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Properties

Label 16.0.62794115988...3517.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 157^{5}$
Root discriminant $72.84$
Ramified primes $37, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![261513, -4431, 310369, -119420, 74239, -59939, 26932, -14243, 9667, -995, 2135, -320, 178, -63, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 63*x^13 + 178*x^12 - 320*x^11 + 2135*x^10 - 995*x^9 + 9667*x^8 - 14243*x^7 + 26932*x^6 - 59939*x^5 + 74239*x^4 - 119420*x^3 + 310369*x^2 - 4431*x + 261513)
 
gp: K = bnfinit(x^16 - 5*x^15 + 14*x^14 - 63*x^13 + 178*x^12 - 320*x^11 + 2135*x^10 - 995*x^9 + 9667*x^8 - 14243*x^7 + 26932*x^6 - 59939*x^5 + 74239*x^4 - 119420*x^3 + 310369*x^2 - 4431*x + 261513, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 14 x^{14} - 63 x^{13} + 178 x^{12} - 320 x^{11} + 2135 x^{10} - 995 x^{9} + 9667 x^{8} - 14243 x^{7} + 26932 x^{6} - 59939 x^{5} + 74239 x^{4} - 119420 x^{3} + 310369 x^{2} - 4431 x + 261513 \)

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:  \(627941159888163345951926403517=37^{12}\cdot 157^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.84$
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:  $37, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{24} a^{13} + \frac{1}{12} a^{12} - \frac{7}{24} a^{11} + \frac{7}{24} a^{10} + \frac{5}{12} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{7}{24} a^{6} + \frac{1}{3} a^{5} - \frac{1}{12} a^{4} + \frac{1}{24} a^{3} - \frac{1}{4} a^{2} + \frac{1}{3} a + \frac{1}{8}$, $\frac{1}{63168} a^{14} - \frac{13}{9024} a^{13} - \frac{2143}{9024} a^{12} + \frac{1475}{4512} a^{11} + \frac{7591}{63168} a^{10} - \frac{1055}{5264} a^{9} + \frac{283}{658} a^{8} - \frac{16271}{63168} a^{7} - \frac{25747}{63168} a^{6} + \frac{8843}{31584} a^{5} + \frac{15115}{63168} a^{4} + \frac{7767}{21056} a^{3} - \frac{17}{31584} a^{2} - \frac{5967}{21056} a - \frac{1267}{3008}$, $\frac{1}{22522545335042396890854346553074176} a^{15} - \frac{18115219271794675723978528325}{2815318166880299611356793319134272} a^{14} - \frac{20253841599161923241119698217819}{1608753238217314063632453325219584} a^{13} - \frac{59052456111209041366410132230311}{3217506476434628127264906650439168} a^{12} + \frac{1908466413989651968095319673067269}{22522545335042396890854346553074176} a^{11} + \frac{1902596674303402047783698207511569}{22522545335042396890854346553074176} a^{10} - \frac{165006727037988787949796315151465}{804376619108657031816226662609792} a^{9} - \frac{6778791910768795586296400341078511}{22522545335042396890854346553074176} a^{8} + \frac{36384730329572218834717093284401}{201094154777164257954056665652448} a^{7} - \frac{1631092246328878484701645383499985}{7507515111680798963618115517691392} a^{6} + \frac{3216332610201480931609690563406445}{22522545335042396890854346553074176} a^{5} - \frac{3610447522591984200266687653142021}{11261272667521198445427173276537088} a^{4} - \frac{3524674233194860023489605164088867}{22522545335042396890854346553074176} a^{3} - \frac{136772888057012463668766392176935}{1072502158811542709088302216813056} a^{2} - \frac{2749492768604088576622934048486611}{11261272667521198445427173276537088} a + \frac{496464343850243267379062259412247}{1072502158811542709088302216813056}$

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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 361345201.166 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.63242590255441.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$157$157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.4.3.4$x^{4} + 19625$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.0.1$x^{4} - x + 15$$1$$4$$0$$C_4$$[\ ]^{4}$
157.4.2.1$x^{4} + 1727 x^{2} + 887364$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$