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Properties

Label 16.12.2316572384...6704.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{56}\cdot 3^{8}\cdot 7^{2}$
Root discriminant $24.99$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -120, 464, -224, -832, 1040, -504, 484, -304, -232, 336, -112, -16, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 16*x^13 - 112*x^12 + 336*x^11 - 232*x^10 - 304*x^9 + 484*x^8 - 504*x^7 + 1040*x^6 - 832*x^5 - 224*x^4 + 464*x^3 - 120*x^2 + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 16*x^13 - 112*x^12 + 336*x^11 - 232*x^10 - 304*x^9 + 484*x^8 - 504*x^7 + 1040*x^6 - 832*x^5 - 224*x^4 + 464*x^3 - 120*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 16 x^{13} - 112 x^{12} + 336 x^{11} - 232 x^{10} - 304 x^{9} + 484 x^{8} - 504 x^{7} + 1040 x^{6} - 832 x^{5} - 224 x^{4} + 464 x^{3} - 120 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23165723849659414216704=2^{56}\cdot 3^{8}\cdot 7^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.99$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{97} a^{14} + \frac{3}{97} a^{13} - \frac{18}{97} a^{12} + \frac{46}{97} a^{11} - \frac{2}{97} a^{10} - \frac{32}{97} a^{9} - \frac{46}{97} a^{8} + \frac{38}{97} a^{7} - \frac{13}{97} a^{6} - \frac{5}{97} a^{5} + \frac{20}{97} a^{4} - \frac{43}{97} a^{3} + \frac{34}{97} a^{2} - \frac{11}{97} a + \frac{22}{97}$, $\frac{1}{434671841248223} a^{15} + \frac{1047719754708}{434671841248223} a^{14} + \frac{125972993448047}{434671841248223} a^{13} - \frac{7301703994862}{434671841248223} a^{12} + \frac{22778618046070}{434671841248223} a^{11} - \frac{156336016021424}{434671841248223} a^{10} + \frac{104591139052891}{434671841248223} a^{9} - \frac{47208370202974}{434671841248223} a^{8} + \frac{31810195151516}{434671841248223} a^{7} + \frac{182143972032186}{434671841248223} a^{6} - \frac{144296409873476}{434671841248223} a^{5} + \frac{17844041612196}{434671841248223} a^{4} + \frac{117702616426727}{434671841248223} a^{3} + \frac{172952267756037}{434671841248223} a^{2} + \frac{64546428791254}{434671841248223} a + \frac{175842801434427}{434671841248223}$

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:  $13$
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:  \( 595719.771812 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.18432.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\)

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$