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Properties

Label 16.8.12430715807...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 5^{14}\cdot 1361^{2}$
Root discriminant $57.00$
Ramified primes $2, 5, 1361$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1330079, 4631752, -6557196, 5180512, -2949992, 1471192, -501424, 7916, 74193, -33008, 6644, 436, -682, 144, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 144*x^13 - 682*x^12 + 436*x^11 + 6644*x^10 - 33008*x^9 + 74193*x^8 + 7916*x^7 - 501424*x^6 + 1471192*x^5 - 2949992*x^4 + 5180512*x^3 - 6557196*x^2 + 4631752*x - 1330079)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 144*x^13 - 682*x^12 + 436*x^11 + 6644*x^10 - 33008*x^9 + 74193*x^8 + 7916*x^7 - 501424*x^6 + 1471192*x^5 - 2949992*x^4 + 5180512*x^3 - 6557196*x^2 + 4631752*x - 1330079, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 144 x^{13} - 682 x^{12} + 436 x^{11} + 6644 x^{10} - 33008 x^{9} + 74193 x^{8} + 7916 x^{7} - 501424 x^{6} + 1471192 x^{5} - 2949992 x^{4} + 5180512 x^{3} - 6557196 x^{2} + 4631752 x - 1330079 \)

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:  \(12430715807334400000000000000=2^{40}\cdot 5^{14}\cdot 1361^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.00$
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:  $2, 5, 1361$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3307174219549013488287576076865638218019} a^{15} - \frac{1646679785147497116609254174912088406693}{3307174219549013488287576076865638218019} a^{14} - \frac{630432233958221923965482886433443757145}{3307174219549013488287576076865638218019} a^{13} - \frac{463919455140691347878084256175867759311}{3307174219549013488287576076865638218019} a^{12} + \frac{1117612915859088144682957983143031378112}{3307174219549013488287576076865638218019} a^{11} - \frac{1166494068412457255252610847948832861843}{3307174219549013488287576076865638218019} a^{10} + \frac{1181820171817201189744130043855262389400}{3307174219549013488287576076865638218019} a^{9} + \frac{244517349728853224291564959538804430140}{3307174219549013488287576076865638218019} a^{8} + \frac{1358735431565452664349160022959527994839}{3307174219549013488287576076865638218019} a^{7} - \frac{53593524874289170316032984265356225948}{174061801028895446751977688256086222001} a^{6} + \frac{359252882613352487623618824887598710640}{3307174219549013488287576076865638218019} a^{5} + \frac{1086468675846488101051510477861254605984}{3307174219549013488287576076865638218019} a^{4} + \frac{1229710246827065213744161661358626632903}{3307174219549013488287576076865638218019} a^{3} - \frac{883803130246305488319293883842588401402}{3307174219549013488287576076865638218019} a^{2} + \frac{1155896870823948115586754516748339912984}{3307174219549013488287576076865638218019} a - \frac{1608849334711697287069071562530632475594}{3307174219549013488287576076865638218019}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 61238652.0232 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
1361Data not computed