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Properties

Label 16.0.24622884053...8933.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{10}\cdot 37^{7}$
Root discriminant $21.72$
Ramified primes $11, 37$
Class number $1$
Class group Trivial
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![407, -407, -484, -88, 1585, -1110, 327, -914, 1024, -338, 67, -109, 47, 10, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 4*x^14 + 10*x^13 + 47*x^12 - 109*x^11 + 67*x^10 - 338*x^9 + 1024*x^8 - 914*x^7 + 327*x^6 - 1110*x^5 + 1585*x^4 - 88*x^3 - 484*x^2 - 407*x + 407)
 
gp: K = bnfinit(x^16 - 3*x^15 - 4*x^14 + 10*x^13 + 47*x^12 - 109*x^11 + 67*x^10 - 338*x^9 + 1024*x^8 - 914*x^7 + 327*x^6 - 1110*x^5 + 1585*x^4 - 88*x^3 - 484*x^2 - 407*x + 407, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 4 x^{14} + 10 x^{13} + 47 x^{12} - 109 x^{11} + 67 x^{10} - 338 x^{9} + 1024 x^{8} - 914 x^{7} + 327 x^{6} - 1110 x^{5} + 1585 x^{4} - 88 x^{3} - 484 x^{2} - 407 x + 407 \)

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:  \(2462288405368583548933=11^{10}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.72$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{37} a^{12} + \frac{1}{37} a^{11} + \frac{8}{37} a^{10} - \frac{11}{37} a^{9} + \frac{7}{37} a^{8} + \frac{10}{37} a^{7} - \frac{9}{37} a^{6} + \frac{8}{37} a^{5} + \frac{11}{37} a^{4} + \frac{12}{37} a^{3} + \frac{3}{37} a^{2}$, $\frac{1}{407} a^{13} - \frac{5}{407} a^{12} + \frac{2}{407} a^{11} + \frac{89}{407} a^{10} + \frac{184}{407} a^{9} - \frac{69}{407} a^{8} + \frac{5}{407} a^{7} + \frac{25}{407} a^{6} - \frac{4}{11} a^{5} - \frac{128}{407} a^{4} - \frac{13}{37} a^{3} - \frac{5}{37} a^{2}$, $\frac{1}{2849} a^{14} - \frac{1}{2849} a^{13} + \frac{4}{2849} a^{12} + \frac{526}{2849} a^{11} - \frac{912}{2849} a^{10} - \frac{796}{2849} a^{9} - \frac{117}{2849} a^{8} + \frac{1079}{2849} a^{7} + \frac{1382}{2849} a^{6} + \frac{677}{2849} a^{5} + \frac{808}{2849} a^{4} + \frac{41}{259} a^{3} - \frac{125}{259} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{94095304765} a^{15} + \frac{3211849}{18819060953} a^{14} - \frac{1095254}{8554118615} a^{13} + \frac{755976663}{94095304765} a^{12} - \frac{34692455849}{94095304765} a^{11} + \frac{14322325404}{94095304765} a^{10} - \frac{28330369471}{94095304765} a^{9} + \frac{4966786469}{94095304765} a^{8} + \frac{2041284801}{94095304765} a^{7} + \frac{932870249}{8554118615} a^{6} + \frac{12493847649}{94095304765} a^{5} + \frac{119995487}{94095304765} a^{4} - \frac{2988415464}{8554118615} a^{3} + \frac{527680994}{1710823723} a^{2} + \frac{35054903}{231192395} a - \frac{7079096}{33027485}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17049.9697335 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.4477.1, 8.0.741610573.1

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$37$37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$