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Properties

Label 16.0.85965144788...4773.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 157^{5}$
Root discriminant $114.39$
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![14157391, 32060193, 29685377, 14668620, 4531102, 1336286, 390010, -35431, -33125, 15756, 1396, -2576, 116, 100, -13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 13*x^14 + 100*x^13 + 116*x^12 - 2576*x^11 + 1396*x^10 + 15756*x^9 - 33125*x^8 - 35431*x^7 + 390010*x^6 + 1336286*x^5 + 4531102*x^4 + 14668620*x^3 + 29685377*x^2 + 32060193*x + 14157391)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 100*x^13 + 116*x^12 - 2576*x^11 + 1396*x^10 + 15756*x^9 - 33125*x^8 - 35431*x^7 + 390010*x^6 + 1336286*x^5 + 4531102*x^4 + 14668620*x^3 + 29685377*x^2 + 32060193*x + 14157391, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 13 x^{14} + 100 x^{13} + 116 x^{12} - 2576 x^{11} + 1396 x^{10} + 15756 x^{9} - 33125 x^{8} - 35431 x^{7} + 390010 x^{6} + 1336286 x^{5} + 4531102 x^{4} + 14668620 x^{3} + 29685377 x^{2} + 32060193 x + 14157391 \)

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:  \(859651447886895620608187246414773=37^{14}\cdot 157^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{13482} a^{14} - \frac{1871}{13482} a^{13} - \frac{37}{13482} a^{12} - \frac{1102}{2247} a^{11} + \frac{4453}{13482} a^{10} - \frac{295}{1498} a^{9} + \frac{181}{963} a^{8} - \frac{3587}{13482} a^{7} - \frac{143}{963} a^{6} - \frac{70}{963} a^{5} + \frac{454}{6741} a^{4} - \frac{16}{6741} a^{3} + \frac{451}{6741} a^{2} + \frac{1975}{6741} a + \frac{11}{13482}$, $\frac{1}{15156339693583148467160308890203020501029206106} a^{15} - \frac{37238638530736926329678414097440717836130}{7578169846791574233580154445101510250514603053} a^{14} - \frac{59982625814496945085885136134792053081916621}{360865230799598773027626402147690964310219193} a^{13} + \frac{1531234022338567101595733069635752130645048871}{15156339693583148467160308890203020501029206106} a^{12} - \frac{6943973813918850397944462619765676404887702143}{15156339693583148467160308890203020501029206106} a^{11} - \frac{2615031068182943853549958080055627026915560464}{7578169846791574233580154445101510250514603053} a^{10} + \frac{2712936179838030664691803735342661698205185151}{15156339693583148467160308890203020501029206106} a^{9} - \frac{767776258267959703278424443539931981783258339}{1684037743731460940795589876689224500114356234} a^{8} - \frac{13220751954703750805338499404343975537162931}{1684037743731460940795589876689224500114356234} a^{7} - \frac{33113840700771235871967324402707113193952359}{360865230799598773027626402147690964310219193} a^{6} - \frac{229849607842659958696820704085556711085378292}{842018871865730470397794938344612250057178117} a^{5} - \frac{147066241897904319336466322784478547403695857}{2526056615597191411193384815033836750171534351} a^{4} + \frac{1144756318784736699096779331486698068262697436}{2526056615597191411193384815033836750171534351} a^{3} - \frac{1320690722727308463559957428717156913923662764}{7578169846791574233580154445101510250514603053} a^{2} + \frac{3022665176317583185230233099968941551272157653}{15156339693583148467160308890203020501029206106} a - \frac{7306889767542897085040630519338227428191325245}{15156339693583148467160308890203020501029206106}$

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:  \( 6976478907.16 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$157$157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$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}$