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Properties

Label 16.0.152858736488597841.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 13^{12}$
Root discriminant $11.86$
Ramified primes $3, 13$
Class number $1$
Class group Trivial
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 9, 3, 18, -11, -70, 50, 27, 0, -35, 20, -8, 5, -2, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 2*x^13 + 5*x^12 - 8*x^11 + 20*x^10 - 35*x^9 + 27*x^7 + 50*x^6 - 70*x^5 - 11*x^4 + 18*x^3 + 3*x^2 + 9*x + 9)
 
gp: K = bnfinit(x^16 - 3*x^15 + 3*x^14 - 2*x^13 + 5*x^12 - 8*x^11 + 20*x^10 - 35*x^9 + 27*x^7 + 50*x^6 - 70*x^5 - 11*x^4 + 18*x^3 + 3*x^2 + 9*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 3 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} + 20 x^{10} - 35 x^{9} + 27 x^{7} + 50 x^{6} - 70 x^{5} - 11 x^{4} + 18 x^{3} + 3 x^{2} + 9 x + 9 \)

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:  \(152858736488597841=3^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.86$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{2}{9} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{5}{18} a^{6} + \frac{1}{18} a^{5} + \frac{1}{3} a^{4} - \frac{5}{18} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{612} a^{14} - \frac{1}{204} a^{13} + \frac{25}{306} a^{12} - \frac{25}{204} a^{11} + \frac{281}{612} a^{10} - \frac{269}{612} a^{9} + \frac{91}{204} a^{8} - \frac{49}{306} a^{7} + \frac{61}{204} a^{6} + \frac{215}{612} a^{5} - \frac{257}{612} a^{4} + \frac{5}{612} a^{3} - \frac{3}{34} a^{2} - \frac{29}{204} a - \frac{21}{68}$, $\frac{1}{29090808} a^{15} - \frac{220}{3636351} a^{14} + \frac{30911}{1711224} a^{13} + \frac{2396455}{29090808} a^{12} + \frac{61229}{855612} a^{11} + \frac{61939}{285204} a^{10} + \frac{254267}{855612} a^{9} + \frac{10925371}{29090808} a^{8} + \frac{1489597}{29090808} a^{7} + \frac{5038673}{14545404} a^{6} + \frac{185173}{427806} a^{5} - \frac{24167}{95068} a^{4} + \frac{9282103}{29090808} a^{3} + \frac{546531}{3232312} a^{2} + \frac{43943}{2424234} a + \frac{521873}{3232312}$

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:  \( -\frac{568145}{9696936} a^{15} + \frac{836833}{4848468} a^{14} - \frac{1882357}{9696936} a^{13} + \frac{107641}{570408} a^{12} - \frac{306095}{808078} a^{11} + \frac{665188}{1212117} a^{10} - \frac{1591706}{1212117} a^{9} + \frac{7072549}{3232312} a^{8} - \frac{4906669}{9696936} a^{7} - \frac{703682}{1212117} a^{6} - \frac{5117413}{1616156} a^{5} + \frac{8874329}{2424234} a^{4} - \frac{2584949}{9696936} a^{3} - \frac{1075787}{9696936} a^{2} - \frac{1160821}{1616156} a + \frac{714583}{3232312} \) (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:  \( 255.175825244 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

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 $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}, \sqrt{13})\), 4.0.117.1 x2, 4.2.507.1 x2, 4.0.19773.1 x2, 4.2.6591.1 x2, 4.0.2197.1, 4.4.19773.1, 8.0.2313441.1, 8.0.390971529.2, 8.0.390971529.1, 8.0.43441281.1 x2, 8.4.390971529.1 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$