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Properties

Label 16.0.93324816857...0625.3
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 $20$ (GRH)
Class group $[2, 10]$ (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![7311616, 140608, -1543984, -200044, 179453, -93589, -6222, 20074, 51, -2326, 51, 106, 35, 22, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 10*x^14 + 22*x^13 + 35*x^12 + 106*x^11 + 51*x^10 - 2326*x^9 + 51*x^8 + 20074*x^7 - 6222*x^6 - 93589*x^5 + 179453*x^4 - 200044*x^3 - 1543984*x^2 + 140608*x + 7311616)
 
gp: K = bnfinit(x^16 - x^15 - 10*x^14 + 22*x^13 + 35*x^12 + 106*x^11 + 51*x^10 - 2326*x^9 + 51*x^8 + 20074*x^7 - 6222*x^6 - 93589*x^5 + 179453*x^4 - 200044*x^3 - 1543984*x^2 + 140608*x + 7311616, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 10 x^{14} + 22 x^{13} + 35 x^{12} + 106 x^{11} + 51 x^{10} - 2326 x^{9} + 51 x^{8} + 20074 x^{7} - 6222 x^{6} - 93589 x^{5} + 179453 x^{4} - 200044 x^{3} - 1543984 x^{2} + 140608 x + 7311616 \)

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$$\chi_{255}(1,·)$, $\chi_{255}(67,·)$, $\chi_{255}(16,·)$, $\chi_{255}(149,·)$, $\chi_{255}(89,·)$, $\chi_{255}(154,·)$, $\chi_{255}(98,·)$, $\chi_{255}(38,·)$, $\chi_{255}(103,·)$, $\chi_{255}(169,·)$, $\chi_{255}(47,·)$, $\chi_{255}(242,·)$, $\chi_{255}(52,·)$, $\chi_{255}(118,·)$, $\chi_{255}(251,·)$, $\chi_{255}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{8} + \frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{2}$, $\frac{1}{2408787893096} a^{13} + \frac{16747250649}{2408787893096} a^{12} - \frac{96787943249}{2408787893096} a^{11} - \frac{87431813889}{2408787893096} a^{10} - \frac{35547506323}{2408787893096} a^{9} - \frac{385023926961}{2408787893096} a^{8} + \frac{228189153061}{2408787893096} a^{7} - \frac{505018880365}{2408787893096} a^{6} - \frac{541615401471}{2408787893096} a^{5} + \frac{1053983563011}{2408787893096} a^{4} + \frac{64679795530}{301098486637} a^{3} + \frac{236924878709}{1204393946548} a^{2} + \frac{87743322531}{1204393946548} a + \frac{5213042677}{23161422049}$, $\frac{1}{62628485220496} a^{14} - \frac{1}{62628485220496} a^{13} + \frac{356592327171}{7828560652562} a^{12} - \frac{241465043791}{3914280326281} a^{11} + \frac{3797558519803}{62628485220496} a^{10} + \frac{1929104844765}{15657121305124} a^{9} - \frac{13809956239197}{62628485220496} a^{8} - \frac{3152138990497}{15657121305124} a^{7} - \frac{7337350228885}{62628485220496} a^{6} + \frac{3198971338167}{15657121305124} a^{5} + \frac{11320228333345}{31314242610248} a^{4} - \frac{16115517515515}{62628485220496} a^{3} - \frac{6495613861645}{62628485220496} a^{2} - \frac{145975993981}{1204393946548} a + \frac{4784014035}{23161422049}$, $\frac{1}{3256681231465792} a^{15} - \frac{1}{3256681231465792} a^{14} - \frac{5}{1628340615732896} a^{13} + \frac{93918160365651}{1628340615732896} a^{12} - \frac{169986951237653}{3256681231465792} a^{11} + \frac{40945451238017}{1628340615732896} a^{10} - \frac{59623887796149}{3256681231465792} a^{9} - \frac{97218833220879}{1628340615732896} a^{8} + \frac{597796398349643}{3256681231465792} a^{7} + \frac{143909318817265}{1628340615732896} a^{6} + \frac{166617430376341}{1628340615732896} a^{5} - \frac{932387900075933}{3256681231465792} a^{4} + \frac{664363285640341}{3256681231465792} a^{3} - \frac{17151743306645}{62628485220496} a^{2} + \frac{292131954433}{602196973274} a - \frac{2155939585}{23161422049}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( -\frac{107955145}{3256681231465792} a^{15} + \frac{5721622685}{3256681231465792} a^{14} + \frac{2560577027}{1628340615732896} a^{13} - \frac{29255844295}{1628340615732896} a^{12} + \frac{119722255805}{3256681231465792} a^{11} + \frac{92517559265}{1628340615732896} a^{10} + \frac{589543046845}{3256681231465792} a^{9} + \frac{2421918328993}{1628340615732896} a^{8} - \frac{13062896410435}{3256681231465792} a^{7} - \frac{940397268095}{1628340615732896} a^{6} + \frac{56680229555075}{1628340615732896} a^{5} - \frac{24824825368475}{3256681231465792} a^{4} - \frac{74341954504713}{3256681231465792} a^{3} + \frac{380541886125}{1204393946548} a^{2} - \frac{6801174135}{23161422049} a - \frac{61750342940}{23161422049} \) (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:  \( 886036.8840713138 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), 4.4.5527125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.5527125.1, \(\Q(\zeta_{5})\), 4.0.36125.1, 4.0.1105425.1, 4.0.44217.1, 8.8.30549110765625.1, 8.0.1305015625.1, 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 ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
17Data not computed