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Properties

Label 16.8.38592250557...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{6}\cdot 89^{12}$
Root discriminant $52.99$
Ramified primes $5, 89$
Class number $1$ (GRH)
Class group Trivial (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![97, 220, 4491, -37956, 85067, -80100, 26395, 13951, -17791, 4909, 2018, -1458, 108, 143, -26, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 26*x^14 + 143*x^13 + 108*x^12 - 1458*x^11 + 2018*x^10 + 4909*x^9 - 17791*x^8 + 13951*x^7 + 26395*x^6 - 80100*x^5 + 85067*x^4 - 37956*x^3 + 4491*x^2 + 220*x + 97)
 
gp: K = bnfinit(x^16 - 5*x^15 - 26*x^14 + 143*x^13 + 108*x^12 - 1458*x^11 + 2018*x^10 + 4909*x^9 - 17791*x^8 + 13951*x^7 + 26395*x^6 - 80100*x^5 + 85067*x^4 - 37956*x^3 + 4491*x^2 + 220*x + 97, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 26 x^{14} + 143 x^{13} + 108 x^{12} - 1458 x^{11} + 2018 x^{10} + 4909 x^{9} - 17791 x^{8} + 13951 x^{7} + 26395 x^{6} - 80100 x^{5} + 85067 x^{4} - 37956 x^{3} + 4491 x^{2} + 220 x + 97 \)

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:  \(3859225055707220942242515625=5^{6}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.99$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{30181647338855289295042533584} a^{15} - \frac{110910234526672122571685697}{1886352958678455580940158349} a^{14} - \frac{257710136381257090034262163}{15090823669427644647521266792} a^{13} - \frac{2510916556687922978066693447}{30181647338855289295042533584} a^{12} + \frac{6228560767775965857980498021}{30181647338855289295042533584} a^{11} - \frac{8497984594777886207061726857}{30181647338855289295042533584} a^{10} + \frac{12352273043681187817255457385}{30181647338855289295042533584} a^{9} - \frac{1885107582485618916646718401}{15090823669427644647521266792} a^{8} + \frac{13289699648976232850984355207}{30181647338855289295042533584} a^{7} - \frac{3847227389989637231050342555}{15090823669427644647521266792} a^{6} - \frac{1673415002991795922342777439}{30181647338855289295042533584} a^{5} + \frac{13978486895158115427355389257}{30181647338855289295042533584} a^{4} - \frac{924332891753918109232075663}{1886352958678455580940158349} a^{3} + \frac{393843623210774938426328975}{7545411834713822323760633396} a^{2} - \frac{1074955802656958892837595533}{30181647338855289295042533584} a + \frac{4022036344475794044499911583}{30181647338855289295042533584}$

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:  $11$
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:  \( 161384680.43 \) (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

\(\Q(\sqrt{89}) \), 4.4.39605.1, 4.4.704969.1, 4.4.3524845.1, 8.4.7842780125.1, 8.4.62122661370125.1, 8.8.12424532274025.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89Data not computed