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Properties

Label 16.16.3488168408...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 13^{4}\cdot 29^{8}$
Root discriminant $34.19$
Ramified primes $5, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -31, -151, 1824, -1826, -5189, 8373, 1082, -6829, 1615, 2053, -828, -226, 140, 0, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 140*x^13 - 226*x^12 - 828*x^11 + 2053*x^10 + 1615*x^9 - 6829*x^8 + 1082*x^7 + 8373*x^6 - 5189*x^5 - 1826*x^4 + 1824*x^3 - 151*x^2 - 31*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 140*x^13 - 226*x^12 - 828*x^11 + 2053*x^10 + 1615*x^9 - 6829*x^8 + 1082*x^7 + 8373*x^6 - 5189*x^5 - 1826*x^4 + 1824*x^3 - 151*x^2 - 31*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 140 x^{13} - 226 x^{12} - 828 x^{11} + 2053 x^{10} + 1615 x^{9} - 6829 x^{8} + 1082 x^{7} + 8373 x^{6} - 5189 x^{5} - 1826 x^{4} + 1824 x^{3} - 151 x^{2} - 31 x + 1 \)

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:  \(3488168408344511962890625=5^{12}\cdot 13^{4}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.19$
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:  $5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{12} a^{5} + \frac{5}{12} a^{4} - \frac{5}{12} a^{3} - \frac{1}{4} a + \frac{5}{12}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{6} a^{9} + \frac{1}{4} a^{6} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{29544} a^{14} - \frac{7}{29544} a^{13} - \frac{379}{14772} a^{12} - \frac{95}{9848} a^{11} - \frac{2437}{29544} a^{10} - \frac{481}{14772} a^{9} - \frac{75}{4924} a^{8} + \frac{4857}{9848} a^{7} + \frac{253}{7386} a^{6} + \frac{10249}{29544} a^{5} - \frac{3857}{14772} a^{4} - \frac{6845}{14772} a^{3} - \frac{4001}{14772} a^{2} + \frac{887}{7386} a + \frac{695}{29544}$, $\frac{1}{30991656} a^{15} + \frac{517}{30991656} a^{14} + \frac{94213}{2582638} a^{13} + \frac{423099}{10330552} a^{12} - \frac{336427}{30991656} a^{11} - \frac{63418}{3873957} a^{10} - \frac{2344969}{15495828} a^{9} - \frac{3584321}{30991656} a^{8} - \frac{929891}{2582638} a^{7} - \frac{13451009}{30991656} a^{6} + \frac{4291529}{15495828} a^{5} + \frac{4082771}{15495828} a^{4} + \frac{1797283}{7747914} a^{3} - \frac{1142641}{5165276} a^{2} + \frac{8842079}{30991656} a + \frac{887453}{15495828}$

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:  \( 10711296.8434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

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 $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1, 8.8.1867663890625.1, 8.8.1867663890625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$