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Properties

Label 16.0.63011844000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 7^{4}$
Root discriminant $23.04$
Ramified primes $2, 3, 5, 7$
Class number $4$
Class group $[2, 2]$
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181, -750, 1256, -680, 202, -600, 1398, -1750, 1585, -1100, 617, -260, 142, -20, 19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 19*x^14 - 20*x^13 + 142*x^12 - 260*x^11 + 617*x^10 - 1100*x^9 + 1585*x^8 - 1750*x^7 + 1398*x^6 - 600*x^5 + 202*x^4 - 680*x^3 + 1256*x^2 - 750*x + 181)
 
gp: K = bnfinit(x^16 + 19*x^14 - 20*x^13 + 142*x^12 - 260*x^11 + 617*x^10 - 1100*x^9 + 1585*x^8 - 1750*x^7 + 1398*x^6 - 600*x^5 + 202*x^4 - 680*x^3 + 1256*x^2 - 750*x + 181, 1)
 

Normalized defining polynomial

\( x^{16} + 19 x^{14} - 20 x^{13} + 142 x^{12} - 260 x^{11} + 617 x^{10} - 1100 x^{9} + 1585 x^{8} - 1750 x^{7} + 1398 x^{6} - 600 x^{5} + 202 x^{4} - 680 x^{3} + 1256 x^{2} - 750 x + 181 \)

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:  \(6301184400000000000000=2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.04$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31919} a^{14} - \frac{7899}{31919} a^{13} - \frac{4182}{31919} a^{12} + \frac{11933}{31919} a^{11} + \frac{1531}{31919} a^{10} - \frac{15356}{31919} a^{9} - \frac{8870}{31919} a^{8} - \frac{5043}{31919} a^{7} - \frac{5061}{31919} a^{6} + \frac{580}{31919} a^{5} + \frac{9857}{31919} a^{4} - \frac{12883}{31919} a^{3} - \frac{11340}{31919} a^{2} + \frac{11668}{31919} a + \frac{12864}{31919}$, $\frac{1}{18162589421832056459} a^{15} - \frac{125014970661116}{18162589421832056459} a^{14} + \frac{7227984736749750344}{18162589421832056459} a^{13} + \frac{299053329480218111}{18162589421832056459} a^{12} + \frac{7057164537199127959}{18162589421832056459} a^{11} + \frac{466930280486391593}{18162589421832056459} a^{10} + \frac{7900558065109163035}{18162589421832056459} a^{9} - \frac{1954547688056742678}{18162589421832056459} a^{8} + \frac{7667699204142331282}{18162589421832056459} a^{7} - \frac{5968673971830162923}{18162589421832056459} a^{6} - \frac{7591253361728910189}{18162589421832056459} a^{5} + \frac{2453936369523889907}{18162589421832056459} a^{4} - \frac{7110953778136220788}{18162589421832056459} a^{3} - \frac{1045714320977896957}{18162589421832056459} a^{2} + \frac{7921465754633851307}{18162589421832056459} a + \frac{2925418125678920380}{18162589421832056459}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3121.7160225 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
2Data not computed
3Data not computed
5Data not computed
$7$7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$