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Properties

Label 16.12.7519232350...1344.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{62}\cdot 113^{4}$
Root discriminant $47.84$
Ramified primes $2, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-196, -672, 1200, 6560, 9112, 6112, -1952, -8000, -4352, 496, 1208, 688, 196, -48, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 48*x^13 + 196*x^12 + 688*x^11 + 1208*x^10 + 496*x^9 - 4352*x^8 - 8000*x^7 - 1952*x^6 + 6112*x^5 + 9112*x^4 + 6560*x^3 + 1200*x^2 - 672*x - 196)
 
gp: K = bnfinit(x^16 - 32*x^14 - 48*x^13 + 196*x^12 + 688*x^11 + 1208*x^10 + 496*x^9 - 4352*x^8 - 8000*x^7 - 1952*x^6 + 6112*x^5 + 9112*x^4 + 6560*x^3 + 1200*x^2 - 672*x - 196, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} - 48 x^{13} + 196 x^{12} + 688 x^{11} + 1208 x^{10} + 496 x^{9} - 4352 x^{8} - 8000 x^{7} - 1952 x^{6} + 6112 x^{5} + 9112 x^{4} + 6560 x^{3} + 1200 x^{2} - 672 x - 196 \)

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:  \(751923235065182967870521344=2^{62}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.84$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} + \frac{1}{20} a^{11} - \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} - \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{1}{10} a^{3} - \frac{1}{10} a^{2} + \frac{1}{5}$, $\frac{1}{140} a^{14} - \frac{1}{140} a^{13} - \frac{3}{140} a^{12} - \frac{17}{140} a^{11} - \frac{11}{140} a^{10} - \frac{2}{35} a^{9} + \frac{3}{35} a^{8} + \frac{2}{35} a^{7} - \frac{3}{70} a^{6} + \frac{1}{10} a^{5} - \frac{31}{70} a^{4} - \frac{1}{2} a^{3} + \frac{13}{70} a^{2} + \frac{13}{35} a - \frac{2}{5}$, $\frac{1}{6767068337213983820} a^{15} + \frac{3048476212591493}{1353413667442796764} a^{14} - \frac{13455771868527144}{1691767084303495955} a^{13} - \frac{54667954587350253}{3383534168606991910} a^{12} - \frac{51638995185059768}{1691767084303495955} a^{11} + \frac{922764429273753}{14216530120197445} a^{10} + \frac{2311008282206703}{241681012043356565} a^{9} - \frac{4435465760408696}{99515710841382115} a^{8} + \frac{28272623453846311}{199031421682764230} a^{7} - \frac{106404271806138771}{676706833721398382} a^{6} - \frac{807718556482264978}{1691767084303495955} a^{5} + \frac{221081011588283227}{1691767084303495955} a^{4} + \frac{36732620659343450}{338353416860699191} a^{3} + \frac{112220045067185708}{241681012043356565} a^{2} - \frac{119159775576235332}{338353416860699191} a + \frac{8596500068914879}{48336202408671313}$

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:  \( 331382337.459 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.242665652224.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.17$x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8:C_2$$[2, 3, 4, 5]$
2.8.31.17$x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8:C_2$$[2, 3, 4, 5]$
113Data not computed