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Properties

Label 16.0.73970906895...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{7}\cdot 79^{10}$
Root discriminant $31.03$
Ramified primes $5, 79$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, 704, -704, 100, -558, 755, -49, -192, -64, 161, -62, 19, -21, 16, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 16*x^14 - 21*x^13 + 19*x^12 - 62*x^11 + 161*x^10 - 64*x^9 - 192*x^8 - 49*x^7 + 755*x^6 - 558*x^5 + 100*x^4 - 704*x^3 + 704*x^2 + 1024)
 
gp: K = bnfinit(x^16 - 6*x^15 + 16*x^14 - 21*x^13 + 19*x^12 - 62*x^11 + 161*x^10 - 64*x^9 - 192*x^8 - 49*x^7 + 755*x^6 - 558*x^5 + 100*x^4 - 704*x^3 + 704*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 16 x^{14} - 21 x^{13} + 19 x^{12} - 62 x^{11} + 161 x^{10} - 64 x^{9} - 192 x^{8} - 49 x^{7} + 755 x^{6} - 558 x^{5} + 100 x^{4} - 704 x^{3} + 704 x^{2} + 1024 \)

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:  \(739709068955222437578125=5^{7}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.03$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{12} + \frac{3}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{16} a^{4} + \frac{7}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{14} + \frac{7}{256} a^{13} - \frac{5}{64} a^{12} + \frac{107}{512} a^{11} - \frac{17}{512} a^{10} - \frac{1}{256} a^{9} - \frac{55}{512} a^{8} + \frac{5}{128} a^{7} - \frac{113}{512} a^{5} + \frac{31}{512} a^{4} - \frac{57}{256} a^{3} - \frac{13}{128} a^{2} - \frac{3}{32} a - \frac{3}{8}$, $\frac{1}{9480468454368544768} a^{15} + \frac{3780804126037367}{4740234227184272384} a^{14} - \frac{36233809015657229}{1185058556796068096} a^{13} - \frac{1115256888565526933}{9480468454368544768} a^{12} + \frac{2296779311175205007}{9480468454368544768} a^{11} + \frac{903069702064547791}{4740234227184272384} a^{10} - \frac{2095193619546087543}{9480468454368544768} a^{9} - \frac{159044491295440707}{2370117113592136192} a^{8} - \frac{3286445051342699}{18516539949938564} a^{7} + \frac{3740200236907787663}{9480468454368544768} a^{6} + \frac{1575767285173055807}{9480468454368544768} a^{5} + \frac{295744055350604055}{4740234227184272384} a^{4} - \frac{343744134553779613}{2370117113592136192} a^{3} - \frac{278620082575332859}{592529278398034048} a^{2} - \frac{5279876708459587}{148132319599508512} a - \frac{2268604950595815}{4629134987484641}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 124272.711036 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-79}) \), 4.0.31205.1, 8.0.4868760125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{6}$ $16$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$79$79.8.4.1$x^{8} + 37446 x^{4} - 493039 x^{2} + 350550729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
79.8.6.2$x^{8} + 395 x^{4} + 56169$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$