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Properties

Label 16.4.37236326108...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{8}\cdot 89^{3}\cdot 379^{4}$
Root discriminant $45.78$
Ramified primes $2, 5, 89, 379$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1862

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56044, -54512, 30716, -30404, -4906, 5362, -7324, 7618, -1475, 1830, 218, 76, 123, -22, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 18*x^14 - 22*x^13 + 123*x^12 + 76*x^11 + 218*x^10 + 1830*x^9 - 1475*x^8 + 7618*x^7 - 7324*x^6 + 5362*x^5 - 4906*x^4 - 30404*x^3 + 30716*x^2 - 54512*x + 56044)
 
gp: K = bnfinit(x^16 - 2*x^15 + 18*x^14 - 22*x^13 + 123*x^12 + 76*x^11 + 218*x^10 + 1830*x^9 - 1475*x^8 + 7618*x^7 - 7324*x^6 + 5362*x^5 - 4906*x^4 - 30404*x^3 + 30716*x^2 - 54512*x + 56044, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 18 x^{14} - 22 x^{13} + 123 x^{12} + 76 x^{11} + 218 x^{10} + 1830 x^{9} - 1475 x^{8} + 7618 x^{7} - 7324 x^{6} + 5362 x^{5} - 4906 x^{4} - 30404 x^{3} + 30716 x^{2} - 54512 x + 56044 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(372363261088299238400000000=2^{16}\cdot 5^{8}\cdot 89^{3}\cdot 379^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.78$
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:  $2, 5, 89, 379$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{12} - \frac{1}{14} a^{9} + \frac{5}{14} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{14} a^{5} - \frac{1}{2} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{14} a^{14} - \frac{1}{14} a^{12} - \frac{1}{14} a^{10} + \frac{2}{7} a^{9} - \frac{1}{2} a^{8} - \frac{3}{7} a^{7} + \frac{5}{14} a^{6} + \frac{3}{7} a^{5} + \frac{3}{14} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{14547814841076683002954558602} a^{15} + \frac{112122895231928445950365781}{4849271613692227667651519534} a^{14} - \frac{13682614825119583047451718}{2424635806846113833825759767} a^{13} + \frac{1864238098394165751976497011}{14547814841076683002954558602} a^{12} - \frac{971378360692607005148683373}{14547814841076683002954558602} a^{11} - \frac{101654502523327447357543017}{4849271613692227667651519534} a^{10} - \frac{158996940603035369160783146}{7273907420538341501477279301} a^{9} - \frac{4062999853635089600375385977}{14547814841076683002954558602} a^{8} - \frac{196377160554908680735690161}{4849271613692227667651519534} a^{7} - \frac{5774084059124234042631827585}{14547814841076683002954558602} a^{6} - \frac{1243459738253533868276780635}{7273907420538341501477279301} a^{5} + \frac{193986955032537225280160799}{4849271613692227667651519534} a^{4} - \frac{353898903208566154737140276}{1039129631505477357353897043} a^{3} - \frac{2850764377265037312509814863}{7273907420538341501477279301} a^{2} - \frac{1067621347914095599229581989}{2424635806846113833825759767} a + \frac{3164077528561284326259346784}{7273907420538341501477279301}$

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:  $9$
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:  \( 8445204.43032 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1862:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 152 conjugacy class representatives for t16n1862 are not computed
Character table for t16n1862 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.22982560000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.6.3.1$x^{6} - 178 x^{4} + 7921 x^{2} - 34543481$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
379Data not computed