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Properties

Label 16.0.14505315669...641.19
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{8}$
Root discriminant $49.84$
Ramified primes $13, 53$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94419, -300456, 295893, -30706, -94847, 11375, 63947, -53150, 21242, -6292, 2178, -741, 221, -46, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 13*x^14 - 46*x^13 + 221*x^12 - 741*x^11 + 2178*x^10 - 6292*x^9 + 21242*x^8 - 53150*x^7 + 63947*x^6 + 11375*x^5 - 94847*x^4 - 30706*x^3 + 295893*x^2 - 300456*x + 94419)
 
gp: K = bnfinit(x^16 + 13*x^14 - 46*x^13 + 221*x^12 - 741*x^11 + 2178*x^10 - 6292*x^9 + 21242*x^8 - 53150*x^7 + 63947*x^6 + 11375*x^5 - 94847*x^4 - 30706*x^3 + 295893*x^2 - 300456*x + 94419, 1)
 

Normalized defining polynomial

\( x^{16} + 13 x^{14} - 46 x^{13} + 221 x^{12} - 741 x^{11} + 2178 x^{10} - 6292 x^{9} + 21242 x^{8} - 53150 x^{7} + 63947 x^{6} + 11375 x^{5} - 94847 x^{4} - 30706 x^{3} + 295893 x^{2} - 300456 x + 94419 \)

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:  \(1450531566903202684958906641=13^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.84$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{117} a^{14} + \frac{1}{39} a^{13} + \frac{49}{117} a^{12} - \frac{4}{9} a^{11} - \frac{16}{117} a^{10} + \frac{10}{39} a^{9} - \frac{4}{13} a^{8} + \frac{2}{9} a^{7} - \frac{10}{117} a^{6} + \frac{55}{117} a^{5} - \frac{40}{117} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{20694450105990590935028398713291939921} a^{15} + \frac{13197494201506749591678606088352318}{6898150035330196978342799571097313307} a^{14} + \frac{526400088473666914428285528473752780}{20694450105990590935028398713291939921} a^{13} + \frac{4913813087981634202472872925639544920}{20694450105990590935028398713291939921} a^{12} + \frac{2360317408075149102008877031472685833}{20694450105990590935028398713291939921} a^{11} + \frac{28506496785432257203439787069561547}{255487038345562851049733317448048641} a^{10} - \frac{1008245697590167220088043879451725477}{2299383345110065659447599857032437769} a^{9} + \frac{5749839454350076155166975532309420585}{20694450105990590935028398713291939921} a^{8} + \frac{792851788327728093458797470238600883}{20694450105990590935028398713291939921} a^{7} - \frac{8828985278247967171338521537114568074}{20694450105990590935028398713291939921} a^{6} - \frac{267181586614093740402046220836716178}{20694450105990590935028398713291939921} a^{5} + \frac{9238666400884090150410307595467933919}{20694450105990590935028398713291939921} a^{4} - \frac{26347707898325417487554226633536987}{1591880777383891610386799901022456917} a^{3} + \frac{180375094351230738905102341616892569}{1591880777383891610386799901022456917} a^{2} + \frac{6856724616669059666141782045770350}{18297480199814846096399998862327091} a - \frac{13760134460647987038469830876845455}{176875641931543512265199989002495213}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $7$
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:  \( 5716630.48735 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.4.8957.1, 4.0.116441.1, 8.4.2929680361933.3, 8.4.2929680361933.5, 8.0.13558506481.2

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
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$