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Properties

Label 16.0.34828517376...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $14.42$
Ramified primes $2, 3, 5$
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![36, 0, 54, 0, 75, 18, -63, 30, -44, 42, 25, -78, 72, -42, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 42*x^13 + 72*x^12 - 78*x^11 + 25*x^10 + 42*x^9 - 44*x^8 + 30*x^7 - 63*x^6 + 18*x^5 + 75*x^4 + 54*x^2 + 36, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 72 x^{12} - 78 x^{11} + 25 x^{10} + 42 x^{9} - 44 x^{8} + 30 x^{7} - 63 x^{6} + 18 x^{5} + 75 x^{4} + 54 x^{2} + 36 \)

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:  \(3482851737600000000=2^{24}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.42$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{60} a^{12} - \frac{1}{5} a^{11} - \frac{7}{30} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} + \frac{1}{5} a^{7} - \frac{2}{15} a^{6} + \frac{2}{5} a^{5} - \frac{29}{60} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{60} a^{13} - \frac{2}{15} a^{11} + \frac{1}{10} a^{10} + \frac{1}{20} a^{9} + \frac{1}{5} a^{8} - \frac{7}{30} a^{7} - \frac{1}{5} a^{6} + \frac{19}{60} a^{5} - \frac{3}{10} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{60} a^{14} + \frac{11}{60} a^{10} - \frac{1}{10} a^{9} - \frac{7}{30} a^{8} - \frac{1}{10} a^{7} - \frac{1}{4} a^{6} - \frac{1}{10} a^{5} - \frac{1}{6} a^{4} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{80336828340} a^{15} - \frac{572925}{77620124} a^{14} - \frac{100294997}{20084207085} a^{13} - \frac{17589571}{26778942780} a^{12} + \frac{472473209}{8926314260} a^{11} + \frac{1311620507}{8926314260} a^{10} - \frac{560544520}{4016841417} a^{9} + \frac{2196852241}{26778942780} a^{8} - \frac{4002857579}{80336828340} a^{7} - \frac{4854117907}{26778942780} a^{6} - \frac{959820444}{2231578565} a^{5} - \frac{3267404971}{26778942780} a^{4} - \frac{2294915356}{6694735695} a^{3} - \frac{105944353}{892631426} a^{2} + \frac{865563837}{2231578565} a + \frac{1603234}{2231578565}$

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:  \( -\frac{3185850}{446315713} a^{15} + \frac{3243449}{116430186} a^{14} - \frac{49757887}{1338947139} a^{13} - \frac{87617969}{2677894278} a^{12} + \frac{345103691}{1338947139} a^{11} - \frac{363363785}{446315713} a^{10} + \frac{643770137}{446315713} a^{9} - \frac{2742687563}{2677894278} a^{8} - \frac{651671911}{1338947139} a^{7} + \frac{1397083097}{1338947139} a^{6} - \frac{311184223}{1338947139} a^{5} + \frac{380673089}{892631426} a^{4} - \frac{151506467}{446315713} a^{3} - \frac{1223953495}{892631426} a^{2} - \frac{13536530}{446315713} a - \frac{44064382}{446315713} \) (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:  \( 2681.6222586 \)
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

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.2.8640.1 x2, 4.2.8640.2 x2, 4.0.5400.2 x2, 4.0.5400.1 x2, 8.0.207360000.2, 8.0.1866240000.5, 8.0.1866240000.9, 8.0.74649600.1 x2, 8.0.1866240000.7 x2, 8.4.1866240000.2 x2, 8.0.29160000.1 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{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}$