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Properties

Label 16.12.1010436359...4832.4
Degree $16$
Signature $[12, 2]$
Discriminant $2^{32}\cdot 113^{7}$
Root discriminant $31.64$
Ramified primes $2, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71, 66, -607, -500, 1750, 1006, -2484, -682, 1829, 34, -610, 66, 56, 8, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 3*x^14 + 8*x^13 + 56*x^12 + 66*x^11 - 610*x^10 + 34*x^9 + 1829*x^8 - 682*x^7 - 2484*x^6 + 1006*x^5 + 1750*x^4 - 500*x^3 - 607*x^2 + 66*x + 71)
 
gp: K = bnfinit(x^16 - 6*x^15 + 3*x^14 + 8*x^13 + 56*x^12 + 66*x^11 - 610*x^10 + 34*x^9 + 1829*x^8 - 682*x^7 - 2484*x^6 + 1006*x^5 + 1750*x^4 - 500*x^3 - 607*x^2 + 66*x + 71, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 3 x^{14} + 8 x^{13} + 56 x^{12} + 66 x^{11} - 610 x^{10} + 34 x^{9} + 1829 x^{8} - 682 x^{7} - 2484 x^{6} + 1006 x^{5} + 1750 x^{4} - 500 x^{3} - 607 x^{2} + 66 x + 71 \)

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:  \(1010436359891525757304832=2^{32}\cdot 113^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.64$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{914048020032648191} a^{15} + \frac{142678555936926414}{914048020032648191} a^{14} - \frac{431180543434392461}{914048020032648191} a^{13} - \frac{127069556297189651}{914048020032648191} a^{12} - \frac{261979364301168292}{914048020032648191} a^{11} - \frac{144243027803863983}{914048020032648191} a^{10} + \frac{308876117251058119}{914048020032648191} a^{9} - \frac{125941295153964898}{914048020032648191} a^{8} - \frac{319084390047771302}{914048020032648191} a^{7} + \frac{268276981984636203}{914048020032648191} a^{6} - \frac{88696316365139935}{914048020032648191} a^{5} + \frac{413119873088243156}{914048020032648191} a^{4} + \frac{439711085750096147}{914048020032648191} a^{3} + \frac{425742725167602907}{914048020032648191} a^{2} + \frac{283776485475037470}{914048020032648191} a + \frac{776788900767286}{12873915775107721}$

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:  \( 3256302.88552 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.5910106112.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
$2$2.8.20.4$x^{8} + 72 x^{4} + 656$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$113$113.4.3.4$x^{4} + 3051$$4$$1$$3$$C_4$$[\ ]_{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.8.4.1$x^{8} + 127690 x^{4} - 1442897 x^{2} + 4076184025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$