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Properties

Label 16.8.34336147906...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $19.21$
Ramified primes $3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 100, 50, -175, -255, -110, 145, 320, 91, -193, -104, 41, 44, 3, -11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 11*x^14 + 3*x^13 + 44*x^12 + 41*x^11 - 104*x^10 - 193*x^9 + 91*x^8 + 320*x^7 + 145*x^6 - 110*x^5 - 255*x^4 - 175*x^3 + 50*x^2 + 100*x + 25)
 
gp: K = bnfinit(x^16 - x^15 - 11*x^14 + 3*x^13 + 44*x^12 + 41*x^11 - 104*x^10 - 193*x^9 + 91*x^8 + 320*x^7 + 145*x^6 - 110*x^5 - 255*x^4 - 175*x^3 + 50*x^2 + 100*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 11 x^{14} + 3 x^{13} + 44 x^{12} + 41 x^{11} - 104 x^{10} - 193 x^{9} + 91 x^{8} + 320 x^{7} + 145 x^{6} - 110 x^{5} - 255 x^{4} - 175 x^{3} + 50 x^{2} + 100 x + 25 \)

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:  \(343361479062744140625=3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.21$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{110} a^{14} + \frac{1}{22} a^{13} - \frac{1}{110} a^{12} - \frac{2}{55} a^{11} - \frac{2}{55} a^{10} + \frac{53}{110} a^{9} + \frac{41}{110} a^{8} + \frac{49}{110} a^{7} - \frac{3}{55} a^{6} + \frac{4}{55} a^{5} - \frac{39}{110} a^{4} + \frac{7}{22} a^{3} - \frac{7}{22} a^{2} - \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{334690003010} a^{15} - \frac{1181881447}{334690003010} a^{14} - \frac{6866491681}{167345001505} a^{13} + \frac{1171829567}{30426363910} a^{12} - \frac{10505652954}{167345001505} a^{11} - \frac{656862421}{66938000602} a^{10} - \frac{5609498223}{66938000602} a^{9} + \frac{38403023959}{167345001505} a^{8} + \frac{80893884243}{334690003010} a^{7} + \frac{79885667218}{167345001505} a^{6} + \frac{6396418423}{66938000602} a^{5} - \frac{101766986339}{334690003010} a^{4} + \frac{8304143330}{33469000301} a^{3} + \frac{4730489641}{66938000602} a^{2} - \frac{16396149637}{33469000301} a + \frac{13772330621}{33469000301}$

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:  \( 19871.1900257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{33}) \), 4.2.12375.1, 4.4.15125.1, \(\Q(\zeta_{15})^+\), 4.2.1375.1, 4.2.2475.1, 4.2.275.1, \(\Q(\sqrt{5}, \sqrt{33})\), 8.4.741200625.1, 8.4.18530015625.1, 8.4.153140625.1, 8.4.153140625.2, 8.4.228765625.1, 8.8.18530015625.1, 8.4.18530015625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$