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Properties

Label 16.0.549378366500390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 11^{8}$
Root discriminant $12.85$
Ramified primes $3, 5, 11$
Class number $1$
Class group Trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 5, -22, 59, 10, -111, 49, 105, -73, -4, -20, 24, -1, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 - x^13 + 24*x^12 - 20*x^11 - 4*x^10 - 73*x^9 + 105*x^8 + 49*x^7 - 111*x^6 + 10*x^5 + 59*x^4 - 22*x^3 + 5*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - x^{13} + 24 x^{12} - 20 x^{11} - 4 x^{10} - 73 x^{9} + 105 x^{8} + 49 x^{7} - 111 x^{6} + 10 x^{5} + 59 x^{4} - 22 x^{3} + 5 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(549378366500390625=3^{8}\cdot 5^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.85$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{30} a^{12} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{11}{30} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{30}$, $\frac{1}{90} a^{13} + \frac{1}{18} a^{11} - \frac{1}{6} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{90} a^{7} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{7}{18} a^{3} - \frac{7}{18} a^{2} - \frac{7}{30} a + \frac{1}{18}$, $\frac{1}{180} a^{14} - \frac{1}{180} a^{13} - \frac{1}{180} a^{12} + \frac{1}{18} a^{11} - \frac{1}{36} a^{10} + \frac{1}{9} a^{9} - \frac{89}{180} a^{8} - \frac{61}{180} a^{7} - \frac{19}{45} a^{6} - \frac{1}{6} a^{5} + \frac{7}{36} a^{4} + \frac{1}{6} a^{3} - \frac{4}{45} a^{2} - \frac{1}{45} a - \frac{29}{180}$, $\frac{1}{1896120} a^{15} + \frac{47}{63204} a^{14} - \frac{913}{189612} a^{13} - \frac{3127}{210680} a^{12} + \frac{31565}{379224} a^{11} + \frac{7151}{126408} a^{10} - \frac{861539}{1896120} a^{9} - \frac{7403}{15801} a^{8} - \frac{151703}{379224} a^{7} + \frac{8477}{41220} a^{6} - \frac{72325}{379224} a^{5} - \frac{183967}{379224} a^{4} - \frac{6134}{79005} a^{3} - \frac{7507}{189612} a^{2} - \frac{16309}{42136} a - \frac{266419}{632040}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{666413}{632040} a^{15} - \frac{59375}{21068} a^{14} - \frac{94233}{105340} a^{13} - \frac{60465}{42136} a^{12} + \frac{1047315}{42136} a^{11} - \frac{546801}{42136} a^{10} - \frac{4923527}{632040} a^{9} - \frac{845483}{10534} a^{8} + \frac{17779199}{210680} a^{7} + \frac{212393}{2748} a^{6} - \frac{3709043}{42136} a^{5} - \frac{730589}{42136} a^{4} + \frac{4195684}{79005} a^{3} - \frac{132513}{21068} a^{2} + \frac{1127521}{210680} a - \frac{303229}{126408} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 535.532332979 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{33})\), 4.0.605.1 x2, 4.0.5445.1 x2, 4.2.275.1 x2, 4.2.2475.1 x2, 8.0.741200625.1, 8.0.9150625.1, 8.0.741200625.3, 8.0.29648025.1 x2, 8.0.6125625.1 x2, 8.0.741200625.2 x2, 8.4.741200625.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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}$