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Properties

Label 16.0.15478121650...0533.2
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 157^{7}$
Root discriminant $137.04$
Ramified primes $37, 157$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1236119787, 325469946, 418519900, -54019469, 49678870, -4208135, -288961, 1212, 25265, 89874, -9624, 1879, 315, -45, -5, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 5*x^14 - 45*x^13 + 315*x^12 + 1879*x^11 - 9624*x^10 + 89874*x^9 + 25265*x^8 + 1212*x^7 - 288961*x^6 - 4208135*x^5 + 49678870*x^4 - 54019469*x^3 + 418519900*x^2 + 325469946*x + 1236119787)
 
gp: K = bnfinit(x^16 - 7*x^15 - 5*x^14 - 45*x^13 + 315*x^12 + 1879*x^11 - 9624*x^10 + 89874*x^9 + 25265*x^8 + 1212*x^7 - 288961*x^6 - 4208135*x^5 + 49678870*x^4 - 54019469*x^3 + 418519900*x^2 + 325469946*x + 1236119787, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 5 x^{14} - 45 x^{13} + 315 x^{12} + 1879 x^{11} - 9624 x^{10} + 89874 x^{9} + 25265 x^{8} + 1212 x^{7} - 288961 x^{6} - 4208135 x^{5} + 49678870 x^{4} - 54019469 x^{3} + 418519900 x^{2} + 325469946 x + 1236119787 \)

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:  \(15478121650083338314369033920290533=37^{12}\cdot 157^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.04$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{33} a^{14} + \frac{13}{33} a^{13} - \frac{1}{11} a^{12} + \frac{2}{11} a^{11} - \frac{4}{11} a^{10} - \frac{8}{33} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{2}{33} a^{6} - \frac{16}{33} a^{5} - \frac{5}{11} a^{4} - \frac{8}{33} a^{3} - \frac{3}{11} a^{2} + \frac{16}{33} a + \frac{3}{11}$, $\frac{1}{285636228632796134390750752093494084259581026180908587754980696447} a^{15} + \frac{2016236221163846523066580654167492796973206390767226302639572681}{285636228632796134390750752093494084259581026180908587754980696447} a^{14} + \frac{109143846640745284014525246640399101209625434470602998265011074333}{285636228632796134390750752093494084259581026180908587754980696447} a^{13} - \frac{13556159955193096151357531173763956806191687951016378347723107412}{95212076210932044796916917364498028086527008726969529251660232149} a^{12} + \frac{4066002501497300059034934649001860832124925689291538333726627617}{10579119578992449421879657484944225342947445414107725472406692461} a^{11} - \frac{16481050198136740631800170280662482506290477816136145632031252180}{285636228632796134390750752093494084259581026180908587754980696447} a^{10} + \frac{3674718066285989482268982154540014501830172740526636535718356205}{10579119578992449421879657484944225342947445414107725472406692461} a^{9} + \frac{1363000527430599321573454708849315867793856150744045966785842417}{2885214430634304387785361132257516002622030567483925128838188853} a^{8} + \frac{62014542191722801439024280208113990105241803445405085159556928682}{285636228632796134390750752093494084259581026180908587754980696447} a^{7} - \frac{17248362698946416037359664612764997483507577408359396634004117399}{95212076210932044796916917364498028086527008726969529251660232149} a^{6} + \frac{6684865677440655021470426564499725456155539490100410725956877853}{25966929875708739490068250190317644023598275107355326159543699677} a^{5} + \frac{107235338122145413202401432089715061551687916535336449834866000711}{285636228632796134390750752093494084259581026180908587754980696447} a^{4} - \frac{6284617311122874329360415934195829018527239871390693968588019376}{285636228632796134390750752093494084259581026180908587754980696447} a^{3} - \frac{5776181463866787998076958728112895188224203578481015194529666057}{285636228632796134390750752093494084259581026180908587754980696447} a^{2} - \frac{140283967187378901881310454722908905939079527157173018446977674535}{285636228632796134390750752093494084259581026180908587754980696447} a - \frac{23849695558317984850427652306044888983789091468140851811814321653}{95212076210932044796916917364498028086527008726969529251660232149}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 17632859130.2 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$$[\ ]^{2}$
157.4.3.3$x^{4} + 785$$4$$1$$3$$C_4$$[\ ]_{4}$
157.4.2.1$x^{4} + 1727 x^{2} + 887364$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
157.4.2.2$x^{4} - 157 x^{2} + 147894$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$