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Properties

Label 16.0.164812863625282197.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 13^{3}\cdot 109^{4}$
Root discriminant $11.91$
Ramified primes $3, 13, 109$
Class number $1$
Class group Trivial
Galois group 16T972

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -2, -7, 10, 3, -26, 21, 25, -41, 7, 5, 2, 5, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 5*x^14 + 5*x^13 + 2*x^12 + 5*x^11 + 7*x^10 - 41*x^9 + 25*x^8 + 21*x^7 - 26*x^6 + 3*x^5 + 10*x^4 - 7*x^3 - 2*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^16 - x^15 - 5*x^14 + 5*x^13 + 2*x^12 + 5*x^11 + 7*x^10 - 41*x^9 + 25*x^8 + 21*x^7 - 26*x^6 + 3*x^5 + 10*x^4 - 7*x^3 - 2*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 5 x^{14} + 5 x^{13} + 2 x^{12} + 5 x^{11} + 7 x^{10} - 41 x^{9} + 25 x^{8} + 21 x^{7} - 26 x^{6} + 3 x^{5} + 10 x^{4} - 7 x^{3} - 2 x^{2} + 3 x + 1 \)

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:  \(164812863625282197=3^{12}\cdot 13^{3}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.91$
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:  $3, 13, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{221825} a^{15} - \frac{42982}{221825} a^{14} + \frac{50737}{221825} a^{13} + \frac{34583}{221825} a^{12} + \frac{37404}{221825} a^{11} - \frac{95544}{221825} a^{10} - \frac{69554}{221825} a^{9} - \frac{35092}{221825} a^{8} + \frac{101102}{221825} a^{7} + \frac{86709}{221825} a^{6} + \frac{8454}{44365} a^{5} - \frac{60117}{221825} a^{4} + \frac{71187}{221825} a^{3} - \frac{56229}{221825} a^{2} - \frac{4728}{221825} a + \frac{22471}{221825}$

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:  \( -\frac{51663}{11675} a^{15} + \frac{75791}{11675} a^{14} + \frac{220494}{11675} a^{13} - \frac{363179}{11675} a^{12} + \frac{78173}{11675} a^{11} - \frac{287128}{11675} a^{10} - \frac{231723}{11675} a^{9} + \frac{2212196}{11675} a^{8} - \frac{2371101}{11675} a^{7} + \frac{62108}{11675} a^{6} + \frac{271273}{2335} a^{5} - \frac{846229}{11675} a^{4} - \frac{108981}{11675} a^{3} + \frac{428977}{11675} a^{2} - \frac{106761}{11675} a - \frac{97373}{11675} \) (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:  \( 250.805825535 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T972:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 35 conjugacy class representatives for t16n972
Character table for t16n972 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.981.1, 8.0.12510693.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
3Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$109$109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.8.0.1$x^{8} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$