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Properties

Label 16.0.29811439152...7861.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{14}\cdot 149^{3}$
Root discriminant $60.21$
Ramified primes $37, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28349, 64115, 80299, 68939, 35483, 5520, -7220, -8125, -2979, -149, 356, 161, 74, 12, -12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 12*x^14 + 12*x^13 + 74*x^12 + 161*x^11 + 356*x^10 - 149*x^9 - 2979*x^8 - 8125*x^7 - 7220*x^6 + 5520*x^5 + 35483*x^4 + 68939*x^3 + 80299*x^2 + 64115*x + 28349)
 
gp: K = bnfinit(x^16 - 3*x^15 - 12*x^14 + 12*x^13 + 74*x^12 + 161*x^11 + 356*x^10 - 149*x^9 - 2979*x^8 - 8125*x^7 - 7220*x^6 + 5520*x^5 + 35483*x^4 + 68939*x^3 + 80299*x^2 + 64115*x + 28349, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 12 x^{14} + 12 x^{13} + 74 x^{12} + 161 x^{11} + 356 x^{10} - 149 x^{9} - 2979 x^{8} - 8125 x^{7} - 7220 x^{6} + 5520 x^{5} + 35483 x^{4} + 68939 x^{3} + 80299 x^{2} + 64115 x + 28349 \)

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:  \(29811439152025391709947927861=37^{14}\cdot 149^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.21$
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:  $37, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{2770037531016685942931430931027} a^{15} - \frac{99995162550131214722283641440}{2770037531016685942931430931027} a^{14} - \frac{73678470128001459852599482197}{2770037531016685942931430931027} a^{13} + \frac{195795905277942397007813801723}{2770037531016685942931430931027} a^{12} - \frac{6946842298368080900912661673}{2770037531016685942931430931027} a^{11} + \frac{238223616605535807054554177366}{2770037531016685942931430931027} a^{10} + \frac{149554856720465848969269398741}{2770037531016685942931430931027} a^{9} + \frac{844275022900534760818477217466}{2770037531016685942931430931027} a^{8} - \frac{101066249122222426868127395082}{2770037531016685942931430931027} a^{7} - \frac{180522057513358544083411273070}{2770037531016685942931430931027} a^{6} + \frac{753257560490555689105674743178}{2770037531016685942931430931027} a^{5} + \frac{88267919500004621344103016150}{395719647288097991847347275861} a^{4} + \frac{111449444210271316636197165809}{395719647288097991847347275861} a^{3} + \frac{1055151212578270414293729391583}{2770037531016685942931430931027} a^{2} + \frac{40497193046824894933847184164}{395719647288097991847347275861} a - \frac{1004067769093824119015746048842}{2770037531016685942931430931027}$

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:  $7$
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:  \( 50044266.1092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

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 t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.382293234941.1

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
Arithmetically equvalently 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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $16$

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
$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}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$