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Properties

Label 16.12.1194077837...0000.2
Degree $16$
Signature $[12, 2]$
Discriminant $2^{24}\cdot 5^{12}\cdot 13^{2}\cdot 29^{7}$
Root discriminant $56.86$
Ramified primes $2, 5, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1606

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16619, -63530, -29884, 128050, 108250, -106780, -103046, 45670, 44539, -9940, -9726, 970, 1070, -30, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 54*x^14 - 30*x^13 + 1070*x^12 + 970*x^11 - 9726*x^10 - 9940*x^9 + 44539*x^8 + 45670*x^7 - 103046*x^6 - 106780*x^5 + 108250*x^4 + 128050*x^3 - 29884*x^2 - 63530*x - 16619)
 
gp: K = bnfinit(x^16 - 54*x^14 - 30*x^13 + 1070*x^12 + 970*x^11 - 9726*x^10 - 9940*x^9 + 44539*x^8 + 45670*x^7 - 103046*x^6 - 106780*x^5 + 108250*x^4 + 128050*x^3 - 29884*x^2 - 63530*x - 16619, 1)
 

Normalized defining polynomial

\( x^{16} - 54 x^{14} - 30 x^{13} + 1070 x^{12} + 970 x^{11} - 9726 x^{10} - 9940 x^{9} + 44539 x^{8} + 45670 x^{7} - 103046 x^{6} - 106780 x^{5} + 108250 x^{4} + 128050 x^{3} - 29884 x^{2} - 63530 x - 16619 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11940778378121216000000000000=2^{24}\cdot 5^{12}\cdot 13^{2}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.86$
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:  $2, 5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2132391210265251551055627418} a^{15} - \frac{150620047183367342468130084}{1066195605132625775527813709} a^{14} + \frac{431890643216317411498150209}{2132391210265251551055627418} a^{13} + \frac{172707118440927672300284647}{1066195605132625775527813709} a^{12} - \frac{244580074879185571252957789}{2132391210265251551055627418} a^{11} + \frac{5636004261993391002979013}{27693392341107163000722434} a^{10} + \frac{216303140716406780060236431}{1066195605132625775527813709} a^{9} + \frac{174317692738379995925399109}{2132391210265251551055627418} a^{8} + \frac{993019842465217819313045011}{2132391210265251551055627418} a^{7} - \frac{277839850652610370434415519}{2132391210265251551055627418} a^{6} + \frac{36425361514651783099089355}{1066195605132625775527813709} a^{5} + \frac{511552586246561293810513367}{2132391210265251551055627418} a^{4} + \frac{31929731650008529150171301}{96926873193875070502528519} a^{3} - \frac{15097735806213672039405891}{2132391210265251551055627418} a^{2} + \frac{168526094799223679632780835}{2132391210265251551055627418} a - \frac{975173912257956747928014817}{2132391210265251551055627418}$

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:  $13$
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:  \( 268995413.67 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1606:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1606 are not computed
Character table for t16n1606 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.97556000000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$