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Properties

Label 16.8.58347649072...3344.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{42}\cdot 7^{4}\cdot 17^{8}\cdot 89^{2}$
Root discriminant $72.51$
Ramified primes $2, 7, 17, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-219964, 1109984, -2273992, 3226520, -3846476, 2661600, -814404, -40336, 214060, -78704, 4456, 2964, -774, 96, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 96*x^13 - 774*x^12 + 2964*x^11 + 4456*x^10 - 78704*x^9 + 214060*x^8 - 40336*x^7 - 814404*x^6 + 2661600*x^5 - 3846476*x^4 + 3226520*x^3 - 2273992*x^2 + 1109984*x - 219964)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 96*x^13 - 774*x^12 + 2964*x^11 + 4456*x^10 - 78704*x^9 + 214060*x^8 - 40336*x^7 - 814404*x^6 + 2661600*x^5 - 3846476*x^4 + 3226520*x^3 - 2273992*x^2 + 1109984*x - 219964, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 96 x^{13} - 774 x^{12} + 2964 x^{11} + 4456 x^{10} - 78704 x^{9} + 214060 x^{8} - 40336 x^{7} - 814404 x^{6} + 2661600 x^{5} - 3846476 x^{4} + 3226520 x^{3} - 2273992 x^{2} + 1109984 x - 219964 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(583476490725942985079960633344=2^{42}\cdot 7^{4}\cdot 17^{8}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.51$
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:  $2, 7, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{36495914998355469419996515683492983299867359154268} a^{15} - \frac{3029352986145204851056792346444419395042716320369}{36495914998355469419996515683492983299867359154268} a^{14} + \frac{3684150397023855173448806980726569730696338018495}{36495914998355469419996515683492983299867359154268} a^{13} - \frac{592236805007144595554759698251238118207343694591}{36495914998355469419996515683492983299867359154268} a^{12} - \frac{2107982651763425886524945886860073433624662284602}{9123978749588867354999128920873245824966839788567} a^{11} + \frac{1768047261031353517743657244559410996546891933171}{9123978749588867354999128920873245824966839788567} a^{10} + \frac{613720022794880224942289375854252074091040499913}{2606851071311104958571179691678070235704811368162} a^{9} - \frac{40074579007402772550032568285817267580099242820}{9123978749588867354999128920873245824966839788567} a^{8} - \frac{581228674791431336135889809670631827892709226535}{18247957499177734709998257841746491649933679577134} a^{7} + \frac{933224667105463031043165792332958638088255373521}{2606851071311104958571179691678070235704811368162} a^{6} - \frac{3450935406412250803120594310897880692790050424845}{18247957499177734709998257841746491649933679577134} a^{5} - \frac{4505382093871402853871470037701746928147625246847}{18247957499177734709998257841746491649933679577134} a^{4} + \frac{734035034153559083166794438372292882906437593400}{9123978749588867354999128920873245824966839788567} a^{3} - \frac{841171276947654287426106388070558255525466783633}{9123978749588867354999128920873245824966839788567} a^{2} + \frac{3577821936888468408595226839337519405433408204675}{9123978749588867354999128920873245824966839788567} a + \frac{231074232394908007318105552644756871726749245}{71842352358967459488182117487190912007612911721}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 1086600250.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.24.32$x^{8} + 20 x^{4} + 164$$8$$1$$24$$Q_8:C_2$$[2, 3, 4]^{2}$
2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$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$$[\ ]_{2}^{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$