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Properties

Label 16.16.7984925229...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{12}\cdot 5^{12}\cdot 41^{8}$
Root discriminant $36.01$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -15, -129, 76, 1425, 1352, -2545, -3229, 1577, 2329, -588, -730, 159, 98, -23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 23*x^14 + 98*x^13 + 159*x^12 - 730*x^11 - 588*x^10 + 2329*x^9 + 1577*x^8 - 3229*x^7 - 2545*x^6 + 1352*x^5 + 1425*x^4 + 76*x^3 - 129*x^2 - 15*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 23*x^14 + 98*x^13 + 159*x^12 - 730*x^11 - 588*x^10 + 2329*x^9 + 1577*x^8 - 3229*x^7 - 2545*x^6 + 1352*x^5 + 1425*x^4 + 76*x^3 - 129*x^2 - 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 23 x^{14} + 98 x^{13} + 159 x^{12} - 730 x^{11} - 588 x^{10} + 2329 x^{9} + 1577 x^{8} - 3229 x^{7} - 2545 x^{6} + 1352 x^{5} + 1425 x^{4} + 76 x^{3} - 129 x^{2} - 15 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7984925229121000000000000=2^{12}\cdot 5^{12}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.01$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{28} a^{13} + \frac{3}{28} a^{12} - \frac{1}{14} a^{10} + \frac{1}{14} a^{9} + \frac{1}{28} a^{8} - \frac{1}{28} a^{7} - \frac{1}{14} a^{6} + \frac{13}{28} a^{5} - \frac{5}{28} a^{4} + \frac{3}{28} a^{3} - \frac{1}{28} a^{2} + \frac{1}{28} a + \frac{2}{7}$, $\frac{1}{28} a^{14} - \frac{1}{14} a^{12} + \frac{5}{28} a^{11} - \frac{3}{14} a^{10} - \frac{5}{28} a^{9} - \frac{1}{7} a^{8} - \frac{3}{14} a^{7} - \frac{1}{14} a^{6} - \frac{1}{14} a^{5} + \frac{11}{28} a^{4} - \frac{3}{28} a^{3} + \frac{11}{28} a^{2} - \frac{1}{14} a - \frac{3}{28}$, $\frac{1}{154073968244} a^{15} - \frac{242095173}{77036984122} a^{14} + \frac{500207305}{38518492061} a^{13} + \frac{13741146423}{154073968244} a^{12} + \frac{1078224241}{5502641723} a^{11} + \frac{8944793255}{154073968244} a^{10} - \frac{18788090165}{77036984122} a^{9} + \frac{7222468139}{38518492061} a^{8} - \frac{5115902459}{38518492061} a^{7} - \frac{86633405}{77036984122} a^{6} - \frac{73156237543}{154073968244} a^{5} - \frac{76869517187}{154073968244} a^{4} - \frac{44629556945}{154073968244} a^{3} + \frac{1542939335}{11005283446} a^{2} - \frac{4504081761}{154073968244} a - \frac{23185549295}{77036984122}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 23347924.4359 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{205}) \), 4.4.336200.1, 4.4.13448.1, 4.4.210125.1 x2, \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.5125.1 x2, 8.8.44152515625.1, 8.8.113030440000.1, 8.8.2825761000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$