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Properties

Label 16.0.39412645190...7209.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{4}\cdot 53^{14}$
Root discriminant $61.27$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31147, -37204, 31792, -31363, 28026, -25678, 19032, -12138, 7272, -4491, 2941, -1372, 364, -58, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 58*x^13 + 364*x^12 - 1372*x^11 + 2941*x^10 - 4491*x^9 + 7272*x^8 - 12138*x^7 + 19032*x^6 - 25678*x^5 + 28026*x^4 - 31363*x^3 + 31792*x^2 - 37204*x + 31147)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 58*x^13 + 364*x^12 - 1372*x^11 + 2941*x^10 - 4491*x^9 + 7272*x^8 - 12138*x^7 + 19032*x^6 - 25678*x^5 + 28026*x^4 - 31363*x^3 + 31792*x^2 - 37204*x + 31147, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 58 x^{13} + 364 x^{12} - 1372 x^{11} + 2941 x^{10} - 4491 x^{9} + 7272 x^{8} - 12138 x^{7} + 19032 x^{6} - 25678 x^{5} + 28026 x^{4} - 31363 x^{3} + 31792 x^{2} - 37204 x + 31147 \)

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:  \(39412645190614083168888797209=13^{4}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.27$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{6}{13} a^{12} + \frac{5}{13} a^{11} + \frac{2}{13} a^{10} - \frac{4}{13} a^{9} - \frac{2}{13} a^{8} + \frac{2}{13} a^{7} + \frac{3}{13} a^{6} - \frac{2}{13} a^{5} + \frac{1}{13} a^{4} + \frac{3}{13} a^{3} - \frac{6}{13} a^{2} - \frac{1}{13} a + \frac{5}{13}$, $\frac{1}{91} a^{14} + \frac{3}{91} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{29}{91} a^{10} + \frac{36}{91} a^{9} - \frac{5}{91} a^{8} - \frac{3}{91} a^{7} + \frac{15}{91} a^{6} - \frac{45}{91} a^{5} + \frac{37}{91} a^{3} - \frac{9}{91} a^{2} + \frac{34}{91} a + \frac{24}{91}$, $\frac{1}{124096322598927337001580588339989} a^{15} + \frac{647006058552184334780172459506}{124096322598927337001580588339989} a^{14} - \frac{138536413725703676198317599082}{9545870969148256692429276026153} a^{13} - \frac{449197040504098223149229232332}{1611640553232822558462085562857} a^{12} - \frac{2778019237693035035032699800888}{17728046085561048143082941191427} a^{11} - \frac{8717411160863054883722255993883}{17728046085561048143082941191427} a^{10} - \frac{49374183530118611895954438813205}{124096322598927337001580588339989} a^{9} - \frac{30881139295259135567298395122255}{124096322598927337001580588339989} a^{8} - \frac{15440715988762841861205727449252}{124096322598927337001580588339989} a^{7} - \frac{33526765080825220117620140851799}{124096322598927337001580588339989} a^{6} - \frac{23900396730213913486230964271}{9545870969148256692429276026153} a^{5} + \frac{15985848479241113429370650255231}{124096322598927337001580588339989} a^{4} + \frac{4429979210747703089528495244805}{9545870969148256692429276026153} a^{3} - \frac{56506268414097098379090389973772}{124096322598927337001580588339989} a^{2} - \frac{11841029503655983271360237938535}{124096322598927337001580588339989} a - \frac{59631447450337756381115317859482}{124096322598927337001580588339989}$

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:  $7$
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:  \( 24692550.7041 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.288136694677.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$