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Properties

Label 16.0.79377267773...0761.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 17^{10}$
Root discriminant $55.43$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3181, -15676, 45084, -84579, 116423, -116912, 87563, -45186, 14638, -541, -1588, 920, -123, -42, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 - 123*x^12 + 920*x^11 - 1588*x^10 - 541*x^9 + 14638*x^8 - 45186*x^7 + 87563*x^6 - 116912*x^5 + 116423*x^4 - 84579*x^3 + 45084*x^2 - 15676*x + 3181)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 42*x^13 - 123*x^12 + 920*x^11 - 1588*x^10 - 541*x^9 + 14638*x^8 - 45186*x^7 + 87563*x^6 - 116912*x^5 + 116423*x^4 - 84579*x^3 + 45084*x^2 - 15676*x + 3181, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 42 x^{13} - 123 x^{12} + 920 x^{11} - 1588 x^{10} - 541 x^{9} + 14638 x^{8} - 45186 x^{7} + 87563 x^{6} - 116912 x^{5} + 116423 x^{4} - 84579 x^{3} + 45084 x^{2} - 15676 x + 3181 \)

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:  \(7937726777341695855516080761=13^{14}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.43$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{771} a^{13} + \frac{122}{771} a^{12} + \frac{25}{771} a^{11} - \frac{323}{771} a^{10} + \frac{332}{771} a^{9} - \frac{14}{771} a^{8} - \frac{5}{771} a^{7} - \frac{22}{771} a^{6} - \frac{23}{771} a^{5} + \frac{108}{257} a^{4} - \frac{73}{771} a^{3} + \frac{317}{771} a^{2} - \frac{182}{771} a - \frac{368}{771}$, $\frac{1}{1370089675881} a^{14} - \frac{7}{1370089675881} a^{13} - \frac{9854798138}{456696558627} a^{12} + \frac{177386366575}{1370089675881} a^{11} + \frac{79164310717}{1370089675881} a^{10} - \frac{651773571475}{1370089675881} a^{9} - \frac{39223956371}{456696558627} a^{8} - \frac{172623147164}{456696558627} a^{7} + \frac{205617582643}{1370089675881} a^{6} - \frac{341802481001}{1370089675881} a^{5} - \frac{141065072302}{1370089675881} a^{4} - \frac{441735540613}{1370089675881} a^{3} + \frac{24723348131}{1370089675881} a^{2} + \frac{42722342941}{152232186209} a - \frac{29417606588}{1370089675881}$, $\frac{1}{12330807082929} a^{15} - \frac{1}{4110269027643} a^{14} + \frac{7753223189}{12330807082929} a^{13} + \frac{1871698788139}{12330807082929} a^{12} - \frac{1018529133970}{12330807082929} a^{11} - \frac{615362049335}{1370089675881} a^{10} - \frac{5406303536212}{12330807082929} a^{9} - \frac{1873757530807}{4110269027643} a^{8} - \frac{4792627623242}{12330807082929} a^{7} + \frac{3769949289332}{12330807082929} a^{6} - \frac{332163508646}{4110269027643} a^{5} + \frac{2864374227337}{12330807082929} a^{4} - \frac{4466404901384}{12330807082929} a^{3} - \frac{17727814909}{12330807082929} a^{2} - \frac{5283219669554}{12330807082929} a + \frac{3960612071893}{12330807082929}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 5337893.19324 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.37349.1, 4.0.634933.1, 4.0.2873.1, 8.4.89093921102069.2, 8.4.308283464021.1, 8.0.403139914489.3

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$17$17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$