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Properties

Label 16.8.78345218527...7104.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{38}\cdot 3^{10}\cdot 13^{6}$
Root discriminant $26.97$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -8, 104, 240, -164, -752, -524, 4, 10, -32, 136, 140, 18, -24, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 24*x^13 + 18*x^12 + 140*x^11 + 136*x^10 - 32*x^9 + 10*x^8 + 4*x^7 - 524*x^6 - 752*x^5 - 164*x^4 + 240*x^3 + 104*x^2 - 8*x - 2)
 
gp: K = bnfinit(x^16 - 10*x^14 - 24*x^13 + 18*x^12 + 140*x^11 + 136*x^10 - 32*x^9 + 10*x^8 + 4*x^7 - 524*x^6 - 752*x^5 - 164*x^4 + 240*x^3 + 104*x^2 - 8*x - 2, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 24 x^{13} + 18 x^{12} + 140 x^{11} + 136 x^{10} - 32 x^{9} + 10 x^{8} + 4 x^{7} - 524 x^{6} - 752 x^{5} - 164 x^{4} + 240 x^{3} + 104 x^{2} - 8 x - 2 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(78345218527771024687104=2^{38}\cdot 3^{10}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.97$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{37} a^{14} + \frac{16}{37} a^{13} + \frac{17}{37} a^{12} - \frac{12}{37} a^{11} + \frac{3}{37} a^{10} + \frac{13}{37} a^{9} - \frac{10}{37} a^{8} + \frac{13}{37} a^{7} - \frac{8}{37} a^{6} + \frac{7}{37} a^{5} + \frac{14}{37} a^{4} + \frac{15}{37} a^{3} + \frac{15}{37} a^{2} + \frac{5}{37} a + \frac{5}{37}$, $\frac{1}{722434666627733} a^{15} + \frac{1721100377882}{722434666627733} a^{14} - \frac{210644676390654}{722434666627733} a^{13} + \frac{239891758839953}{722434666627733} a^{12} - \frac{15122103597630}{722434666627733} a^{11} + \frac{310125843014735}{722434666627733} a^{10} - \frac{296765779694046}{722434666627733} a^{9} - \frac{89093826242258}{722434666627733} a^{8} + \frac{330408540926731}{722434666627733} a^{7} - \frac{97753742323797}{722434666627733} a^{6} - \frac{283079963462061}{722434666627733} a^{5} + \frac{4516294001861}{13630842766561} a^{4} + \frac{78644752516214}{722434666627733} a^{3} + \frac{335791374851879}{722434666627733} a^{2} - \frac{146959776535056}{722434666627733} a + \frac{45157922153760}{722434666627733}$

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:  $11$
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:  \( 480160.681981 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.29952.1, 4.4.7488.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.3588489216.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$