  \(933248168570425273681640625=3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.49$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(137,·)$, $\chi_{255}(203,·)$, $\chi_{255}(13,·)$, $\chi_{255}(16,·)$, $\chi_{255}(149,·)$, $\chi_{255}(89,·)$, $\chi_{255}(152,·)$, $\chi_{255}(217,·)$, $\chi_{255}(154,·)$, $\chi_{255}(157,·)$, $\chi_{255}(208,·)$, $\chi_{255}(169,·)$, $\chi_{255}(251,·)$, $\chi_{255}(188,·)$, $\chi_{255}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{354} a^{10} + \frac{5}{177} a^{8} + \frac{35}{354} a^{6} - \frac{1}{118} a^{5} + \frac{25}{177} a^{4} - \frac{5}{118} a^{3} + \frac{25}{354} a^{2} - \frac{5}{118} a + \frac{65}{177}$, $\frac{1}{354} a^{11} + \frac{5}{177} a^{9} + \frac{35}{354} a^{7} - \frac{1}{118} a^{6} + \frac{25}{177} a^{5} - \frac{5}{118} a^{4} + \frac{25}{354} a^{3} - \frac{5}{118} a^{2} + \frac{65}{177} a$, $\frac{1}{1416} a^{12} - \frac{121}{708} a^{8} - \frac{15}{118} a^{7} - \frac{41}{472} a^{6} - \frac{43}{118} a^{5} + \frac{587}{1416} a^{4} - \frac{33}{118} a^{3} - \frac{217}{472} a^{2} - \frac{17}{118} a - \frac{415}{1416}$, $\frac{1}{656586456} a^{13} + \frac{134503}{656586456} a^{12} - \frac{36410}{27357769} a^{11} - \frac{18454}{27357769} a^{10} - \frac{20852623}{328293228} a^{9} + \frac{65872307}{328293228} a^{8} + \frac{11717127}{218862152} a^{7} - \frac{52788159}{218862152} a^{6} - \frac{146264989}{656586456} a^{5} + \frac{124604417}{656586456} a^{4} - \frac{24914981}{218862152} a^{3} - \frac{2121895}{218862152} a^{2} - \frac{178360879}{656586456} a - \frac{326024617}{656586456}$, $\frac{1}{25606871784} a^{14} + \frac{7}{25606871784} a^{13} - \frac{363221}{1969759368} a^{12} + \frac{2887367}{6401717946} a^{11} - \frac{3162469}{12803435892} a^{10} + \frac{49379831}{1422603988} a^{9} - \frac{974031905}{25606871784} a^{8} + \frac{482490331}{25606871784} a^{7} - \frac{92162777}{12803435892} a^{6} + \frac{481922169}{2845207976} a^{5} - \frac{2483788787}{12803435892} a^{4} - \frac{5779462849}{25606871784} a^{3} - \frac{120492781}{6401717946} a^{2} + \frac{11437776391}{25606871784} a - \frac{12665018137}{25606871784}$, $\frac{1}{6134100528985416} a^{15} - \frac{31333}{6134100528985416} a^{14} - \frac{4497667}{6134100528985416} a^{13} + \frac{81797498428}{255587522041059} a^{12} + \frac{851565437719}{1022350088164236} a^{11} + \frac{2799237385793}{3067050264492708} a^{10} - \frac{46598950785401}{6134100528985416} a^{9} - \frac{1135171026603817}{6134100528985416} a^{8} - \frac{28864993777331}{511175044082118} a^{7} + \frac{1361523139165337}{6134100528985416} a^{6} + \frac{753254527699145}{1533525132246354} a^{5} + \frac{2535751928467123}{6134100528985416} a^{4} + \frac{379973940550355}{3067050264492708} a^{3} + \frac{807596554691869}{2044700176328472} a^{2} - \frac{228415942147709}{2044700176328472} a - \frac{268812288081659}{766762566123177}$

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

Class group and class number

$C_{10}\times C_{10}$, which has order $100$ (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:  \( 76496.79137714463 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), 4.0.614125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.614125.1, 4.4.325125.1, \(\Q(\zeta_{15})^+\), 4.0.44217.1, 4.0.1105425.1, 8.0.377149515625.1, 8.8.105706265625.1, 8.0.1221964430625.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$3$3.4.2.2$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}$
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}$
5Data not computed
17Data not computed