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Properties

Label 16.0.93492089436...3456.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{12}$
Root discriminant $15.33$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $(C_2^2\times C_4):C_2$ (as 16T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -48, 264, -736, 1268, -1536, 1344, -888, 534, -400, 396, -336, 222, -104, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 222*x^12 - 336*x^11 + 396*x^10 - 400*x^9 + 534*x^8 - 888*x^7 + 1344*x^6 - 1536*x^5 + 1268*x^4 - 736*x^3 + 264*x^2 - 48*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 104*x^13 + 222*x^12 - 336*x^11 + 396*x^10 - 400*x^9 + 534*x^8 - 888*x^7 + 1344*x^6 - 1536*x^5 + 1268*x^4 - 736*x^3 + 264*x^2 - 48*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 104 x^{13} + 222 x^{12} - 336 x^{11} + 396 x^{10} - 400 x^{9} + 534 x^{8} - 888 x^{7} + 1344 x^{6} - 1536 x^{5} + 1268 x^{4} - 736 x^{3} + 264 x^{2} - 48 x + 4 \)

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:  \(9349208943630483456=2^{44}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.33$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{50} a^{14} - \frac{1}{10} a^{12} + \frac{3}{25} a^{11} - \frac{7}{50} a^{9} - \frac{1}{5} a^{8} - \frac{9}{25} a^{7} - \frac{3}{25} a^{4} + \frac{8}{25} a^{3} - \frac{4}{25} a^{2} - \frac{3}{25} a - \frac{3}{25}$, $\frac{1}{2326674250} a^{15} + \frac{12786931}{2326674250} a^{14} + \frac{606783}{20231950} a^{13} - \frac{289678612}{1163337125} a^{12} - \frac{108460557}{1163337125} a^{11} - \frac{479354207}{2326674250} a^{10} + \frac{5148563}{50579875} a^{9} - \frac{224441514}{1163337125} a^{8} + \frac{165350121}{1163337125} a^{7} - \frac{1507091}{46533485} a^{6} + \frac{439814197}{1163337125} a^{5} - \frac{62308332}{232667425} a^{4} + \frac{440253494}{1163337125} a^{3} - \frac{104573727}{1163337125} a^{2} + \frac{446116029}{1163337125} a + \frac{422528807}{1163337125}$

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:  \( \frac{1231553122}{232667425} a^{15} - \frac{8963149014}{232667425} a^{14} + \frac{329320446}{2023195} a^{13} - \frac{100799441498}{232667425} a^{12} + \frac{200904491966}{232667425} a^{11} - \frac{269526250754}{232667425} a^{10} + \frac{12811488836}{10115975} a^{9} - \frac{563992726687}{465334850} a^{8} + \frac{456132945352}{232667425} a^{7} - \frac{30642446606}{9306697} a^{6} + \frac{1105173642768}{232667425} a^{5} - \frac{1099749493964}{232667425} a^{4} + \frac{31050449684}{9306697} a^{3} - \frac{70787699844}{46533485} a^{2} + \frac{74289427752}{232667425} a - \frac{6868615966}{232667425} \) (order $12$)
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:  \( 4738.13967163 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T37):

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.4.13824.1, \(\Q(\zeta_{12})\), 4.0.13824.1, 8.0.191102976.5, 8.0.191102976.3, 8.0.5308416.2

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed