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Properties

Label 16.0.96826519964...0000.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $20.49$
Ramified primes $2, 3, 5, 7$
Class number $4$
Class group $[4]$
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85, -460, 1184, -1870, 2098, -1798, 1118, -436, 153, -56, 4, 2, 3, -2, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 - 2*x^13 + 3*x^12 + 2*x^11 + 4*x^10 - 56*x^9 + 153*x^8 - 436*x^7 + 1118*x^6 - 1798*x^5 + 2098*x^4 - 1870*x^3 + 1184*x^2 - 460*x + 85)
 
gp: K = bnfinit(x^16 - x^14 - 2*x^13 + 3*x^12 + 2*x^11 + 4*x^10 - 56*x^9 + 153*x^8 - 436*x^7 + 1118*x^6 - 1798*x^5 + 2098*x^4 - 1870*x^3 + 1184*x^2 - 460*x + 85, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} - 2 x^{13} + 3 x^{12} + 2 x^{11} + 4 x^{10} - 56 x^{9} + 153 x^{8} - 436 x^{7} + 1118 x^{6} - 1798 x^{5} + 2098 x^{4} - 1870 x^{3} + 1184 x^{2} - 460 x + 85 \)

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:  \(968265199641600000000=2^{16}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.49$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{545} a^{14} - \frac{223}{545} a^{13} - \frac{48}{545} a^{12} - \frac{33}{109} a^{11} + \frac{251}{545} a^{10} + \frac{54}{545} a^{9} - \frac{244}{545} a^{8} - \frac{108}{545} a^{7} - \frac{269}{545} a^{6} + \frac{74}{545} a^{5} + \frac{12}{109} a^{4} - \frac{232}{545} a^{3} - \frac{81}{545} a^{2} - \frac{26}{109} a + \frac{29}{109}$, $\frac{1}{195414759826284815} a^{15} - \frac{176392120984283}{195414759826284815} a^{14} - \frac{85617473417851408}{195414759826284815} a^{13} + \frac{11701502671412231}{39082951965256963} a^{12} + \frac{15942726660491341}{195414759826284815} a^{11} - \frac{25056604255019376}{195414759826284815} a^{10} + \frac{39042361788797771}{195414759826284815} a^{9} - \frac{41994724511019108}{195414759826284815} a^{8} + \frac{24750848583746691}{195414759826284815} a^{7} + \frac{61538128623692874}{195414759826284815} a^{6} - \frac{967004214854210}{39082951965256963} a^{5} - \frac{40373347953460902}{195414759826284815} a^{4} - \frac{71669475319650071}{195414759826284815} a^{3} + \frac{10077148291230937}{39082951965256963} a^{2} + \frac{158796925044230}{640704130577983} a + \frac{3941152376045213}{39082951965256963}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( \frac{15261825484}{308119712695} a^{15} + \frac{3633902710}{61623942539} a^{14} - \frac{128903418}{308119712695} a^{13} - \frac{38640319501}{308119712695} a^{12} - \frac{1230632511}{308119712695} a^{11} + \frac{44771337328}{308119712695} a^{10} + \frac{116824994037}{308119712695} a^{9} - \frac{146384762069}{61623942539} a^{8} + \frac{1407230318023}{308119712695} a^{7} - \frac{4680278915932}{308119712695} a^{6} + \frac{10898773369858}{308119712695} a^{5} - \frac{12446493500298}{308119712695} a^{4} + \frac{12533259602932}{308119712695} a^{3} - \frac{8239711424247}{308119712695} a^{2} + \frac{515544567472}{61623942539} a - \frac{11324866614}{61623942539} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7906.86966275 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{35}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{-15}, \sqrt{-21})\), \(\Q(i, \sqrt{35})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(i, \sqrt{21})\), 8.0.31116960000.4, 8.0.1244678400.3, 8.0.1244678400.2

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$