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Properties

Label 16.0.76741107731...1773.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 37^{9}$
Root discriminant $15.15$
Ramified primes $3, 37$
Class number $2$
Class group $[2]$
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![1, -1, 11, -8, 36, -56, 78, -232, 243, -52, 24, -113, 113, -62, 25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 62*x^13 + 113*x^12 - 113*x^11 + 24*x^10 - 52*x^9 + 243*x^8 - 232*x^7 + 78*x^6 - 56*x^5 + 36*x^4 - 8*x^3 + 11*x^2 - x + 1)
 
gp: K = bnfinit(x^16 - 7*x^15 + 25*x^14 - 62*x^13 + 113*x^12 - 113*x^11 + 24*x^10 - 52*x^9 + 243*x^8 - 232*x^7 + 78*x^6 - 56*x^5 + 36*x^4 - 8*x^3 + 11*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 25 x^{14} - 62 x^{13} + 113 x^{12} - 113 x^{11} + 24 x^{10} - 52 x^{9} + 243 x^{8} - 232 x^{7} + 78 x^{6} - 56 x^{5} + 36 x^{4} - 8 x^{3} + 11 x^{2} - x + 1 \)

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:  \(7674110773159501773=3^{10}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.15$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{25654707823} a^{15} + \frac{11176118827}{25654707823} a^{14} - \frac{7139599251}{25654707823} a^{13} + \frac{2904395810}{25654707823} a^{12} - \frac{7615310348}{25654707823} a^{11} - \frac{11929243590}{25654707823} a^{10} - \frac{4098596426}{25654707823} a^{9} - \frac{11269718565}{25654707823} a^{8} + \frac{6736938563}{25654707823} a^{7} + \frac{5183809453}{25654707823} a^{6} - \frac{10768195156}{25654707823} a^{5} - \frac{88192063}{361333913} a^{4} + \frac{3344378356}{25654707823} a^{3} - \frac{2061751527}{25654707823} a^{2} + \frac{4627845612}{25654707823} a + \frac{10899487920}{25654707823}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1006782}{793993} a^{15} - \frac{6041273}{793993} a^{14} + \frac{18865349}{793993} a^{13} - \frac{41659784}{793993} a^{12} + \frac{65262103}{793993} a^{11} - \frac{31544995}{793993} a^{10} - \frac{38170058}{793993} a^{9} - \frac{60083607}{793993} a^{8} + \frac{180930597}{793993} a^{7} - \frac{43630033}{793993} a^{6} - \frac{34167189}{793993} a^{5} - \frac{319901}{11183} a^{4} + \frac{3181511}{793993} a^{3} + \frac{2731739}{793993} a^{2} + \frac{1350365}{793993} a + \frac{1526197}{793993} \) (order $6$)
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:  \( 1010.09524782 \)
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

\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.4102893.1

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 $16$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$37.4.3.3$x^{4} + 74$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$